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Current MSCS Graduate Courses ( Fall 2009, Spring 2010 )

Fall 2009

MATH 502 Metamathematics I (D. Marker)
PREREQUISITES:
Graduate standing or consent of instructor.
DESCRIPTION:
A first graduate course in mathematical logic. We will introduce the fundamental themes of mathematical logic (truth, provability, and computability), discuss their interconnections and examine the power and limits of formal methods. Topics covered will include: Godel's Completeness Theorem, The Compactness Theorem and elementary model theory, Model theory of algebraically closed fields models of computation, Church's Thesis, undecidability, Godel's Incompleteness Theorem.
MATH 507 Model Theory II (A. Medvedev)
PREREQUISITES:
Math 506 or Phil 567.
DESCRIPTION:
Intermediate stability theory, dependence, prime models, isolation, regular types, dimension, weight.
MATH 514 Number Theory I (R. Takloo-Bighash)
PREREQUISITES:
DESCRIPTION:
Introduction to classical, algebraic, and analytic number theory.
MATH 516 Abstract Algebra I (O. Kashcheyeva)
PREREQUISITES:
MATH 330 and 425.
DESCRIPTION:
Structure of groups, Sylow theorems, solvable groups, structure of rings, polynomial rings, projective and injective modules, finitely generated modules over a PID.
MATH 533 Real Analysis I (A. Cheskidov)
PREREQUISITES:
MATH 411 or 414 or the equivalent.
DESCRIPTION:
Introduction to real analysis. Lebesgue measure and integration, differentiation, L-p classes, abstract integration.
MATH 536 Complex Analysis II (P. Shalen)
PREREQUISITES:
MATH 535.
DESCRIPTION:
Normal families, Riemann mapping theorem. Analytic continuation, harmonic and subharmonic functions, Picard theorem, selected topics.
MATH 549 Differentiable Manifolds I (S. Wenger)
PREREQUISITES:
Math 445.
DESCRIPTION:
Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes' theorem, degree theory, vector fields.
MATH 552 Algebraic Geometry I (I. Coskun)
PREREQUISITES:
DESCRIPTION:
Basic commutative algebra, affine and projective varieties, regular and rational maps, function fields, dimension and smoothness, projective curves, schemes, sheaves, and cohomology, positive characteristic.
MATH 569 Advanced Topics in Geometric and Differential Topology (J. Deblois)
PREREQUISITES:
Algebraic topology at the level of fundamental groups and covering spaces. A glancing familiarity with the concepts of Riemannian geometry will be helpful for motivation.
DESCRIPTION:
The main topics will be geometric properties of metric spaces (for example, notions of non-positive curvature), consequences (such as the Cartan-Hadamard theorem), and implications for the structure of groups which act nicely on them. These will be addressed in the setting of cube complexes, which provide both a rich source of examples and a tangible platform for understanding the properties above. Examples of groups acting on cube complexes include (but are not limited to) free groups, free products with amalgamation, Coxeter groups, and some hyperbolic manifold groups.
MATH 574 Applied Optimal Control (S. Yau)
PREREQUISITES:
Math 411 or 427 or consent of instructor.
DESCRIPTION:
Introduction to optimal control theory; calculus of variations, maximum principle, dynamic programming, feedback control, linear systems with quadratic criteria, singular control, optimal filtering, stochastic control. Book: Linear System Theory by Wilson Rugh, Prentice Hall Information and System Science Series.
MATH 576 Boundary Value Problems (I. Nenciu)
PREREQUISITES:
Math 320 and 417 and 481; or consent of instructor.
DESCRIPTION:
Distributions, Green's functions, alternative theorem, regular and singular Sturm-Liouville problems, spectral theory, potential theory, method of images, complex variable methods, equations of evolution.
MATH 578 Asymptotic Methods (C. Knessl)
PREREQUISITES:
MATH 417 and 481, or consent of instructor.
DESCRIPTION:
Asymptotic series, Laplace's method, stationary phase, steepest descent method, Stokes phenomena, uniform expansions, multi-dimensional Laplace integrals, Euler-MacLaurin formula, irregular singular points, WKBJ method.
MATH 581 Special Topics in Fluid Mechanics (R. Abramov)
PREREQUISITES:
Grade of C or better in MATH 580
DESCRIPTION:
Geophysical flows for the atmosphere and ocean, barotropic flow, conserved quantities, chaotic dynamics, empirical statistical theories for geophysical flows, equilibrium statistical mechanics, mean response to small fluctuations.
MATH 589 Teaching and Presentation of Mathematics (S. Hurder)
PREREQUISITES:
None.
DESCRIPTION:
No graduation credit awarded for students enrolled in the Master of Science in the Teaching of Mathematics degree program. Required for teaching assistants in MSCS. Strategies and techniques for effective teaching in college and for mathematical consulting. Observation and evaluation, classroom management, presenting mathematics in multidisciplinary research teams.
MATH 590 Advanced Topics in Applied Mathematics (S. Petrovic)
PREREQUISITES:
Some algebraic (or algebraic geometry) knowledge with interest in statistics and applications; OR some statistical background with interest in algebraic techniques; OR interest in applications of algebraic techniques to statistics and mathematical biology; and mathematical maturity and consent of instructor.
DESCRIPTION:
Algebraic statistics is an emerging field, aimed at solving statistical inference problems using concepts from algebraic geometry and commutative algebra, as well as related computational and combinatorial techniques. It is centered around the observation that many important statistical models correspond to algebraic or semi-algebraic sets of parameters. Computational algebraic geometry can be used to study parameter spaces and other features of statistical models. This course provides an introduction to the algebraic techniques that have emerged as useful tools in statistics and statistical models in biology. Topics will include: introduction to polynomial rings, ideals, and Groebner bases; a survey of applications: graphical and hierarchical models, Markov bases for log-linear models, Markov bases for contingency table analysis, phylogenetic models and the space of trees, and other topics that are emerging this year; basics on current open problems in the area; and crash-courses on the use of some software packages including their limitations and possibilities for improvement.

Spring 2010

MATH 503 Metamathematics II (C. Rosendal)
PREREQUISITES:
MATH 502 and some familiarity with mathematical logic, automata or Turing machines is useful, but not essential. We shall cover all needed material from the basis and up.
DESCRIPTION:
Basic theory of recursive functions and recursively enumerable sets. The second part of the course will cover a more specialized topic of modern computability theory such as Kolmogorov complexity or Turing degrees. The course will be of broad interests to students in logic, discrete mathematics and theoretical computer science.
MATH 517 Abstract Algebra II (L. Ein)
PREREQUISITES:
MATH 516.
DESCRIPTION:
Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems.
MATH 535 Complex Analysis I ()
PREREQUISITES:
MATH 411 or 427.
DESCRIPTION:
Analytic functions as mappings. Cauchy theory. Power series. Partial fractions. Infinite products.
MATH 546 Advanced Topics in Analysis ()
PREREQUISITES:
Approval of department.
DESCRIPTION:
MATH 547 Algebraic Topology I (D. Groves)
PREREQUISITES:
MATH 330 and 445.
DESCRIPTION:
The fundamental group, cell spaces, introduction to homology, development of singular homology theory, applications, cohomology, and selected topics.
MATH 550 Differentiable Manifolds II (K. Whyte)
PREREQUISITES:
Math 549.
DESCRIPTION:
Vector bundles and classifying spaces, Lie groups and Lie algebras, tensors, Hodge theory, Poincare duality. Topics from elliptic operators, Morse theory, cobordism theory, de Rahm theory, characteristic classes.
MATH 553 Algebraic Geometry II (D. Chen)
PREREQUISITES:
Math 552.
DESCRIPTION:
Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces.
MATH 554 Complex Manifolds I (C. Schnell)
PREREQUISITES:
MATH 517 and 535.
DESCRIPTION:
Holomorphic functions in several variables, Riemann surfaces, Sheaf theory, vector bundles, Stein manifolds, Cartan theorem A and B, Grauert direct image theorem.
MATH 571 Topics in Algebraic Geometry (A. Marian)
PREREQUISITES:
Approval of the department.
DESCRIPTION:
Hilbert Schemes.
MATH 577 Advanced Applied Partial Differential Equations (R. Shvydkoy)
PREREQUISITES:
Math 410 and 417 and 481.
DESCRIPTION:
Quasilinear and nonlinear first order PDE's, shock solutions, second order equations, cylinder and sphere problems, Wave, Laplace and diffusion equations, maximum principles, nonlinear wave motion.
MATH 582 Wave Propagation and Scattering I (B. Akers)
PREREQUISITES:
Math 417 and 481; or consent of the instructor.
DESCRIPTION:
Solutions of wave equations in multiple dimensions, vector, and dyadic waves; separable and nonseparable problems. Representations: Green's function integrals, complex integrals, spectral representations. Approximate solutions.
MATH 586 Computational Finance (D. Nicholls)
PREREQUISITES:
Grade of C or better in MATH 220 and Stat 381; or consent of instructor.
DESCRIPTION:
Introduction to the mathematics of financial derivatives; options, asset price random walks, Black-Scholes model; partial differential techniques for option valuation, binomial models, numerical methods; exotic options, interest-rate derivatives.
MATH 590 Advanced Topics in Applied Mathematics (S. Yau)
PREREQUISITES:
Consent of instructor.
DESCRIPTION:
This is a topic course in Bioinformatics. We do not assume any biology background. We will discuss how to use various mathematical methods to understand DNA sequences and Protein sequences (Amino Acid sequences). In particular, we shall discuss the method to predict coding region of protein in DNA sequence. We shall also discuss a new method to predict the property of newly found protein. The same method will allow us to construct moduli space of genomes which will provide us the phylogenetic relation of all genomes.