**This page is under construction, and this list should be regarded as illustrative rather than definitive.**

**APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE**

**AMCS 602-603: **Algebraic Techniques for Applied Mathematics and Computational Science. Staff. First semester: We begin with an introduction to group theory. The emphasis is on groups as symmetries and transformations of space. After an introduction to abstract groups and the basic facts about finite groups, we turn our attention to compact Lie groups and their representations. In the latter connection we explore the connections between orthogonal polynomials, classical transcendental functions and group representations. This unit is completed with a discussion of finite groups and their applications in coding theory. Second semester: We turn to linear algebra and the structural properties of linear systems of equations relevant to their numerical solution. In this context we introduce eigenvalues and the spectral theory of matrices. Methods appropriate to the numerical solution of very large systems are discussed. We then turn to the problem of solving systems of polynomial equations, introducing basic properties of rings, ideals and modules. This allows us to define Grobner bases and their use in the numerical solution of algebraic equations. The theoretical content of this course is illustrated and supplemented throughout the year with substantial computational examples and assignments.

**AMCS/MATH 608: **Analysis I. (First semester). Prerequisite: Math 360/361 or Math 508/509. Complex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. Introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C^0 and the Riesz-Markov theorem, L^p spaces, Fubini theorem.

**AMCS/MATH 609: **Analysis II. (second Semester). Prerequisite: AMCS/MATH 608 or equivalent. Real analysis: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L^2-theory of the Fourier transform. Functional analysis: normed linear spaces, convexity, the Hahn-Banach Theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, L^p theory for the Fourier transform.

**AMCS/MATH 610: **Functional Analysis. (Prereq: 608 or 609, some elementary complex analysis is essential, usually offered in the Spring) Convexity and the Hahn Banach Theorem. Hilbert Spaces, Banach Spaces, and examples: Sobolev spaces, Holder spaces. The uniform boundedness principle, Baire category theorem, bounded operators, open m\ apping theorem, closed graph theorem and applications. The concepts of duality and dual spaces. The Riesz theory of compact operators and Fredholm theory. Functional calculus and elementary Spectral Theory. Interpolation theorems. Applications to partial differential equations and approximation theory.

Through the Math and Statistics departments we also offer a one year course on Probability and Stochastic processes:

**STAT 530-531/MATH 546-547: **Probability, and Stochastic Processes for Applied Mathematics and Computational Science. Staff. Probability and statistics are the language most appropriate to the discussion of experimental data. This course introduces and places on a firm foundation, methods and ideas arising in probability theory and the theory of stochastic processes, and their applications to problems of empirical science. First semester: Basic concepts of measure theory, as measurement theory, probability theory, random variables, expectation, and independence, the weak and strong laws of large numbers, the central and Poisson limit theorems. Second semester: Measure theory revisited, conditional expectations, Martingales, Brownian motion, diffusion processes and stochastic integration and differential equations. Applications to the analysis of partial differential equations.

**AMCS/MEAM 637:** Mesoscale Modeling and Simulation. (Spring 2013) Dr. David Srolovitz.

**AMCS 701:** Topics in Applied Mathematics (Fall 2012, Vadim Markel). It is difficult to imagine modern research in any physics or engineering field without numerical simulations. The existence and wide availability of high-level languages such as Matlab or Mathematica, and of many other commercial packages of more or less general applicability, helps create an impression that familiarity with technical programming and computational methods is currently either superfluous or obsolete. However, experience shows that this is not the case. Familiarity with a "real" programming language and proficiency in its application to real-life research problems is a very valuable asset in a research career.

This course is aimed at gaining hands-on experience in computer modeling with the use of Fortran - a programming language, which is, contrary to some expectations, alive, well and very modern. The instructor believes that this particular research tool, when mastered, will serve the students a lifetime, allowing them to generate, process and publish data faster and more efficiently, with more confidence in results and less potential for errors. The course will combine computations with some aspects of applied math and physics. The coursework will consist of several programming projects. We will first discuss physical and mathematical aspects of a problem, then discuss various computational approaches, and then implement those in a code. We will start with the following topics:

(1) Aggregation of particles (diffusion-limited and concentration limited aggregation, formation of random fractal aggregates of Witten-Sander Meakin type, computation of density-density correlation, etc.)

(2) EM Waves in chains of polarizable particles (series summation, FT, numerical integration)

(3) Very large systems of equations, e.g., scattering from aggregated spheres (expansion into VSH, translation of VSH, storage schemes for very large matrices [here we have to deal effectively with composite indexes], implementation of CG descent for special storage schemes

(4) Continued fraction expansions with application to scattering (e.g., multiple aggregated spheres) and homogenization.

If we have time (or if the course is continued in future semesters), we will consider additional topics such as the pole expansion of the Mie coefficients, large spheres, etc.

The following courses are sometimes taken by students in the AMCS Masters program:

**AMCS 510:** (x-listed MATH 410) Complex Analysis. (C) Staff. Prerequisite(s): MATH 241 or permission of instructor. Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.**AMCS 520:** (x-listed MATH 420) Ordinary Differential Equations. (C) Staff. Prerequisite(s): MATH 241 or permission of instructor. After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.**AMCS 525:** (x-listed MATH 425) Partial Differential Equations. (A) Staff. Prerequisite(s): MATH 241 or permission of instructor. Knowledge of PHYS 150-151 will be helpful. Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.**AMCS 530:** (x-listed MATH 430) Introduction to Probability. (M) Staff. Prerequisite(s): MATH 240.

Random variables, events, special distributions, expectations, independence, law of large numbers, introduction to the central limit theorem, and applications.**AMCS 532:** (x-listed MATH 432) Game Theory. (C) Staff. A mathematical approach to game theory, with an emphasis on examples of actual games. Topics will include mathematical models of games, combinatorial games, two person (zero sum and general sum) games, non-cooperating games and equilibria.

Below is an extensive listing of courses offered throughout the University, which may be suitable as part of the course program for a graduate student in AMCS. Courses not listed may also be suitable. Each AMCS student's program of study must be approved by the Graduate Group Chair, and simply choosing courses from this list will not guarantee approval.

**BIOCHEMISTRY AND MOLECULAR BIOPHYSICS (MD) {BMB} **

**554. (CHEM555) Macromolecular Crystallography: Methods and Applications. (A)** Marmorstein. Prerequisite(s): undergraduate calculus and trigonometry.

The first half of the course covers the principles and techniques of macromolecular structure determination using X-ray crystallography. The second half of the course covers extracting biological information from X-ray crystal structures with special emphasis on using structures reported in the literature and presented by faculty and students.

**581. (BE 581) Techniques of Magnetic Resonance Imaging. (K)** Wehrli.

Detailed introduction to the physics and engineering of magnetic resonance imaging as applied to diagnostic medicine. Covered are magnetism, spin relaxation, spatial encoding principles, Fourier analysis, imaging pulse sequence and pulse design, contrast mechanisms, chemical shift, flow encoding, diffusion and perfusion and a discussion of the most relevant clinical applications.

**601. Fundamentals of Magnetic Resonance. (A)** Leigh and Reddy.

This course introduces basic theoretical and experimental concepts of magnetic resonance and its applications in biochemistry, biology and medicine. Topics covered include description of the phenomenon of magnetic resonance, and classical and quantum strategies to compute nuclear spin resonses in liquids,solids and biological tissues, polarization transfer and mujltiple quantum effects and their applications in biomedicine. Nuclear spin relaxation in solid-state materials and in biological systems will be discussed. Concepts of magnetic resonance imaging, imaging strategies, image contrast, and diagnostic applications are discussed. The course includes several practicals dealing with the demonstration of NMR hardware and experiments to compute basic NMR parameters on high resolution and clinical MRI scanners. For further details of this course,visit www.mmrrcc.upenn.edu

**603. Advanced Topics in Magnetic Resonance. (K)** Leigh and Reddy.

Advanced topics in theory and applications of magnetic resonance spectroscopy and imaging (Nuclear Magnetic Resonance - NMR; and Electron Spin Resonance - EPR) of biological tissues and solid-state materials to problems in biochemistry, biology, bioengineering and medicine.

**604. (BE 619) Statistical Mechanics. (H)** Schotland. Prerequisite(s): CBE 618 or equivalent.

A modern introduction to statistical mechanics with biophysical applications. Theory of ensembles. Noninteracting systems. Liquid theory. Phase transitions and critical phenomena. Nonequilibrium systems. Applications to reaction kinetics, polymers and membranes.

**611. Advanced X-ray Diffraction Methods. (J)** Van Duyne. Prerequisite(s): BMB 554/CHEM 555 or equivalent,or permission of instructor. Course meets for 8 weeks and is offered for 1/2 credit.

Advanced topics in macromolecular x-ray diffraction. Crystallization, synchrotron data collection, data processing, anomalous diffraction, phasing methods, density modification methods, refinement. Emphasis is on applications and currently available methodology.

**619. Protein Folding. (I)** Axelsen and Englander. Course meets for 8 weeks and is offered for 1/2 credit.

Introduction to the folding of especially soluble proteins but also membrane proteins; critical readings in current literature and important earlier literature; class discussion of papers interspersed with didactic lectures as required. Exposure to equilibium, kinetics, thermodynamics principles and use as they occur in the real literature. Exposure to the range of biophysical technologies as used in the literature.**BIOENGINEERING (EG) {BE} **

**505. Quantitative Human Physiology. (B)** Prerequisite(s): BE 305.

Introduction to human physiology using the quantitative methods of engineering and physical science. Emphasis is on the operation of the major organ systems at both the macroscopic and cellular level.

**510. Biomechanics and Biotransport. (A)** Prerequisite(s): Math through 241; BE 350, BE 324 as pre-or corequisites.

The course is intended as an introduction to continuum mechanics in both solid and fluid media, with special emphasis on the application to biomedical engineering. Once basic principles are established, the course will cover more advanced concepts in biosolid mechanics that include computational mechanics and bio-constitutive theory. Applications of these advanced concepts to current research problems will be emphasized.

**511. Analysis and Design of Bioengineering Signals. (B)** Prerequisite(s): BE 301 or graduate status. Not intended for students with previous courses in digital signal processing.

This is a practically-oriented course in the analysis of biomedical signals focusing on medically significant applications. The specific applications will vary from year to year, but lectures will include the nature of major signals of biomedical importance, diginal signal processing including convolution, digital filtering, wavelet analysis. The course will include student experiments using Matlab and independent projects.

**517. (ESE 517) Optical Imaging. (C)** Prerequisite(s): ESE 310 and ESE 325 or equivalent.

A modern introduction to the physical principles of optical imaging with biomedical applications. Propagation and interference of electromagnetic waves. Geometrical optics and the eikonal. Plane-wave expansions, diffraction and the Rayleigh criterion. Scattering theory and the Born approximation. Introduction to inverse problems. Multiple scattering and radiative transport. Diffusion approximation and physical optics of diffusing waves. Imaging in turbid media. Introduction to coherence theory and coherence imaging. Applications will be chosen from the recent literature in biomedical optics.

**519. Cellular-Level Neural Simulation and Modeling. (M)** Finkel.

Cellular level simulation of neurons at the biophysical level. Topics include cable theory, the Hodgkin-Huxley formalism for different channelspecies , synaptic interactions and plasticity, information measures in network activity, neuromodulation, and applications to modeling neurological disease.

**520. (INSC594) Computational Neuroscience and Neuroengineering. (B)** Finkel.

Computational modeling and simulation of the structure and function of brain circuits. A short survey of the major ideas and techniques in the neural network literature. Particular emphasis on models of hippocampus, basal ganglia and visual cortex. A series of lab exercises introduces techniques of neural simulation.

**539. (ESE 539) Neural Networks, Chaos, and Dynamics: Theory and Application. (B)**

Physiology and anatomy of living neurons and neural networks; Brain organization; Elements of nonlinear dynamics, the driven pendulum as paradigm for complexity, synchronicity, bifurcation, self-organization and chaos; Iterative maps on the interval, period-doubling route to chaos, universality and the Feigenbaum constant, Lyapunov exponents, entropy and information; Geometric characterization of attractors; Fractals and the Mandelbrot set; Neuron dynamics: from Hudgkin-Huxley to integrate and fire, bifurcation neuron; Artificial neural networks and connectionist models, Hopfield (attractor-type) networks,energy functions, convergence theorems, storage capacity, associative memory, pattern classification, pattern completion and error correction, the Morita network; Stochastic networks, simulated annealing and the Boltzmann machine, solution of optimization problems, hardware implementations of neural networks; the problem of learning, algorithmic approaches: Perception learning, back-propagation, Kohonnen's self-organizing maps and other networks; Coupled-map lattices; Selected applications including financial markets.

**546. (BMB 546) Quantitative Image Analysis. (H)**

Most of the time will be spent on different kinds of analysis methods (e.g., intensity measurements and approaches to segmentation) along with brief reviews of necessary mathematical background (e.g., transforms) and examples of specific areas of application (primarily biomedical). While traditional image processing techniques will be reviewed as a means of preparing images for analysis, they will not be a principal focus of this course.

**581. (BMB 581) Techniques of Magnetic Resonance Imaging. (M)**

Detailed survey of the physics and engineering of magnetic resonance imaging as applied to medical diagnosis. Covered are: history of MRI, fundamentals of electromagnetism, spin and magnetic moment, Bloch equations, spin relaxation, image contrast mechanisms, spatial encoding principles, Fourier reconstruction,imaging pulse sequences and pulse design, high-speeding imaging techniques, effects of motion, non-Cartesian sampling strategies, chemical shift encoding, flow encoding, susceptibility boundary effects, diffusion and perfusion imaging.

**583. (BE 483) Molecular Imaging. (C)** Prerequisite(s): BIOL 215 or BE 305 or permission of the instructor.

This course will provide a comprehensive survey of modern medical imaging modalities with an emphasis on the emerging field of molecular imaging. The basic principles of X-ray, computed tomography, nuclear imaging, magnetic resonance imaging, and optical tomography will be reviewed. The emphasis of the course, however, will focus on the concept of contrast media and targeted molecular imaging. Topics to be covered include the chemistry and mechanisms of various contrast agents, approaches to identifying molecular markers of disease, ligand screening strategies, and the basic principles of toxicology and pharmacology relevant to imaging agents.

**584. (MATH584) Mathematics of Medical Imaging and Measurements. (M)** Prerequisite(s): Math through 241 as well as some familiarity with linear algebra and basic physics.

In the last 25 years there as has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object, which cannot be obtained directly. The data that is accessible is typically some collection of weighted averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT). In this contest we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.

**619. (BMB 604) Statistical Mechanics. (C)** Prerequisite(s): CBE 618 or equivalent.

A modern introduction to statistical mechanics with biophysical applications. Theory of ensembles. Noninteracting systems. Liquid theory. Phase transitions and critical phenomena Nonequilibrium systems. Applications to reaction kinetics, polymers and membranes.

**630. (EE 630) Elements of Neural Computation, Complexity and Learning. (M)** Prerequisite(s): A semester course in probability or equivalent exposure to probability (e.g. ESE 530).

Non-linear elements and networks: linear and polynomial threshold elements, sigmoidal units, radial basis functions. Finite (Boolean) problems: acyclic networks; Fourier analysis and efficient computation; projection pursuit; low frequency functions. Network capacity: Feedforward networks; Vapnik-Chervnenkis dimension. Learning theory: Valiant's learning model; the sample complexity of learning. Learning algorithms: Perception training, gradient descent algorithms, stochastic approximation. Learning complexity: the intractability of learning; model selection.

**L/R ****662. (CBE 618, MEAM662) Advanced Molecular Thermodynamics.**

Review of classical thermodynamics. Phase and chemical equilibrium for multicomponent systems. Prediction of thermodynamic functions from molecular properties. Concepts in applied statistical mechanics. Modern theories of liquid mixtures.

**BIOLOGY (AS) {BIOL} **

**438. Systems Biology: Integrative physiology and biomechanics of the muscular system. (M)** Rome. Prerequisite(s): 1 year physics, 1 year chemistry, and BIOL 251 or 215.

The course will focus on muscle function from the level of molecules to whole animal locomotion. At each level of organization, muscle function will be explored from mechanical and energetic viewpoints. The course will include lectures, demonstrations, and several guest expert lectures. Students will also be introduced to realistic musculo-skeletal modelling and forward dynamic simulations to explore integrated function.

**535. Mathematical Ecology. (K)** Dunham.

Survey and development of mathematical theories in ecology, particularly theories of population growth, predation, and competition, as well as other topics of current interest.

**536. (CIS 536, GCB 536) Computational Biology. (B)** Kim. Prerequisite(s): College level introductory biology required; undergraduate or graduate level statistics taken previously or concurrently required; molecular biology and/or genetics encouraged; programming experience encouraged.

Introductory computational biology course designed for both biology students and computer science, engineering students. The course will cover fundamentals of algorithms, statistics, and mathematics as applied to biological problems. In particular, emphasis will be given to biological problem modeling and understanding the algorithms and mathematical procedures at the "pencil and paper" level. That is, practical implementation of the algorithms is not taught but principles of the algorithms are covered using small sized examples. Topics to be covered are: genome annotation and string algorithms, pattern search and statistical learning, molecular evolution and phylogenetics, functional genomics and systems level analysis.

**537. (CIS 635, GCB 537) Advanced Computational Biology. (A)** Staff. Prerequisite(s): BIOL 536 or permission of instructor.

Discussion of special research topics.

**540. (CAMB541, MOLB541) Genetic Analysis. (B)** Poethig. Prerequisite(s): BIOL 221 or permission of instructor.

The logic and methodology of genetic analysis in plants and animals. This lecture course will focus on the use of mutations to study gene function and higher order biological processes, methods for reporting and manipulating gene expression, and analysis of the genetic basis of natural variation.

**556. Advanced Statistics. (B)** Staff. Prerequisite(s): BIOL 446 or equivalent, and permission of instructor.

Advanced statistical methods, including multivariate techniques (in particular discriminant functions, principal components, multiple correlation, and regression) and the design and analysis of experiments.

**SM ****571. Proteomics. (B)** Rea/ Blair/Speicher.

Biology 571 is concerned with recent developments in the identification and characterization of proteins using high-sensitivity, high-resolution mass spectrometric (MS) techniques. Several new MS technologies, including matrix-assisted laser desorption/ionization (MALDI), electrospray ionization (ESI) and fast atom/ion bombardment (FAB) are making the study of biomolecules routine. Working from the original literature (and some work that has yet to be published), this course will emphasize how, in what is now the post-genomic functional genomic era, this technology has given rise to and sustains proteomics in all its guises, from the study of single molecules to the temporal and spatial definition of the total protein complement, the 'proteome', of a cell.

**BIOSTATISTICS (BSTA) **

**621. (STAT432, STAT512) Statisical Inference I. (B)** Faculty. Prerequisite(s): BSTA 620.

Statistical inference including estimation, confidence intervals, hypothesis tests and non-parametric methods.

**622. (STAT550) Statistical Inference II. (A)** Brown. Prerequisite(s): BSTA 621.

Statistical inference including estimation, confidence intervals, hypothesis tests and non-parametric methods.

**630. Statistical Methods for Data Analysis I. (A)** Shults and Putt. Prerequisite(s): Multivariable calculus and linear algebra, BSTA 620 (may be taken concurrently).

This first course in statistical methods for data analysis is aimed at first year Biostatistics degree candidates. It focuses on the analysis of continuous data, and includes descriptive statistics, such as central tendencies, dispersion measures, shapes of a distribution, graphical representations of distributions, transformations, and testing for goodness of fit for a distribution. Populations, samples, hypotheses of differences and equivalence, and errors will be defined. One and two sample t-tests, analysis of variance, correlation, as well as non-parametric tests and correlations will be covered.

Estimation, including confidence intervals, and robust methods will be discussed. The relationship between outcome variables and explanatory variables will be examined via regression analysis, including single linear regression, multiple regression, model fitting and testing, partial correlation, residuals, multicolinearity. Examples of medical and biologic data will be used throughout the course, and use of computer software demonstrated.

**631. Statistical Methods and Data Analysis II. (B)** Gimotty. Prerequisite(s): linear algebra, calculus, BSTA 630, BSTA 620, BSTA 621 (may be taken concurrently).

This is the second half of the methods sequence and focuses on categorical data and survival data. Topics in categorical data to be covered include defining rates, incidence and prevalence, the chi-squared test, Fisher's exact test and its extension, relative risk and odds-ratio, sensitivity, specificity, predictive values, logistic regression with goodness of fit tests, ROC curves, Mantel-Haenszel test, McNemar's test, the Poisson model, and the Kappa statistic. Survival analysis will include defining the survival curve, censoring, and the hazard function, the Kaplan-Meier estimate, Greenwood's formula and confidence bands, the log rank test, and Cox's proportional hazards regression models. Examples of medical and biologic data will be used throughout the course, and use of computer software demonstrated.

**651. Introduction to Linear Models and Generalized Linear Models. (B)** Tu. Prerequisite(s): linear algebra, calculus, BSTA 630, BSTA 620, BSTA 621 (may be taken concurrently).

This course extends the content on linear models in BSTA 630 and BSTA 631 to more advanced concepts and applications of linear models. Topics include the matrix approach to linear models including regression and analysis of variance, general linear hypothesis, estimability, polynomial, piecewise, ridge, and weighted regression, regression and collinearity diagnostics, multiple comparisons, fitting strategies, simple experimental designs (block designs, split plot), random effects models, Best Linear Unbiased Prediction. In addition, generalized linear models will be introduced with emphasis on the binomial, logit and Poisson log-linear models. Applications of methods to example data sets will be emphasized.

**820. (STAT552) Statistical Inference III. (B)** Faculty. Prerequisite(s): To be advised.

Statistical inference including estimation, confidence intervals, hypothesis tests and non-parametric methods.

**CHEMICAL AND BIOMOLECULAR ENGINEERING (EG) {CBE} **

**502. Nonlinear Methods in Chemical Engineering. (M)** Seider, Ungar, Sinno.

Steady and time-periodic bifurcation and strange attractors for lumped and distributed parameter systems. Singularity theory and imperfect bifurcation to characterize solution and bifurcation diagrams. Applications in reactor analysis, heat transfer, turbulence, process control, etc.

**508. Probability and Statistics for Biotechnology. (C)**

This course is designed as an overview of probability and statistics including linear regression, correlation, and multiple regression. The program will also include statistical quality control and analysis of variance with attention to method of analysis, usual method of computation, test on homogeneity of variances, simplifying the computations, and multi-factor analysis.

**617. (ESE 617, MEAM613) Control of Nonlinear Systems. (A)** Seider.

PID control of nonlinear systems; steady-state, periodic and chaotic attractors. Multiple-input, multiple-output systems; decoupling methods and decentralized control structures. Digital control; z-transforms, implicit model control, impact of uncertainties. Constrained optimization; quadratic dynamic matrix control. Nonlinear predictive control. Transformations for input/output linearized controllers.

**L/R ****618. (BE 662, MEAM662) Advanced Molecular Thermodynamics. (A)** Glandt, Discher.

Review of classical thermodynamics. Phase and chemical equilibrium for multicomponent systems. Prediction of thermodynamic functions from molecular properties. Concepts in applied statistical mechanics. Modern theories of liquid mixtures.

**619. Application of Thermodyanics to Chemical Engineering II. (B)** Glandt.

An introduction to statistical mechanics and its applications in chemical engineering. Ensembles. Monatomic and polatomic ideal gases. Ideal lattices; adsorption and polymer elasticity. Imperfect gases. Dense liquids. Computer simulation techniques. Interacting lattices.

**621. Advanced Chemical Kinetics and Reactor Design. (B)** Gorte, Perlmutter, Vohs.

Mechanisms of chemical reactions. Transition state theory. Langmuir-Hinshelwood Kenetics. Absorption and cataysis. Simple and complex reaction schemes. Design of idealized reactors. Fluidized reactors. Solid-gas reactions. Residence time distributions. Reaction and diffusion in solid catalysts. Reactor stability and control.

**L/R ****640. (MEAM570) Transport Processes I. (A)** Diamond, Sinno.

The course provides an unified introduction to momentum, energy (heat), and mass transport processes. The basic mechanisms and the constitutive laws for the various transport processes will be delineated, and the conservation equations will be derived and applied to internal and external flows featuring a few examples from mechanical, chemical, and biological systems. Reactive flows will also be considered.

**641. Transport Processes II. (K)** Crocker.

A continuation of CHE 640, with additional emphasis on heat and mass transport.

**701. Scattering Methods/Colloidal and Macromolecular Systems. (M)**

The scattering of light, x-rays and neutrons in (1) the characterization of macromolecules in solution and the solid state, (2) the study of solid-state polymer morphology, and (3) the characterization of inorganic, organic and biological systems of colloidal dimensions. Both theory and experimental methods will be covered.

**CHEMISTRY (AS) {CHEM} **

**521. Statistical Mechanics I. (A)** Prerequisite(s): CHEM 222.

Principles of statistical mechanics with applications to systems of chemical interest.

**522. Statistical Mechanics II. (B)** Prerequisite(s): CHEM 521.

A continuation of CHEM 521. The course will emphasize the statistical mechanical description of systems in condensed phases.

**523. Quantum Chemistry I. (A)** Prerequisite(s): CHEM 222.

The principles of quantum theory and applications to atomic systems.

**524. Quantum Chemistry II. (B)** Prerequisite(s): CHEM 523.

Approximate methods in quantum theory and applications to molecular systems.

**525. Molecular Spectroscopy. (A)**

A modern introduction to the theory of the interaction of radiation and matter and the practice of molecular spectroscopy. Conventional microwave, magnetic resonance, optical, photoelectron, double-resonance, and laser spectroscopic techniques will be included.

**526. Chemical Dynamics. (B)**

Theoretical and experimental aspects of important rate processes in chemistry.

**555. (BMB 554) Macromolecular Crystallography: Methods and Applications. (A)**

The first half of the course covers the principles and techniques of macro- molecular structure determination using X-ray crystallography. The second half of the course covers extracting biological information from X-ray crystal structures with special emphasis on using structures reported in the recent literature and presented by the students.

**559. BIOMOLEC IMAGING. (B)** Dmochowski.

This course considers the noninvasive, quantitative, and repetitive imaging of targeted macromolecules and biological processes in living cells and organisms.Imaging advances have arisen from new technologies, probe chemistry, molecular biology, and genomic information. This course covers the physical principles underlying many of the latest techniques, and defines experimental parameters such as spatial and temporal resolution, gain, noise, and contrast. Applications to cellular and in vivo imaging are highlighted for confocal, two-photon, and force microscopies; single-molecule, CARS, and fluorescence correlation spectroscopy; FRET and fluorescence bleaching; mass spectroscopy; MRI, PET and SPECT. The role of molecular imaging agents comprised of proteins, organic or inorganic materials is widely discussed.

**COMPUTER & INFORMATION SCIENCE (CIS) **

**L/R ****502.**** Analysis of Algorithms. (C)** Prerequisite(s): CIT 594 or equivalent.

An investigation of several major algorithms and their uses in areas including list manipulation, sorting, searching, selection and graph manipulation. Efficiency and complexity of algorithms. Compexity Classes.

**510. (CSE 410) Curves and Surfaces: Theory and Applications. (M)** Prerequisite(s): Basic knowledge of linear algebra, calculus, and elementary geometry. CIS 560 is not required.

The course is about mathematical and algorithmic techniques used for geometric modeling and geometric design, using curves and surfaces. There are many applications in computer graphics as well as in robotics, vision, and computational geometry. Such techniques are used in 2D and 3D drawing and plot, object silhouettes, animating positions, product design (cars, planes, buidlings), topographic data, medical imagery, active surfaces of proteins, attribute maps (color, texture, roughness), weather data, art, etc. Three broad classes of problems will be considered: approximating curved shapes, using smooth curves or surfaces. Interpolating curved shapes, using smooth curves or surfaces. Rendering smooth curves or surfaces.

**511. Theory of Computation. (C)** Prerequisite(s): Basic notions of discrete algebra.

Finite automata (deterministic and nondeterministic) regular graphs, regular expressions, regular grammars, (Nerode congruence), the "pumping lemma", closure properties. Context-free languages. Standard forms: removal of e-rules, chain rules, reduced grammars. Chomsky Normal Form. Context-free languages as fixed points (Ginsburg and Rose's Theorem). Greibach Normal Form (using Rosenkrantz's matrix method). Ogden's Lemma and the "pumping lemma". Pushdown automata (PDA's). Equivalence of PDA's and context-free grammars. Brief sketch of top-down and bottom-up (nondeterministic) parsing. Deterministic PDA's. Closure properties. Partial recursive functions, Turing machines and RAM programs. Primitive recursion. Minimization. Equivalence of the models. Church/Turing's thesis. Acceptable Codings. A Universal RAM program. Undecidability of the halting problem. Recursively enumerable sets (RE sets).

**SM ****518.**** (PHIL412) Topics in Logic; Finite Model Theory and Descriptive Complexity. (C)**

This course will examine the expressive power of various logical languages over the class of finite structures. The course begins with an exposition of some of the fundamental theorems about the behavior of first-order logic in the context of finite structures, in particular, the Ehrenfeucht-Fraisse Theorem and the Trahktenbrot Theorem. The first of these results is used to show limitations on the expressive power of first-order logic over finite structures while the second result demonstrates that the problem of reasoning about finite structures using first-order logic is surprisingly complex. The course then proceeds to consider various extensions of first-order logic including fixed-point operators, generalized quantifiers, infinitary languages, and higher-order languages. The expressive power of these extensions will be studied in detail and will be connected to various problems in the theory of computational complexity. This last motif, namely the relation between descriptive and computational complexity, will be one of the main themes of the course.

**520. Artificial Intelligence and Machine Learning. (A)** Prerequisite(s): Elementary probability, calculus, and linear algebra. Basic programming experience.

This course will provide a survey of mathematical methods and programming techniques in artificial intelligence, pattern recognition, machine learning, and neural computation. Topics include: inference and learning in probabilistic graphical models; autonomous agents, decision-making, and reinforcement learning; neural networks and biologically inspired models of computation; statistical methods for prediction, clustering, and dimensionality reduction; and applications to vision, robotics, speech, and natural language processing.

**530. Computational Linguistics. (A)**

Computational approaches to the problem of understanding and producing written and spoken natural language, including speech processing, syntactic parsing, statistical and corpus-based techniques, semantic interpretation, discourse meaning, and the role of pragmatics and world knowledge. It is recommended that students have some knowledge of logic, basic linguistics, and programming.

**535. (BIOL535, GCB 535) Introduction to Bioinformatics. (A)**

The course covers methods used in computational biology, including the statistical models and algorithms used and the biological problems which they address. Students will learn how tools such as BLAST work, and will use them to address real problems. The course will focus on sequence analysis problems such as exon, motif and gene finding, and on comparative methods but will also cover gene expression and proteomics.

**536. (BIOL536, GCB 536) Computational Biology. (A)** Prerequisite(s): Math 104/114 or equivalent, BIOL 221 or equivalent, or permission of the instructor.

Computational problems in molecular biology, including sequence search and analysis, informatics, phylogenetic reconstruction, genetic mapping and optimization.

**537. Biomedical Image Analysis. (C)** Prerequisite(s): Math through multivariate calculus (MATH 241), programming experience, as well as some familiarity with linear algebra, basic physics, and statistics.

This course covers the fundamentals of advanced quantitative image analysis that apply to all of the major and emerging modalities in biological/biomaterials imaging and in vivo biomedical imaging. While traditional image processing techniques will be discussed to provide context, the emphasis will be on cutting edge aspects of all areas of image analysis (including registration, segmentation, and high-dimensional statistical analysis). Significant coverage of state-of-the-art biomedical research and clinical applications will be incorporated to reinforce the theoretical basis of the analysis methods.

**558. (LING525) Computer Analysis and Modeling of Biological Signals and Systems. (B)** Prerequisite(s): Undergraduate-level knowledge of linear algebra.

A graduate course intended to introduce the use of signal and image processing tools for analyzing and modeling biological systems. We present a series of fundamental examples drawn from areas of speech analysis/synthesis, computer vision, and modeling of biological perceptual systems. Students learn the material through lectures and via a set of computer exercises developed in MATLAB.

**560. (CSE 460) Computer Graphics. (A)** Prerequisite(s): One year programming experience (C, JAVA, C++).

A thorough introduction to computer graphics techniques, including 3D modeling, rendering, and animation. Topics cover: geometric transformations, geometric algorithms, software systems (OpenGL), 3D object models (surface and volume), visible surface algorithms, image synthesis, shading and mapping, ray tracing, radiosity, global illumination, photon mapping, anti-aliasing, animation techniques, and virtual environments.

**571. (PHIL411) Recursion Theory. (A)**

The course covers the basic theory of recursive and recursively enumerable sets and the connection between this theory and a variety of decision problems of interest in a computational setting. The course will then proceed to an exposition of recursion theoretic reducibilities. Elementary results about degrees of unsolvability are established. The theory of arithmetical, analytical, and projective hierarchies will be presented. The study of functionals at this point will provide an entry into the computationally important subject of recursion at higher types. Basic parts of the theory of inductive definitions and monotone operators will be presented. If time and interest permit, this theory will be applied to the analysis of the semantical paradoxes. The course will conclude with an investigation of the lower levels of the analytical and projective hierarchies. Applications to the degrees of unsolvability of various logical systems will be presented, connections between the hierarchies and predicative formal systems will be established, and the relation between the theory of the projective hierarchy and topics in classical descriptive set theory will be indicated.

**SM ****572.**** (PHIL413) Set Theory. (C)**

This course is an introduction to set theory. It will begin with a study of Zermelo-Fraenkel set theory (ZF) as a partial description of the cumulative hierarchy of sets. Elementary properties of cardinal and ordinal numbers will be developed in ZF. The inner model of constructible sets will be used to extablish the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF. The method of forcing will be introduced to establish the independence of the continuum hypothesis from ZF and other independence results. Large cardinals and their bearing on the resolution of questions about the continuum will be considered.

**610. (MATH676) Advanced Geometric Methods in Computer Science. (B)** Prerequisite(s): CIS 510 or coverage of equivalent material.

The purpose of this course is to present some of the advanced geometric methods used in geometric modeling, computer graphics, computeer vision, etc. The topics may vary from year to year, and will be selected among the following subjects (nonexhaustive list): Introduction to projective geometry with applications to rational curves and surfaces, control points for rational curves, retangular and triangular rational patches, drawing closed rational curves and surfaces; Differential geometry of curves (curvature, torsion, osculating planes, the Frenet frame, osculating circles, osculating spheres); Differential geometry of surfaces (first fundamental form, normal curvature, second fundamental form, geodesic curvature, Christoffel symbols, principal curvatures, Gaussian curvature, mean curvature, the Gauss map and its derivative dN, the Dupin indicatrix, the Theorema Egregium equations of Codazzi-Mainadi, Bonnet's theorem, lines of curvatures, geodesic torsion, asymptotic lines, geodesic lines, local Gauss-Bonnet theorem).

**613. (ESE 617, MEAM613) Nonlinear Control Theory. (M)** Prerequisite(s): A sufficient background to linear algebra (ENM 510/511 or equivalent) and a course in linear control theory (MEAM 513 or equivalent), or written permission of the instructor.

The course studies issues in nonlinear control theory, with a particular emphasis on the use of geometric principles. Topics include: controllability, accessibility, and observability, and observability for nonlinear systems; Forbenius' theorem; feedback and input/outpub linearizaiton for SISO and MIMO systems; dynamic extension; zero dynamics; output tracking and regulation; model matching disturbance decoupling; examples will be taken from mechanical systems, robotic systems, including those involving nonholonomic constraints, and active control of vibrations.

**635. (BIOL537, GCB 537) Advanced Computational Biology. (A)** Prerequisite(s): Biol 536 or permission of the instructor.

Discussion of special research topics.

**SM ****639.**** Statistical approaches to Natural Language Understanding. (C)**

This course examines the recent development of corpus-based techniques in natural language processing, focussing on both statistical and primarily symbolic learning techniques. Particular topics vary from year to year.

**COGNITIVE SCIENCE (COGS) {COGS} **

**501-502 ****Mathematical Foundations of Language & Communication I and II** Cross-listed with PSYC 502 and LING 546

A two-semester Mathematical Foundations sequence will provide all students with basic mathematical modeling and algorithmic tools, while still providing sufficient challenges for the most advanced. These two courses (course numbers to be announced soon), will be taught in a computer/media lab setting and will cover relevant aspects of a wide range of mathematical topics that are directly relevant to animal, human or machine communication, or that provide prerequisites for these topics. Examples of topics directly relevant to communication include information theory, game theory, and formal language theory. Examples of important topics include signal processing, machine learning, and probabilistic models. These two semesters obviously cannot substitute entirely for the dozen or more semesters that normally would be required to cover a similar range of topics. However, they can give students the ability to understand and implement algorithms from published descriptions, especially given appropriate libraries of basic functions, and to discuss alternative approaches with experts in a well-informed manner. It is clearly not the case that every LCS-IGERT students will use every mathematical or algorithmic topic from this course in his or her research. However, applications are often unexpected, and fortune favors the prepared. In addition, this background will enable students to make sense of a wide range of courses and readings that might otherwise be inaccessible. Finally, the shared experience of this course will help IGERT students to establish a personal as well as conceptual basis for future collaborations. Each semester of this two-semester sequence will be co-taught by two faculty members (Liberman and Kahana will teach the first semester, in Spring of 2006). Because of the diversity of topics and of the students' backgrounds, the two-semester course sequence will be organized into a series of "modules", each designed to explicate a core mathematical and algorithmic topic. Each module will deal with specific problems of the type that IGERT students need to solve and will be as self-contained as possible, although of course one module will often require understanding of concepts and techniques taught in another.

**ELECTRICAL & SYSTEMS ENGINEERING (EG) {ESE} **

**500. Linear Systems Theory. (A)** Prerequisite(s): Open to graduates and undergraduates that have taken undergraduate courses in linear algebra and differential equations.

This graduate level course focuses on linear system theory in time domain based on linear operators. The course introduces the fundamental mathematics of linear spaces, linear operator theory, and then proceeds with existence and uniqueness of solutions of differential equations, the fundamental matrix solution and state transition matrix for time-varying linear systems. It then focuses on the fundamental concepts of stability, controllability, and observability, feedback, pole placement, observers, output feedback, kalman filtering, linear quadratic regulator. Special topics such as optimal control, robust, geometric linear control will be considered as time permits.

**502. Introduction to Spatial Analysis. (B)** Prerequisite(s): ESE 302 or equivalent.

The course is designed to introduce students to modern statistical methods for analyzing spatial data. These methods include nearest-neighbor analyses of spatial point patterns, variogram and kriging analyses of continuous spatial data, and autoregression analyses of area data. The underlying statistical theory of each method is developed and illustrated in terms of selected GIS applications. Students are also given some experience with ARCMAP, JMPIN, and MATLAB software.

**504. (OPIM910) Introduction to Optimization Theory. (A)** Prerequisite(s): Linear Algebra.

Introduction to mathematical programming for students who would like to be intelligent and sophisticated consumers of mathematical programming theory but do not plan to specialize in this area. Integer and nonlinear programming are covered, including the fundamentals of each area together with a sense of the state-of-the-art and expected directions of future progress.

**505. (ESE 406, MEAM513) Control of Systems. (B)**

Basic methods for analysis and design of feedback control in systems. Applications to practical systems. Methods presented include time response analysis, frequency response analysis, root locus, Nyquist and Bode plots, and the state-space approach.

**507. Adaptive Complex Systems. (M)** Prerequisite(s): ESE 301 and ESE 304 or equivalent.

This course examines the characteristics and behavior of systems which adapt to changes in the environment. Observation of adaptive systems reveals the existence of some of the following properties and/or capabilities: learning, memory, sensing, acting, directing attention, having redundant sub-systems, communicating, predicting the future, creativity, being able to choose, recreating, willing to protect its descendants, dreaming, taking risks, reasoning, cooperating, forming conceptual domain by abstracting and modeling the world, recognizing the constraints, etc. This course will explore methodologies that are being used for designing man-made systems (infrastructures, corporations, government, etc.). The question that will be explored is why and how an adaptive (biological or otherwise) system responds to changes in the environment. We are interested in learning how that understanding can inform the design of man-made systems.

**509. (TCOM503) Waves, Fibers and Antennas for Telecommunications. (A)**

This course is designed to provide an understanding of the physical aspects of telecommunications systems. This includes an understanding of waves and wave propagation, basic optics, the operation of optical fibers and fiber communication systems, an introduction to optical networks, free-space optical communications, and an understanding of simple antennas and arrays and their use in wireless communication.

**510. Electromagnetic and Optical Theory. (A)**

This course reviews electrostatics, magnetostatics, electric and magnetic materials, induction, Maxwell's equations, potentials and boundary-value problems. Topics selected from the areas of wave propagation, wave guidance, antennas, and diffraction will be explored with the goal of equipping students to read current research literature in electromagnetics, microwaves, and optics.

**511. Modern Optics and Image Understanding. (B)** Prerequisite(s): ESE 310, graduate standing, or permission of the instructor.

The goal of this course is to provide a unified approach to modern optics, image formation, analysis, and understanding that form the theoretical basis for advanced imaging systems in use today in science, medicine and technology. The emphasis is on imaging systems that employ electromagnetic energy but the principles covered can be extended to systems employing other forms of radiant energy such as acoustical.

**517. (BE 517) Optical Imaging. (A)** Prerequisite(s): ESE 310 and 325 or equivalent.

A modern introduction to the physical principles of optical imaging with biomedical applications. Propagation and interference of electromagnetic waves. Geometrical optics and the eikonal. Plane-wave expansions, diffraction and the Rayleigh criterion. Scattering theory and the Born approximation. Introduction to inverse problems. Multiple scattering and radiative transport. Diffusion approximation and physical optics of diffusing waves. Imaging in turbid media. Introduction to coherence theory and coherence imaging. Applications will be chosen from the recent literature in biomedical optics.

**530. Elements of Probability Theory and Random Processes. (A)** Prerequisite(s): A semester of undergraduate probability at the level of STAT 430 or ESE 301.

This rapidly moving course provides a formal framework for the development of fundamental ideas in probability theory. This course is a prerequisite for subsequent courses in communication theory and telecommunications such as ESE 576 and TCOM 501. The course is also suitable for students seeking a rigourous and broad graduate-level exposure to probabalistic ideas and principles with applications in diverse settings.

Topics covered are taken from: discrete and continuous probability spaces; combinatorial probabilities; conditional probability and indepence; Bayes rules and the theorem of total probability; the inclusion-exclusion principle, Bonferroni's inequalities, the Poisson paradigm, probability sieves, and the Lovascz local lemma; arithmetic and lattice distributions; the central term and the tails of the binomial, Poisson approximation; densities in one and more dimensions; characterizations of the uniform, exponential and normal densities; probability spaces, random variables and distribution functions; transformations, random number generation; independent random variables, Borel's normal law; measures of central tendency---mean, median, mode, mathematical expectation; the monotone convergence theorem and its applications; additivity and monotonicity of expectations; moments; the inequalities of Markov, Chebyshev, Chernoff, and Talagrand; concentration phenomena and applications; limit theorems, the weak and strong laws; generating functions, recurrent events, Blackwell's theorem; characteristic functions, the central limit theorem.

**531. Digital Signal Processing. (A)**

This course covers the fundamentals of real-time processing of discrete-time signals and digital systems. Specific topics covered are: review of signals and linear system representations; convolution and discrete Fourier transforms; Z-transforms; frequency response of lienar discrete-time systems; sampling and analog/digital conversion; finite and infinite impulse response filters; digital filter design; fast Fourier transfers and applications; adaptive filtering algorithms; wavelet transforms. Projects requiring implementation of specific digital signal processing algorithms will also be assigned.

**539. (BE 539) Neural Networks, Chaos, and Dynamics: Theory and Application. (B)**

Physiology and anatomy of living neurons and neural networks; Brain organization; Elements of nonlinear dynamics, the driven pendulum as paradigm for complexity, synchronicity, bifurcation, self-organization and chaos; Iterative maps on the interval, period-doubling route to chaos, universality and the Feigenbaum constant, Lyapunov exponents, entropy and information; Geometric characterization of attractors; Fractals and the Mandelbrot set; Neuron dynamics: from Hudgkin-Huxley to integrate and fire, bifurcation neuron; Artificial neural networks and connectionist models, Hopfield (attractor-type) networks, energy functions, convergence theorems, storage capacity, associative memory, pattern classification, pattern completion and error correction, the Morita network; Stochastic networks, simulated annealing and the Boltzmann machine, solution of optimization problems, hardware implementations of neural networks; the problem of learning, algorithmic approaches: Perception learning, back-propagation, Kohonnen's self-organizing maps and other networks; Coupled-map lattices; Selected applications including financial markets.

**575. (TCOM511) Introduction to Wireless Systems. (A)** Prerequisite(s): Undergraduate linear systems and elementary probability theory.

System/Network Design, cellular concepts, resource management, radio management, radio channel propagation fundamentals, modulation, fading countermeasure, diversity, coding, spread spectrum, multiple access techniques.

**576. Digital Communication Systems. (B)** Prerequisite(s): Undergraduate linear systems, probability, random processes.

Sampling, source coding, and capacity. Quantization and coding of speech and video. Baseband data transmission: line coding, intersymbol interference, equalization, digital modulation schemes, spectral efficiency. Error control coding; block and convolutional codes, Viterbi Algorithm.

**603. Simulation Modeling and Analysis. (B)** Prerequisite(s): Probability (undergraduate level) and one computer language.

This course provides a study of discrete-event systems simulation. Some areas of application include: queuing systems, inventory systems, reliability systems and Monte-Carlo systems. The course examines many of the discrete and continuous probability distributions used in simulation studies as well as the Poisson process. Long-run measurements of performances of queuing systems, steady-state behavior of infinite and finite-population queuing systems and network of queues are also examined. Fundamental to most simulation studies is the ability to generate reliable random numbers. The course investigates the basic properties of random numbers and techniques used for the generation of pseudo-random numbers. In addition, the course examines techniques used to test pseudo-random numbers for uniformity and independence. These include the Kolmogorov-Smirnov and chi-squared tests, runs tests, gap tests, and poker tests. Random numbers are used to generate random samples and the course examines the inverse-transform, convolution, composition and acceptance/rejection methods for the generation of random samples for many different types of probability distributions.

Finally, since most inputs to simulation are probabilistic instead of deterministic in nature, the course examines some techniques used for identifying the probabilistic nature of input data. These include identifying distributional families with sample data, then using maximum-likelihood methods for parameter estimating within a given family and then testing the final choice of distribution using chi-squared goodness-of-fit tests.

**605. Modern Convex Optimization. (B)** Prerequisite(s): Knowledge of linear algebra and willingness to do programming. Exposure to numerical computing, optimization, and application fields is helpful but not required.

This course concentrates on recognizing and solving convex optimization problems that arise in engineeering. Topics include: convex sets, functions, and optimization problems. Basis of convex analysis. Linear, quadratic, geometric, and semidefinite programming. Optimality conditions, duality theory, theorems of alternative, and applications. Interior-point methods, ellipsoid algorithm and barrier methods, self-concordance. Applications to signal processing, control, digital and analog circuit design, computation geometry, statistics, and mechanical engineering.

**610. Electromagnetic and Optical Theory II. (M)**

This course covers exact, approximate and numerical methods of wave propagation, radiation, diffraction and scattering with an emphasis on bringing students to a point of contributing to the current research literature. Topics are chosen from a list including analytical and numerical techniques, waves in complex media and metamaterials, photonic bandgap structures, imaging, miniaturized antennas, high-impedance ground plans, and fractal electrodynamics.

**617. (CBE 617, CIS 613, MEAM613) Non-Linear Control Theory. (M)** Prerequisite(s): Undergraduate Control course.

This courses focuses on nonlinear systems, planar dynamical systems, Poincare Bendixson Theory, index theory, bifurcations, Lyapunov stability, small-gain theorems, passivity, the Poincar map, the center manifold theorem, geomentric control theory, and feedback linearization.

**630. Elements of Neural Computation, Complexity, and Learning. (M)** Prerequisite(s): A semester course in probability or equivalent exposure to probability (e.g. ESE 530).

Non-linear elements and networks: linear and polynomial threshold elements, sigmoidal units, radial basis functions. Finite (Boolean) problems: acyclic networks; Fourier analysis and efficient computation; projection pursuit; low frequency functions. Network capacity: Feedforward networks; Vapnik-Chervnenkis dimension. Learning theory: Valiant's learning model; the sample complexity of learning. Learning algorithms: Perception training, gradient descent algorithms, stochastic approximation. Learning complexity: the intractability of learning; model selection.

**632. Random Process Models and Optimum Filtering. (M)** Prerequisite(s): ESE 530 or equivalent.

Convergence, continuity, stationarity and second order properties of random processes. Spectral representation. Markov processes, Wiener and Poisson processes. Karhunen-Loeve expansion. Optimum filtering: matched and Wiener filtering, finite observations, spectral factorization. Kalman filtering. Basic concepts of parameter estimation and hypothesis testing.

**650. Learning in Robotics. (A)** Prerequisite(s): Students will need permission from the instructor. They will be expected to have a good mathematical background with knowledge of machine learning techniques at the level of CIS 520, signal processing techniques at the level of ESE 531, as weill as have some robotics experience.

This course will cover the mathematical fundamentals and applications of machine learning algorithms to mobile robotics. Possible topics that will be discussed include probabalistic generative models for sensory feature learning. Bayesian filtering for localization and mapping, dimensionality reduction techniques formotor control, and reinforcement learning of behaviors. Students are expected to have a solid mathematical background in machine learning and signal processing, and will be expected to implement algorithms on a mobile robot platform for their course projects. Grading will be based upon course project assignments as well as class participation.

**674. Information Theory. (M)** Prerequisite(s): ESE 530 or equivalent exposure to probability theory.

Deterministic and probabalistic information. The pigeon-hole principle. Entropy, relative entropy, and mutual information. Random processes and entropy rate. The asymptotic equipartition property. Optimal codes and data compression. Channel capacity. Source channel coding. The ubiquitous nature of the theory will be illustrated with a selection of applications drawn from among: universal source coding, vector quantization, network communication, the stock market, hypothesis testing, algorithmic computation and Kolmogorov complexity, and thermodynamics.

**ENGINEERING AND APPLIED SCIENCE (EG) {EAS} **

**ENGINEERING MATHEMATICS (ENM)**

**L/R ****502. (ENM 402) Numerical Methods and Modeling. (B)** Sinno. Prerequisite(s): Knowledge of a computer language, Math 240 and 241; ENM 510 is highly recommended; or their equivalents.

Numerical modeling using effective algorithms with applications to problems in engineering, science, and mathematics, and is intended for graduate and advanced undergraduate students in these areas. Interpolation and curve fitting, numerical integration, solution of ordinary and partial differential equations by finite difference, and finite element methods. Includes use of representative numerical software packages such as MATLAB PDE Toolbox.

**503. Introduction to Probability and Statistics. (A)** Prerequisite(s): MATH 240 or equivalent.

Introduction to probability. Expectation. Variance. Covariance. Joint probability. Moment generating functions. Stochastic models and applications. Markov chains. Renewal processes. Queuing models. Statistical inference. Linear regression. Computational probability. Discrete-event simulation.

**504. Logic and Computation in Algebra. (B)** Prerequisite(s): Discrete mathematics, algebra and set theory (CSE 260, CSE 261), CIS 511 and CIS 500 strong recommended as corequisites.

An introduction to universal algebra, equational reasoning, lambda calculus and computation by term rewriting. Provides a strong foundation for further studies in computational logic, programming languages, and computational linguistics. Universal algebra, trees and algebraic terms, unification, equational logic, rewrite systems, applications to automated deduction, lambda calculus, combinatory logic, simple types. Applications to programming languages. Connections with computability theory.

**508. Engineering Math. (A)** Staff.

**510. Foundations of Engineering Mathematics - I. (A)** Prerequisite(s): MATH 240, MATH 241 or equivalent.

This is the first course of a two semester sequence, but each course is self contained. Over the two semesters topics are drawn from various branches of applied mathematics that are relevant to engineering and applied science. These include: Linear Algebra and Vector Spaces, Hilbert spaces, Higher-Dimensional Calculus, Vector Analysis, Differential Geometry, Tensor Analysis, Optimization and Variational Calculus, Ordinary and Partial Differential Equatins, Initial-Value and Boundary-Value Problems, Green's Functions, Special Functions, Fourier Analysis, Integral Transforms and Numerical Analysis. For the 2004-2005 Academic Year, the fall course emphasizes the study of Hilbert spaces, ordinary and partial differential equations, the initial-value, boundary-value problem, and related topics.

**511. Foundations of Engineering Mathematics - II. (B)** Prerequisite(s): Math 240, Math 241 or equivalent.

This is the second course of a two semester sequence, but each course is self contained. Over the two semesters topics are drawn from various branches of applied mathematics that are relevant to engineering and applied science. These include: Linear Algebra and Vector Spaces, Hilbert spaces, Higher-Dimensional Calculus, Vector Analysis, Differential Geometry, Tensor Analysis, Optimization and Variational Calculus, Ordinary and Partial Differential Equations, Initial-Value and Boundary-Vaue Problems, Green's Functions, Special Functions, Fourier Analysis, Integral Transforms and Numerical Analysis. For the 2004-2005 Academic Year, the spring course emphasizes the study of higher-dimensional calculus, vector analysis, linear algebra, tensor analysis, numerical methods and related topics.

**600. Functional Analysis. (A)** Prerequisite(s): ENM 500, ENM 501 or ENM 510, ENM 511 or equivalent.

This course teaches the fundamental concepts underlying metric spaces, normed spaces, vector spaces, and inner-product spaces. It begins with a discussion of the ideals of convergence and completeness in metric spaces and then uses these ideas to develop the Banach fixed-point theorem and its applications to linear equations, differential equations and integral equations. The course moves on to a study of normed spaces, vector spaces, and Banach spaces and operators defined on vector spaces, as well as functional defined between vector spaces and fields. The course then moves to the study of inner product spaces, Hilbert spaces, orthogonal complements, direct sums, and orthonormal sets. Applications include the study of Legendre, Hermite, Laguerre, and Chebyshev polynomials, and approximation methods in normed spaces. The course then concludes with a study of eigenvalues and eigenspaces of linear operators and spectral theory in finite-dimensional vector spaces.

**601. Special Topics in Engineering Mathematics - Nonlinear Dyanics and Chaos. (B)** Prerequisite(s): Permission of Instructor.

This course provides an introduction to how simulation can be used for studying systems that evolve in discrete intervals of time, and systems that do not depend on time such as Monte-Carlo systems. The discrete systems and Monte-Carlo systems that the course focuses on are: queuing systems, reliability systems, inventory systems, and warfare analysis. Emphasis is placed on the mathematics needed to develop the simulations. Consequently, the course covers the common probability distributions encountered in simulations, basic queuing theory, the generation of pseudo-random numbers and pseudo-random variants, and fitting statistical distributions to input data. Throughout the course, examples of simulations using Microsoft Excel or equivalent software are provided. The course ends with the analysis of a small warfare simulation project that uses most of the topics covered by the course.

Discrete Dynamical Systems: One-Dimensional Maps, Fixed Points and Cobwebs, The Liapunov Exponent, Universality and Feigenbaum's Number, Renormalization Theory, Fractals, Countable and Uncountable Sets, The Cantor Middle-Thirds Set, Self-Similar Fractals and Their Dimensions, The von Koch Curve, Box Dimension and Multifractals.

**603. Introduction to Probability, Random Variables, and Random Functions. (B)** Prerequisite(s): MATH 240, MATH 241 or equivalent.

Foundations of probability theory. Random variables. Distribution functions. Expected values. Characteristic functions. Sequences of random variables. Elements of random processes.

**GENOMICS AND COMPUTATIONAL BIOLOGY (MD) {GCB} **

**531. Introduction to Genome Science. (A)** W.Ewens M.Bucan.

This course serves as an introduction to the main laboratory and theoretical aspects of genomics and computational biology. The main topics discussed center around the analysis of sequences (annotation, alignment, homology, gene finding, variation between sequences, SNP's) and the functional analysis of genes (expression levels, proteomics, screens for mutants), together with a discussion of gene mapping, linkage disequilibrium and integrative genomics.

**535. (CIS 535) Introduction to Computational Biology. (A)** S.Master H.Hannenhalli. Prerequisite(s): Introductory Biology and Introductory Programming. Course to be held in Room 305 of the Towne Building.

The course provides a broad overview of bioinformatics and computational biology as applied to biomedical research. Course material will be geared towards answering specific biological questions ranging from detailed analysis of a single gene through whole-genome analysis, transcriptional profiling, and systems biology. The relevant principles underlying these methods will be addressed at a level appropriate for biologists without a background in computational sciences. This course should enable students to integrate modern bioinformatics tools into their research program.

Should I take the course? This course will emphasize hands-on experience with application to current biological research problems. However, it is not intended for computer science students who want to learn about biologically motivated algorithmic problems; GCB/CIS/BIO536 would be more appropriate for such individuals. The course will assume a solid knowledge of modern biology. An advanced undergraduate course such as BIO421 or a graduate course in Biology such as BIOL526 (Experimental Principles in Cell and Molecular Biology), BIOL527 (Advanced Molecular Biology and Genetics), BIOL528 (Advanced Molecular Genetics), BIOL540 (Genetic Systems), or equivalent, is a prerequisite.

**536. (BIOL536, CIS 536) Computational Biology. (M)**

An introductory computational biology course designed for computational scientists. The course will cover fundamentals of algorithms, statistics, and mathematics as applied to biological problems. In particular, emphasis will be given to biological problem modeling. Students will be expected to learn the basic algorithms underlying computational biology, basi c mathematical / statistical proofs and molecular biology. Topics to be cover ed are genome annotation and string algorithms, pattern search and statistical learning, molecular evolution and phylogenetics and small molecule folding.

**537. (BIOL537, CIS 635) Advanced Computational Biology. (A)** S.Kannan J.Kim. Course to be held in Room 109 of Leidy Laboratories.

A discussion of special research topics.

**SM ****752. (CAMB752) Genomics. (B)** Drs. Riethman and Cheung.

Recent advances in molecular biology, computer science, and engineering have opened up new possibilities for studying the biology of organisms. Biologists now have access to the complete set of cellular instructions encoded in the DNA of specific organisms, including dozens of bacterial species, the yeast Saccharomyces cerevisiae, the nematode C. elegans, and the fruit fly Drosophila melanogaster.

The goals of the course are to 1) introduce the basic principles involved in mapping and sequencing genomes, 2) familiarize the students with new instrumentation, informatics tools, and laboratory automation technologies related to genomics; 3) teach the students how to access the information and biological materials that are being developed in genomics, and 4) examine how these new tools and resources are being applied to specific research problems.

**MATERIALS SCIENCE AND ENGINEERING (EG) {MSE} **

**520. Structure of Materials. (A)** Prerequisite(s): Permission of the Undergraduate Curriculum Chair and Instructor.

Description of Crystal Structure-Symmetry, Point and Space Groups. Structures of different material types-glasses, polymers, semiconductors, ceramics and metals. Relationship between bonding and structural types. Methods of structure determination. Diffraction of x-rays and neutrons--x-ray methods. Microstructures of solids. Topology of granular structures. Grain boundary structures. Fractal description of microstructures.

**530. Thermodynamics and Phase Equilibria. (A)** Worrell, Winey. Prerequisite(s): Permission of the Undergraduate Curriculum Chair and Instructor.

Review of fundamental thermodynamic laws and criteria for equilibrium. Reaction equilibria in multicomponent systems. Free energies of mixing solutions, liquids, solids, and polymers. Binary and ternary phase diagrams. Surfaces and interfaces.

**540. (MSE 440) Phase Transformations. (B)** Chen. Prerequisite(s): Permission of the Undergraduate Curriculum Chair and Instructor.

The atomic structure of condensed matter is dependent upon temperature, pressure, thermal history and other variables. In this course, the science of such structural transitions is treated. The topics discussed include introduction to statistical mechanics, theory of nucleation and growth kinetics, solidification, diffusionless solid state transformations, and microscopic theory of phase transition.

**566. Physical Properties of Ceramics. (A)** Prerequisite(s): MSE 360 or MSE 560 and a good foundation in solid state physics are prerequisites for this class.

This course will focus on the properties of inorganic compounds considered to be ceramics. Optical, dielectric and magnetic properties of oxides are treated in depth and illustrated with laboratory demonstrations and experiments. Strategies for mechanical property optimization are examined.

**570. (ESE 514) Physics of Materials I. (C)** Fischer. Prerequisite(s): Undergraduate physics and math thru modern physics and differential equations.

Failures of classical physics and the historical basis for quantum theory. Postulates of wave mechanics; uncertainty principle, wave packets and wave-particle duality. Schrodinger equation and operators; eigenvalue problems in 1 and 3 dimensions (barriers, wells, hydrogen, atom). Perturbation theory; scattering of particles and light. Free electron theory of metals; Drude and Sommerfeld models, dispersion relations and optical properties of solids. Extensive use of computer-aided self-study will be made.

**571. (ESE 515) Physics of Materials II. (M)** Fischer. Prerequisite(s): MSE 570 or equivalent.

Failures of free electron theory. Crystals and the reciprocal lattice wave propagation in periodic media; Bloch's theorem. One-electron band structure models: nearly free electrons, tight binding. Semiclassical dynamics and transport. Cohesive energy, lattic dynamic and phonons. Dielectric properties of insulators. Homeogenous semiconductors and p-n junctions. Experimental probes of solid state phenomena; photoemission, energy loss spectroscopy, neturon scattering. As time permits, special topics selected from the following: correlation effects, semiconductor alloys and heterostructures, amorphous semiconductors, electro-active polymers.

**580. (MSE 430) Polymers and Biomaterials. (B)** Prerequisite(s): MSE 260 or equivalent course in thermodynamics or physical chemistry (such as BE 223, CHE 231, MEAM 203).

This course focuses on synthesis, characterization, microstructure, rheology, and structure-property relationships of polymers, polymer directed composites and their applications in biotechnology. Topical coverage includes: polymer synthesis and functionalizaiton; polymerizaiton kinetics; structure of glassy, crystalline, and rubbery polymers; thermodynamics of polymer solutions and blends, and crystallization; liquid crystallinity, microphase separation in block copolymers; polymer directed self-assembly of inorganic materials; biological applications of polymeric materials. Case studies include thermodynamics of block copolymer thin films and their applications in nanolithography, molecular templating of sol-gel growth using block copolymers as templates; structure-property of conducting and optically active polymers; polymer degradation in drug delivery; cell adhesion on polymer surface in tissue engineering.

**581. Advanced Polymer Physics. (A)** Winey/Composto. Prerequisite(s): MSE 430 or equivalent.

Advanced polymer physics includes the topics of polymer chain statistics, thermodynamics, rubber elasticity, polymer morphology, fracture, and chain relaxation. Rigorous derivations of select theories will be presented along with experimental results for comparison. Special topics, such as liquid crystalline polymers, blends and copolymers, will be presented throughout the course. Special topics, such as liquid crystallintiy, nanostructures, and biopolymer diffusion, will be investigated by teams of students using the current literature as a resource.

**660. Atomistic Modeling in Materials Science. (M)** Vitek.

Why and what to model: Complex lattice structures, structures of lattice defects, crystal surfaces, interfaces, liquids, linking structural studies with experimential observations, computer experiments. Methods: Molecular statics, molecular dynamics, Monte Carlo. Evaluation of physical quantities employing averages, fluctuations, correlations, autocorrelations, radial distribution function, etc. Total energy and interatomic forces: Local density functional theory and abinitio electronic structure calculations, tight-binding methods, empirical potentials for metals, semiconductors and ionic crystals.

**670. Statistical Mechanics of Solids. (A)**

This course constitutes an introduction to statistical mechanics with an emphasis on application to crystalline solids. Ensemble theory, time and ensemble averages and particle statistics are developed to give the basis of statistical thermodynamics. The theory of the thermodynamic properties of solids is presented in the harmonic approximation anharmonic properties are treated by the Mie-Gruneisen method. Free electron theory in metals and semiconductors is given in some detail, with the transport properties being based on conditional transition probabilities and the Boltzmann transport equation. The theory of order-disorder alloys is treated by the Bragg-Williams, Kirkwood and quasi-chemical methods.

**MATHEMATICS (AS) {MATH} **

**500. Geometry-Topology, Differential Geometry. (M)** Staff. Prerequisite(s): Math 240/241.

Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.

**501. Geometry-Topology, Differential Geometry. (M)** Staff. Prerequisite(s): Math 500 or with the permission of the instructor.

Review of 2- and 3-dimensional vector calculus, differential geometry of curves and surfaces, Gauss-Bonnet theorem, elementary Riemannian geometry, knot theory, degree theory of maps, transversality.

**L/L ****502. Abstract Algebra. (A)** Staff. Prerequisite(s): Math 240. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

An introduction to groups, rings, fields and other abstract algebraic systems, elementary Galois Theory, and linear algebra -- a more theoretical course than Math 370.

**L/L ****503. Abstract Algebra. (B)** Staff. Prerequisite(s): Math 502 or with the permission of the instructor. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.

Continuation of Math 502.

**L/L ****508. Advanced Analysis. (A)** Staff. Prerequisite(s): Math 240/241. Math 200/201 also recommended.

Construction of real numbers, the topology of the real line and the foundations of single variable calculus. Notions of convergence for sequences of functions. Basic approximation theorems for continuous functions and rigorous treatment of elementary transcendental functions. The course is intended to teach students how to read and construct rigorous formal proofs. A more theoretical course than Math 360.

**L/L ****509. Advanced Analysis. (B)** Staff. Prerequisite(s): Math 508 or with the permission of the instructor. Linear algebra is also helpful.

Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods.

**513. (CSE 313, MATH313) Computational Linear Algebra.** Staff.

A number of important and interesting problems in a wide range of disciplines within computer science are solved by recourse to techniques from linear algebra. The goal of this course will be to introduce students to some of the most important and widely used algorithms in matrix computation and to illustrate how they are actually used in various settings. Motivating applications will include: the solution of systems of linear equations, applications matrix computations to modeling geometric transformations in graphics, applications of the Discrete Fourier Transform and related techniques in digital signal processing, the solution of linear least squares optimization problems and the analysis of systems of linear differential equations. The course will cover the theoretical underpinnings of these problems and the numerical algorithms that are used to perform important matrixcomputations such as Gaussian Elimination, LU Decomposition and Singular Value Decomposition.

**530. Mathematics of Finance. (M)** Staff. Prerequisite(s): Math 240, Stat 430.

This course presents the basic mathematical tools to model financial markets and to make calculations about financial products, especially financial derivatives. Mathematical topics covered: stochastic processes, partial differential equations and their relationship. No background in finance is assumed.

**542. Calculus of Variations. (M)** Staff. Prerequisite(s): Math 241.

Introduction to calculus of variations. The topics will include the variation of a functional, the Euler-Lagrange equations, parametric forms, end points, canonical transformations, the principle of least action and conservation laws, the Hamilton-Jacobi equation, the second variation.

**547. (STAT531) Stochastic Processes.** Staff.

**582. Applied Mathematics and Computation. (M)** Staff. Prerequisite(s): Math 240-241. Math 312, Math 360. Knowledge of Math 412 and Math 508 is recommended.

This course offers first-hand experience of coupling mathematics with computing and applications. Topics include: Random walks, randomized algorithms, information theory, coding theory, cryptography, combinatorial optimization, linear programming, permutation networks and parallel computing. Lectures will be supplemented by informal talks by guest speakers from industry about examples and their experience of using mathematics in the real world.

**583. Applied Mathematics and Computation. (M)** Staff. Prerequisite(s): Math 582 or with the permission of the instructor.

Continuation of Math 582.

**584. (BE 584) The Mathematics of Medical Imaging and Measurement. (M)** Staff. Prerequisite(s): Math 241, knowledge of linear algebra and basic physics.

In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly.The data which is accesible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT).In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.

**585. The Mathematics of Medical Imaging and Measurement. (M)** Staff. Prerequisite(s): Math 584 or with the permission of the instructor.

Continuation of Math 584.

**590. Advanced Applied Mathematics. (M)** Staff. Prerequisite(s): Math 241.

This course offers first-hand experience of coupling mathematics with applications. Topics will vary from year to year. Among them are: Random walks and Markov chains, permutation networks and routing, graph expanders and randomized algorithms, communication and computational complexity, applied number theory and cryptography.

**591. Advanced Applied Mathematics. (M)** Staff. Prerequisite(s): Math 590 or with the permission of the instructor.

Continuation of Math 590.

**594. (PHYS500) Advanced Methods in Applied Mathematics. (M)** Staff. Prerequisite(s): Math 241 or Permission of Instructor. Physics 151 would be helpful for undergraduates.

Introduction to mathematics used in physics and engineering, with the goal of developing facility in classical techniques. Vector spaces, linear algebra, computation of eigenvalues and eigenvectors, boundary value problems, spectral theory of second order equations, asymptotic expansions, partial differential equations, differential operators and Green's functions, orthogonal functions, generating functions, contour integration, Fourier and Laplace transforms and an introduction to representation theory of SU(2) and SO(3). The course will draw on examples in continuum mechanics, electrostatics and transport problems.

**600. Topology and Geometric Analysis. (A)** Staff. Prerequisite(s): Math 500/501 or with the permission of the instructor.

Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.

**601. Topology and Geometric Analysis. (B)** Staff. Prerequisite(s): Math 600 or with the permission of the instructor.

Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.

**602. Algebra. (A)** Staff. Prerequisite(s): Math 370/371 or Math 502/503.

Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal ideals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem, elementary representation theory. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations.

**603. Algebra. (B)** Staff. Prerequisite(s): Math 602 or with the permission of the instructor.

Continuation of Math 602.

**608. Real Analysis. (C)** Staff. Corequisite(s): Math 600/601.

Lebesgue measure and integral, Borel measures, convergence theorems. Banach spaces, Hahn-Banach Theorem, Lp-spaces, Riesz-Fischer theorem, Stone-Weierstrass theorem, Radon-Nikodym theorem. Applications to Fourier series and integrals, Plancherel Theorem, Distributions, convolutions and mollifiers. Partitions of unity. Applications to P.D.E.'s

**609. Complex Analysis. (C)** Staff. Corequisite(s): Math 600/601.

Complex numbers, analytic functions, Cauchy's theorem and consequences, isolated singularities, analytic continuation, open mapping theorem, infinite series and products, harmonic and subharmonic functions, maximum principle, fractorial linear transformations, geometric and local properties of analytic functions, Weierstrauss Theorem, normal families, residues, conformal mapping, Riemann mapping theorem, branch points, second order linear O.D.E.'s.

**618. Algebraic Topology, Part I. (A)** Staff. Prerequisite(s): Math 600/601 or with the permission of the instructor.

Homotopy groups, Hurewicz theorem, Whitehead theorem, spectral sequences. Classification of vector bundles and fiber bundles. Characteristic classes and obstruction theory.

**619. Algebraic Topology, Part I. (B)** Staff. Prerequisite(s): Math 618 or with the permission of the instructor.

Rational homotopy theory, cobordism, K-theory, Morse theory and the h-corbodism theorem. Surgery theory.

*Algebra *

**620. Algebraic Number Theory. (M)** Staff. Prerequisite(s): Math 602/603.

Dedekind domains, local fields, basic ramification theory, product formula, Dirichlet unit theory, finiteness of class numbers, Hensel's Lemma, quadratic and cyclotomic fields, quadratic reciprocity, abelian extensions, zeta and L-functions, functional equations, introduction to local and global class field theory. Other topics may include: Diophantine equations, continued fractions, approximation of irrational numbers by rationals, Poisson summation, Hasse principle for binary quadratic forms, modular functions and forms, theta functions.

**621. Algebraic Number Theory. (M)** Staff. Prerequisite(s): Math 620 or with the permission of the instructor.

Continuation of Math 620.

**622. Complex Algebraic Geometry. (M)** Staff. Prerequisite(s): Math 602/603 and Math 609.

Algebraic geometry over the complex numbers, using ideas from topology, complex variable theory, and differential geometry. Topics include: Complex algebraic varieties, cohomology theories, line bundles, vanishing theorems, Riemann surfaces, Abel's theorem, linear systems, complex tori and abelian varieties, Jacobian varieties, currents, algebraic surfaces, adjunction formula, rational surfaces, residues.

**623. Complex Algebraic Geometry. (M)** Staff. Prerequisite(s): Math 622 or with the permission of the instructor.

Continuation of Math 622.

**624. Algebraic Geometry. (M)** Staff. Prerequisite(s): Math 602/603.

Algebraic geometry over algebraically closed fields, using ideas from commutative algebra. Topics include: Affine and projective algebraic varieties, morphisms and rational maps, singularities and blowing up, rings of functions, algebraic curves, Riemann Roch theorem, elliptic curves, Jacobian varieties, sheaves, schemes, divisors, line bundles, cohomology of varieties, classification of surfaces.

**625. Algebraic Geometry. (M)** Staff. Prerequisite(s): Math 624 or with the permission of the instructor.

Continuation of Math 624.

**626. Commutative Algebra. (M)** Staff. Prerequisite(s): Math 602/603.

Topics in commutative algebra taken from the literature. Material will vary from year to year depending upon the instructor's interests.

**627. Commutative Algebra. (M)** Staff. Prerequisite(s): Math 602/603.

Topics in commutative algebra taken from the literature. Material will vary from year to year depending upon the instructor's interests.

**628. Homological Algebra. (M)** Staff. Prerequisite(s): Math 602/603.

Complexes and exact sequences, homology, categories, derived functors (especially Ext and Tor). Homology and cohomology arising from complexes in algebra and geometry, e.g. simplicial and singular theories, Cech cohomology, de Rham cohomology, group cohomology, Hochschild cohomology. Projective resolutions, cohomological dimension, derived categories, spectral sequences. Other topics may include: Lie algebra cohomology, Galois and etale cohomology, cyclic cohomology, l-adic cohomology. Algebraic deformation theory, quantum groups, Brauer groups, descent theory.

**629. Homological Algebra. (M)** Staff. Prerequisite(s): Math 628 or with the permission of the instructor.

Continuation of Math 628.

*Algebraic and Differential Topology *

**630. Differential Topology. (M)** Staff. Prerequisite(s): Math 600/601.

Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory. The instructor may elect to cover other topics such as Morse Theory, h-cobordism theorem, characteristic classes, cobordism theories.

**631. Differential Topology. (M)** Staff. Prerequisite(s): Math 630 or with the permission of the instructor.

Continuation of Math 630.

**632. Topological Groups. (M)** Staff. Prerequisite(s): Math 600/601 and Math 602/603.

Fundamentals of topological groups. Haar measure. Representations of compact groups. Peter-Weyl theorem. Pontrjagin duality and structure theory of locally compact abelian groups.

**633. Topological Groups. (M)** Staff. Prerequisite(s): Math 632 or with the permission of the instructor.

Continuation of Math 632.

**638. Algebraic Topology, Part II. (C)** Staff. Prerequisite(s): Math 618/619.

Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.

**639. Algebraic Topology, Part II. (C)** Staff. Prerequisite(s): Math 638 or with the permission of the instructor.

Continuation of Math 638.

*Classical Analysis *

**640. Ordinary Differential Equations. (M)** Staff. Prerequisite(s): Math 508/509.

The general existence and uniqueness theorems for systems of ordinary differential equations and the dependence of solutions on initial conditions and parameters appearing in the equation. The proofs of existence and uniqueness are related to numerical algorithms for finding approximate solutions for systems of ODE's. We consider special properties of constant coefficient and linear systems. We then present the theory of linear equations with analytic coefficients, the theories of singular points, indicial roots and asymptotic solutions. We then turn to boundary value problems for second order equations with an emphasis on the eigenfunction expansions associated with self adjoint boundary conditions and the Sturm comparison theory. The remaining time is devoted to topics; for example: Hamiltonian systems and symplectic geometry, singular boundary value problems, perturbation theory, the Lyapounov-Schmidt theory and the Poincare-Bendixson theorem, the equations of mathematical physics, the calculus of variations, symmetries of ODE's and transformation groups.

**641. Ordinary Differential Equations. (M)** Staff. Prerequisite(s): Math 640 or with the permission of the instructor.

Continuation of Math 640

**644. Partial Differential Equations. (M)** Staff. Prerequisite(s): Math 600/601, Math 608/609.

Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo-differential operators, and non-linear problems. Sobolev spaces and the theory of distributions will be developed as needed.

**645. Partial Differential Equations. (M)** Staff. Prerequisite(s): Math 600/601, Math 608/609.

Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo-differential operators, and nonlinear problems. Sobolev spaces and the theory of distributions will be developed as needed.

**646. Several Complex Variables. (M)** Staff. Prerequisite(s): Math 600/601, Math 608/609.

Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.

**647. Several Complex Variables. (M)** Staff. Prerequisite(s): Math 646 or with the permission of the instructor.

Continuation of Math 646.

*Functional Analysis *

**650. Lie Algebras. (M)** Staff. Prerequisite(s): Math 602/603.

Connections with Lie groups, universal enveloping algebras, Poincare-Birkhoff-Witt Theorem, Lie and Engels theorems, free Lie algebras, Killing form, semisimple algebras, root systems, Dynkin diagrams, classification of complex simple Lie algebras, representation theory of Lie algebras, cohomology of Lie algebras.

**651. Lie Algebras. (M)** Staff. Prerequisite(s): Math 650 or with the permission of the instructor.

Continuation of Math 650.

**652. Operator Theory. (M)** Staff.

Subject matter may include spectral theory of operators in Hilbert space, C*-algebras, von Neumann algebras.

**653. Operator Theory. (M)** Staff.

Subject matter may include spectral theory of operators in Hilbert space, C*-algebras, von Neumann algebras.

**654. Lie Groups. (M)** Staff. Prerequisite(s): Math 600/601, Math 602/603.

Connection of Lie groups with Lie algebras, Lie subgroups, exponential map. Algebraic Lie groups, compact and complex Lie groups, solvable and nilpotent groups. Other topics may include relations with symplectic geometry, the orbit method, moment map, symplectic reduction, geometric quantization, Poisson-Lie and quantum groups.

**655. Lie Groups. (M)** Staff. Prerequisite(s): Math 654 or with the permission of the instructor.

Continuation of Math 654.

**656. Representation of Continuous Groups. (M)** Staff.

Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups.

**657. (PHYS657) Representation of Continuous Groups. (M)** Staff.

Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups.

*Differential Geometry *

**660. Differential Geometry. (M)** Staff. Prerequisite(s): Math 600/601, Math 602/603.

Riemannian metrics and connections, geodesics, completeness, Hopf-Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet-Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions, the soul theorem, Gromov-Hausdorff convergence.

**661. Differential Geometry. (M)** Staff. Prerequisite(s): Math 660 or with the permission of the instructor.

Continuation of Math 660.

*Other Subjects *

**676. (CIS 610) Advanced Geometric Methods in Computer Science. (M)** Staff. Prerequisite(s): Math 312 or Math 412, or with the permission of the instructor.

Advanced geometric methods used in geometric modeling,computer graphics, computer vision, and robotics.

**678. (MATH440, MATH441) Combinatorial Analysis and Graph Theory. (M)** Staff.

Generating functions, enumeration methods, Polya's theorem, combinatorial designs, discrete probability, extremal graphs, graph algorithms and spectral graph theory, combinatorial and computational geometry.

**680. Applied Linear Analysis. (M)** Staff. Prerequisite(s): Math 241 and one semester of: Math 360/361 or Math 508/509.

Application of techniques from linear algebra to real problems in economics, engineering, physics, etc. and the difficulties involved in their implementation. Particular emphasis is placed on solving equations, the eigenvalue problem for symmetric matrices and the metric geometry of spaces of matrices. Applications to problems such as options pricing, image reconstruction, airplane and ship design, oil prospecting, etc. (these topics will vary from year to year). Analysis of the numerical algorithms available to solve such problems, rates of convergence, accuracy and stability.

**681. Applied Linear Analysis. (M)** Staff. Prerequisite(s): Math 680 or with the permission of the instructor.

Continuation of Math 680.

**692. Numerical Analysis. (M)** Staff. Prerequisite(s): Math 320/321.

A study of numerical methods for matrix problems, ordinary and partial differential equations, quadrature and the solution of algebraic or transcendental equations. Emphasis will be on the analysis of those methods which are particularly suited to automatic high-speed computation.

**693. Numerical Analysis. (M)** Staff. Prerequisite(s): Math 692 or with the permission of the instructor.

Continuation of Math 692.

**694. (MATH724, PHYS654) Mathematical Foundations of Theoretical Physics. (M)** Staff.

Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.

**L/R ****695. (PHYS655) Mathematical Foundations of Theoretical Physics. (M)** Staff.

Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.

**696. (PHYS656) Topics in Mathematical Physics and String Theory. (M)** Staff. Prerequisite(s): Math 694 or permission of the instructor.

This interdisciplinary course discusses advanced topics in mathematical physics. Topics may include elliptic operators, heat kernels, complexes and the Atiyah-Singer index theorem, Feynman graphs and anomalies, computing Abelian and non-Abelian anomalies, and the relation of anomalies to the index theorem.

**697. (PHYS657) Topics in Mathematical Physics and String Theory. (M)** Staff. Prerequisite(s): Math 696 or permission of the instructor.

Continuation of Math 696. Topics may include the family index theorem, equivariant cohomology and loop spaces, the homological algebra of BRST invariance and the Wess-Zumino consistency condition, the descent equations, and worldsheet anomalies in string theory.

**MECHANICAL ENGINEERING AND APPLIED MECHANICS (EG) {MEAM} **

**513. (ESE 406, ESE 505) Modern Feedback Control Theory. (M)** Prerequisite(s): ESE 210, Junior Standing. Sophomores required permission.

Basic methods for analysis and design of feedback control in systems. Applications to practical systems. Methods presented include time response analysis, frequency response analysis, root locus, Nyquist and Bode plots, and the state-space approach.

**L/R ****527. (MEAM427) Finite Element Analysis. (M)** Prerequisite(s): MATH 241 and PHYS 151.

Today's robots replace, assist, or entertain humans in many tasks. Recent examples of robots are planetary rovers, robot pets, medical surgical assistive devices, and semi-autonomous ground vehicles for search and rescue operations. The goal of this class is to introduce the students to the common kinematic, dynamic, and computational principles and practical examples that are representative of today's robotic systems. The three main topics are coordinate system transformations and kinematics, control of mobile robots, and motion planning of robotic systems. The laboratory component includes simulation exercises, programming and

Numerical modeling using effective algorithms with applications to problems in engineering, science, and mathematics. Interpolation and curve fitting, numerical integration. Introduction to the solution of ordinary and partial (the course emphasis) differential equations by finite difference, and, with more emphasis, finite element methods including element formulation and assembly. If time permits: optimizaiton: Monte Carlo simulation.

**528. Advanced Kinematics. (M)** Prerequisite(s): Multivariate calculus, introductory abstract algebra, mathematical maturity.

Differential geometry, Lie groups and rigid body kinematics, Lie algebra, quaternions and dual number algebra, geometry of curves and ruled surfaces, trajectory generation and motion planning, applications to robotics and spatial mechanisms.

**530. (MEAM630) Continuum Mechanics. (A)** Prerequisite(s): Multivariable Calculus, Linear Algebra, Partial Differential Equations.

This course serves as a basic introduction to the Mechanics of continuous media, and it will prepare the student for more advanced courses in solid and fluid mechanics. The topics to be covered include: Tensor algebra and calculus, Lagrangian and Eulerian kinematics, Cauchy and Piola-Kirchhoff stresses, General principles: conservation of mass, conservation of linear and angular momentum, energy and the first law of thermodynamics, entropy and the second law of thermodynamics; constitutive theory, ideal fluids, Newtonian and non-Newtonian fluids, finite elasticity, linear elasticity, materials with microstructure.

**533. (MEAM433) Advanced Heat and Mass Transfer. (M)** Prerequisite(s): MEAM 302 and MEAM 333 or equivalent.

This course follows a first general course in heat transfer, to give further understanding of the basic mechanisms, the kinds of transport processes and of engineering applications, design and methodology. More generalized formulations, treatment, and results for conductive, convective, radiative and combined transport will be given. Extensive use of computers for numerical solutions of complex problems and computer-aided education. Several specific design applications will be considered in detail, such as the following: heat exchangers, thermal stressing, solar collectors, electronic equipment cooling, cooling towers, environmental discharges, engine cooling and structure icing.

**535. Advanced Dynamics. (A)**

Rigid body kinematics; Newtonian formulations of laws of motion; concepts of momentum, energy and inertia properties; generalized coordinates, holonomic and nonholonomic constraints. Generalized forces, principle of virtual work, D'Alembert's principle. Lagrange's equations of motion and Hamilton's equations. Conservation laws and integrals of motion. Friction, impulsive forces and impact. Applications to systems of rigid bodies.

**536. (MEAM436) Viscous Fluid Flow. (M)** Prerequisite(s): MEAM 302. This course may be taken by M.S.E. students for credit. M.S.E. students will be required to do some extra work, they will be graded on a different grade scale than undergraduate students, and they will be required to demonstrate a higher level of maturity in their class assignments. MEAM doctoral candidates will not be permitted to count this course as a part of their degree requirements.

Review of the fundamental laws of fluid mechanics. Analysis and discussion of the theory of incompressible viscous flow. Dimensional reasoning, similarity, Stokes approximations, laminar boundary layer theory, methods for non-similar boundary layers, approximate techniques, stability and turbulence.

**L/R ****561. Thermodynamics I. (A)** Prerequisite(s): Undergraduate thermodynamics.

Advanced classical equilibrium thermodynamics and fundamentals of nonequilibrium thermodynamics. Exergy analysis.

**L/R ****570. (CBE 640) Transport Processes I. (A)** Diamond, Sinno.

The course provides a unified introduction to momentum, energy (heat), and mass transport processes. The basic mechanisms and the constitutive laws for the various transport processes will be delineated, and the conservation equations will be derived and applied to internal and external flows featuring a few examples from mechanical, chemical, and biological systems. Reactive flows will also be considered.

**571. Advanced Topics in Transport Phenomena. (C)** Prerequisite(s): Either MEAM 570, MEAM 642, CHE 640 or equivalent, or Written permission of the Instructor.

The course deals with advanced topics in transport phenomena and is suitable for graduate students in mechanical, chemical and bioengineering who plan to pursue research in areas related to transport phenomena or work in an industrial setting that deals with transport issues. Topics include: Multi-component transport processes; Electrokinetic phenomena; Phase change at interfaces: Solidification, melting, condensation, evaporation, and combustion; Radiation heat transfer: properties of real surfaces, non-participating media, gray medium approximation, participating media transport, equation of radiative transfer, optically thin and thick limits, Monte-Carlo methods: Microscale energy transport in solids; microstructure, electrons, phonons, interactions of photons with electrons, phonons and surfaces; microscale radiation phenomena.

**613. (CBE 617, CIS 613, ESE 617) Nonlinear Control Theory. (M)** Prerequisite(s): Undergraduate Controls Course.

This course focuses on nonlinear systems, planar dynamical systems, Poincare Bendixson Theory, index theory, bifurcations, Lyapunov stability, small-gain theorems, passivity, the Poincar map, the center manifold theorem, geometric control theory, and feedback linearization.

**620. Advanced Topics in Applied Mathematics I. (M)** Prerequisite(s): Graduate standing in engineering and MEAM 535 or ESE 500 or CIS 580 or equivalent.

Geometry of rigid body displacements, coordinate systems and transformations; Kinematics of spatial mechanisms, direct and inverse kinematics for serial chain linkages, velocity and acceleration analysis; Dynamics, trajectory generation and control of manipulators; Motion planning and control of robotic systems.

**630. (MEAM530) Advanced Continuum Mechanics. (A)** Prerequisite(s): One graduate level course in applied mathematics and one in either fluid or solid mechanics.

This course is a more advanced version of MEAM 530. The topics to be covered include: tensor algebra and calculus, Lagrangian and Eulerian kinematics; Cauchy and Piola-Kirchhoff stresses. General principles: conservation of mass, conservation of linear and angular momentum, energy and the first law of yhermodynamics, entropy and the second law of thermodynamics. Constitutive theory, ideal fluids, Newtonian and non-Newtonian fluids, finite elasticity, linear elasticity, materials with microstructure.

**631. Advanced Elasticity. (M)** Prerequisite(s): MEAM 519 or permission of instructor.

Reciprocal theorem. Uniqueness. Variational theorems. Rayleigh-Ritz, Galerkin, and weighted residue methods. Three-dimensional solutions and potentials. Papkovitch-Neuber formulation. Problems of Boussinesq and Mindlin. Hertz theory of contact stress.

**642. Fluid Mechanics I. (B)**

Fluid mechanics as a vector field theory; basic conservation laws, constitutive relations, boundary conditions, Bernoulli theorems, vorticity theorems, potential flow. Viscous flow; large Reynolds number limit; boundary layers.

**643. Fluid Mechanics II. (A)**

Waves, one-dimensional gas dynamics. Transition, turbulence. Small Reynolds number limit: Stokes' flow. Compressible potential flow. Method of characteristics. Rotating flows. Stratified flows. Jets.

**644. Fluid Mechanics III. (B)**

Theory of hydrodynamic discontinuities: contact and gas dynamic. Shock structure. Higher order boundary layer theory. Stability theory. Compressible boundary layers or introduction to kinetic theory.

**645. Fluid Mechanics IV. (A)**

Gas kinetic theory: Boltzmann equation. H-theorem, equilibrium solutions, transport coefficients. Rarified gas dynamics, methods of approximate solution to Boltzmann equation. Continuum limit: Navier-Stokes equations.

**646. Computational Mechanics. (M)** Prerequisite(s): ENM 510, ENM 511, and one graduate level introductory course in mechanics. FORTRAN or C programming experience is necessary.

The course is divided into two parts. The the course introduces general numerical techniques for elliptical partial differential equations - finite difference method, finite element method and spectral method. The second part of the course concentrates on the finite volume method. SIMPLER formulation for the Navier-Stokes equations will be fully described in the class. Students will be given chances to modify a program specially written for this course to solve some practical problems in heat transfer and fluid flows.

**647. Non-Newtonian Fluid Dynamics. (M)** Prerequisite(s): ENM 510 and MEAM 642 or 530.

This in an introductory course in rheology - study of flow and deformation of matter. The course will describe the rheological behavior of polymers, low-molecular weight synthetic fluids and particulate suspensions. The course will concentrate on continuum modellng of mechanical behavior of polymeric fluids under different flow conditions. The material covered in the course will be of interest to students in mechanical engineering, chemical engineering, materials science and bioengineering.

**661. Advanced Thermodynamics Seminar. (M)** Upon demand.

Classical statistical mechanics as developed by Gibbs and Boltzmann. The H-theoremand approach to equilibrium. Fluctuations, application to ideal and real gases and to chemical equilibrium, quantized systems, theory of specific heats, Maxwell Boltzmann, Bose-Einstein and Fermi-Dirac Statistics, mean-free path phenomena diffusion, the Boltzmann equation and transport phenomena.

**L/R ****662. (BE 662, CBE 618) Advanced Molecular Thermodynamics. (A)**

Review of classical thermodynamics. Phase and chemical equilibrium for multicomponent systems. Prediction of thermodynamic functions from molecular properties. Concepts in applied statistical mechanics. Modern theories of liquid mixtures.

**664. Heat Conduction and Mass Diffusion. (A)** Prerequisite(s): ENM 510 or equivalent, and undergraduate level heat and/or mass transfer.

Advanced modeling and solutions of heat conduction and mass diffusion, with emphasis on the similarities and analogies between these phenomena. Analytical and numerical solutions, including the use of available general and specific software for the solution of the associated differential equations. Inverse problem solution techniques. Applications including basic and combined problems as well as moving interfaces, effects of energy sources and chemical reactions, interfacial contact resistance, advanced insulation, thermal stresses, composite materials, and microscale and non-continuum systems.

**665. Heat Transfer II: Convection. (B)** Prerequisite(s): Undergraduate level heat transfer and MEAM 642 or permission of instructor.

Development of formulations governing forced, buoyancy induced, and phase change transport and convective motions with emphasis on the underlying conservation principles. Following the delineation of the different kinds of transport, the principal models, and methods applicable for each kind are discussed.

**666. Heat Transfer III: Radiation. (M)** Prerequisite(s): MEAM 664 and 665.

Introduction, black body radiation, radiation to and from a surface element, radiative heat exchange among surfaces separated by a non-participating medium, radiation and conduction in non-participating media, radiation and convection in non-participating media, introduction to radiative heat transfer in participating media.

**PHYSICS (AS) {PHYS} **

**500. (MATH594) Mathematical Methods of Physics. (C)**

A discussion of those concepts and techniques of classical analysis employed in physical theories. Topics include complex analysis. Fourier series and transforms, ordinary and partial equations, Hilbert spaces, among others.

**514. Mechanics, Fluids, Chaos. (B)**

A general introduction to linear and nonlinear dynamical systems with an emphasis on astrophysical systems. Lagrangian and Hamiltonian formulations. Celestial mechanics. Equilibria and stability. Orbits. Resonances. Galactic dynamics. Intended for graduate students and advanced undergraduates.

**516. Electromagnetic Theory II. (B)**

Electrostatics and magnetostatics, Maxwell's equations, electromagnetic waves, interactions of radiation with matter, etc.

**518. Introduction to Condensed Matter Physics. (B)** Prerequisite(s): Undergraduate training in quantum mechanics and statistical thermodynamics.

An introduction to condensed matter physics designed primarily for advanced undergraduate and graduate students desiring a compact survey of the field. Band theory of solids, phonons, electrical magnetic and optical properties of matter, and superconductivity.

**531. Quantum Mechanics I. (A)** Prerequisite(s): A minimum of one semester of quantum mechanics at the advanced undergraduate level.

Wave mechanics, complementarity and correspondence principles, semi-classical (WKB) approximation, bound state techniques, periodic potentials, angular momentum, scattering theory, phase shift analysis, and resonance phenomena.

**532. Quantum Mechanics II. (B)** Prerequisite(s): PHYS 531.

Spin and other two dimensional systems, matrix mechanics, rotation group, symmetries, time independent and time dependent perturbation theory, and atomic and molecular systems.

**580. (BCHE580) Biological Physics. (C)** Goulian. Prerequisite(s): PHYS 401 or CHEM 221-222 (may be taken concurrently) or familiarity with basic statistical mechanics and thermodynamics. Recommended: Basic background in chemistry and biology.

A survey of basic biological processes at all levels of organization (molecule, cell, organism, population) in the light of simple ideas from physics. Both the most ancient and the most modern physics ideas can help explain emergent aspects of life, i.e., those which are largely independent of specific details and cut across many different classes of organisms. Topics may include thermal physics, entropic forces, free energy transduction, structure of biopolymers, molecular motors, cell signaling and biochemical circuits, nerve impulses and neural computing, populations and evolution, and the origins of life on Earth and elsewhere.

**581. (PHYS401) Thermodynamics. (A)**

**601. Introduction to Field Theory. (A)**

Elementary relativistic quantum field theory of scalar, fermion, and Abelian gauge fields. Feynman Diagrams.

**611. Statistical Mechanics. (A)** Prerequisite(s): PHYS 401, 531, or equivalent.

Introduction to the canonical structure and formulation of modern statistical mechanics. The thermodynamic limit. Entropic and depletion forces. Gas and liquid theory. Phase transitions and critical phenomena. The virial expansion. Quantum statistics. Path integrals, the Fokker-Planck equation and stochastic processes.

**632. Relativistic Quantum Field Theory. (M)** Prerequisite(s): PHYS 601.

Advanced topics in field theory, including renormalization theory.

**633. Relativistic Quantum Field Theory. (M)** Prerequisite(s): PHYS 632.

A continuation of PHYS 632, dealing with non-Abelian gauge theories.

**STATISTICS (WH) {STAT} **

**510. (BSTA620, STAT430) Probability. (A)** Small. Prerequisite(s): A one year course in calculus.

Probability. Elements of matrix algebra. Discrete and continuous random variables and their distributions. Moments and moment generating functions. Joint distributions. Functions and transformations of random variables. Law of large numbers and the central limit theorem. Point estimation: sufficiency, maximum likelihood, minimum variance. Confidence intervals.

**511. Statistics. (B)** Ewens. Prerequisite(s): STAT 510.

Tests of hypotheses. Examples of normal means and variances. Neyman-Pearson lemma. Generalized likelihood ratio tests. Ordinary least squares estimation. Inference in linear models: hypothesis tests and confidence statements. Bivariate normal distribution and correlation. Analysis of variance for one- and two-way layouts. Categorical data. Generalized least squares and autocorrelated disturbances. Lagged-variable models. Simultaneous equations models and introductory topics in econometrics.

**512. (BSTA621, STAT432) Mathematical Statistics. (B)** Staff. Prerequisite(s): STAT 430 or 510 or equivalent.

An introductory course in the mathematical theory of statistics. Topics include estimation, confidence intervals, hypothesis testing, decision theory models for discrete data, and nonparametric statistics.

**530. (MATH546) Probability. (A)** Pemantle. Prerequisite(s): STAT 430 or 510 or equivalent.

Measure theory and foundations of Probability theory. Zero-one Laws. Probability inequalities. Weak and strong laws of large numbers. Central limit theorems and the use of characteristic functions. Rates of convergence. Introduction to Martingales and random walk.

**531. (MATH547) Stochastic Processes. (B)** Pemantle. Prerequisite(s): STAT 530.

Markov chains, Markov processes, and their limit theory. Renewal theory. Martingales and optimal stopping. Stable laws and processes with independent increments. Brownian motion and the theory of weak convergence. Point processes.

**540. Statistical Methods and Computation. (A)** Buja. Prerequisite(s): Permission of instructor.

Introduction to the computational aspects of statistical methodology, with emphasis on applications development. Assignments include algorithm analysis and program development.

**541. Statistical Methods. (B)** Buja. Prerequisite(s): STAT 431 or 511 or equivalent.

Multiple linear regression, logit and probit regression, analysis of variance, experimental design, log-linear models, goodness-of-fit.

**550. (BSTA622) Mathematical Statistics. (A)** Small. Prerequisite(s): STAT 431 or 511 or equivalent.

Decision theory and statistical optimality criteria, sufficiency, invariance, estimation and hypothesis testing theory, large sample theory, information theory.

**551. Introduction to Linear Statistical Models. (B)** Morrison. Prerequisite(s): STAT 550.

Properties of the multivariate and spherical normal distributions, quadratic forms, estimation and testing in the linear model with applications to analysis of variance and regression models, generalized inverses, and simultaneous inference.

**552. (BSTA820) Advanced Topics in Mathematical Statistics. (A)** Staff. Prerequisite(s): STAT 550 and 551.

A continuation of STAT 550.

**900. Advanced Probability. (M)** Staff. Prerequisite(s): STAT 531 or equivalent.

The topics covered will change from year to year. Typical topics include the theory of large deviations, percolation theory, particle systems, and probabilistic learning theory.

**901. (OPIM931) Stochastic Processes II. (M)** Staff. Prerequisite(s): OPIM 930 or equivalent.

Martingales, optimal stopping, Wald's lemma, age-dependent branching processes, stochastic integration, Ito's lemma.

**910. Forecasting and Time Series Analysis. (I)** Staff. Prerequisite(s): STAT 511 or 541 or equivalent.

Fourier analysis of data, stationary time series, properties of autoregressive moving average models and estimation of their parameters, spectral analysis, forecasting. Discussion of applications to problems in economics, engineering, physical science, and life science.

**915. Nonparametric Inference. (M)** Staff. Prerequisite(s): STAT 511 or equivalent.

Statistical inference when the functional form of the distribution is not specified. Nonparametric function estimation, density estimation, survival analysis, contingency tables, association, and efficiency.

**927. Bayesian Statistical Theory and Methods. (M)** Zhao. Prerequisite(s): STAT 551.

A course in Bayesian statistical theory and methods. Axiomatic developments of utility theory and subjective probability, and elements of Bayesian theory.

**932. (BSTA653) Survival Models and Analysis Methods for Medical and Biological Data. (M)** Zhao. Prerequisite(s): STAT 551.

Parametric models, nonparametric methods for one-and two-sample problems, proportional hazards model, inference based on ranks. Problems will be considered from clinical trials, toxicology and tumorigenicity studies, and epidemiological studies.

**933. Analysis of Categorical Data. (M)** Rosenbaum. Prerequisite(s): STAT 541 and 551.

Likelihood equations for log-linear models, properties of maximum likelihood estimates, exact and approximate conditional inference, computing algorithms, weighted least squares methods, and conditional independence and log-linear models. Applied topics, including interpretation of log-linear and logit model parameters, smoothing of tables, goodness-of-fit, and incomplete contingency tables.

**940. Advanced Inference I. (M)** Staff. Prerequisite(s): STAT 551.

The topics covered will change from year to year. Typical topics include sequential analysis, nonparametric function estimation, robustness, bootstrapping and applications decision theory, likelihood methods, and mixture models.

**941. Advanced Inference II. (M)** Staff. Prerequisite(s): STAT 940.

A continuation of STAT 940.

**TELECOMMUNICATIONS (EG) {TCOM}**

**500. (TCOM400) Introduction to Networks and Protocols. (A)** Prerequisite(s): Undergraduate probability and analysis. Course open to Seniors and Graduate Students in SEAS and Wharton. All others need permission of the instructor.

This is an introductory course on packet networks and associated protocols that form the basis of today's communication infrastructure, with a particular emphasis on IP based networks such as the Internet. The course introduces the various design and implementation choices that are behind the development of modern networks, and emphasizes basic analytical understanding in motivating those choices. Topics are covered in a mostly "bottom-up" approach, starting with a brief review of physical layer issues such as digital transmission, error correction and error recovery strategies. This is then followed by a discussion of link layer aspects, including multiple access control (MAC) strategies, local area networks (Ethernet, token rings, and 802.11 wireless LANs), and general store-and-forward packet switching. Network layer solutions, including IP addressing, naming, and routing are covered next, before exploring transport layer and congestion control solutions such as TCP. Finally, basic approaches for quality-of-service and network security are examined. Specific applications and aspects of data compression and streaming may also be covered.

**501. Networking - Theory and Fundamentals. (B)** Prerequisite(s): ESE 530 or STAT 530 or equivalent.

Stochastic processes are introduced as drivers of queues to provide an analytical platform for the analysis of delays in networks models. Topics covered are selected from: Delay models in the network layer; the Poisson process; renewal processes, rewards, and the renewal theorem; Little's law; Markov chains; semi-Markov processes; Markov processes; ergodicity, limit laws and stationary distributions; M/M/1, M/M/m, M/M/m/m queues; alternating renewal processes and fluid flow models; M/G/1, G/M/1, G/G/1 queues; the Pollaczek-Khinchin formulae; priority classes; time-reversibility; networks of queues; Jackson networks.

**502. Advanced Networking Protocols. (B)** Prerequisite(s): TCOM 500 or equivalent.

The course delves into the details of the many protocols whose combined operation is behind modern data networks. It starts with reviewing issues associated with naming and addressing, and in particular solutions that work at the Internet scale. This is followed by an in-depth review of the Internet's "control plane," namely the different routing protocols that govern packet forwarding decisions, including unicast (RIP2, EIGRP, OSPF, BGP, etc.) and multicast (DVMRP, CBT, PIM, etc.) routing protocols. The challenges associated with implementing efficient packet forwarding decisions are then discussed and illustrated through several representative techniques and algorithms. Next, the course introduces technologies that implement advanced functionalities over IP networks, including signaling protocols, e.g., RSVP, used to request service guarantees from the network, and protocols such as MPLS and MP-BGP that enable the efficient deployment of virtual private networks and traffic engineering solutions. If time permits, topics related to service classes and traffic management, as well as interactions between IP and other networking technologies, e.g., ATM, may also be covered.

**503. (ESE 509) Waves, Fibers and Antennas for Telecommunications. (A)** Faculty.

This course is designed to provide an understanding of the physcial aspects of telecommunications systems. This includes an understanding of waves and wave propagation, basic optics, the operation of optical fibers and fiber communication systems, an introduction to optical networks, free-space optical communications, and an understanding of simple antennas and arrays and their use in wireless communications.

**601. Advanced Networking Modeling and Analysis. (C)**

Traffic management and call admission: traffic characterization traffic shaping, admission control, statistical multiplexing, effective bandwidth. Scheduling: fair queuing, rate-controlled service disciplines. Buffer management: pushout, threshold, random early detection, sharing mechanisms (complete partitioning, complete sharing, hybrids), coupling buffer management and scheduling. Markov decision process and application in resource allocation (memory, badwidth allocation). Switching: input queuing, output queuing, shared memory, combined input/output queuing. Maximum throughout in input queued switches, emulating output queuing with input queuing via speedup. Building larger switches: CIOS networks, banyan netowrks, etc. TCP modeling.