# Graduate Courses

MSCS 500-Level Graduate Courses

Below is a list of courses we expect to be offering in the semesters ahead.

## Spring 2023 Planned Graduate Courses Heading link

Math 504: Set Theory | Naive and axiomatic set theory. Independence of the continuum hypothesis and the axiom of choice. Course Information: Same as PHIL 565. Prerequisite(s): MATH 430 or MATH 502 or PHIL 562. | Tom Benhamou | ||

Math 512: Advanced Topics in Logic | Advanced topics in modern logic; e.g. large cardinals, infinitary logic, model theory of fields, o-minimality, Borel equivalence relations. Course Information: Same as PHIL 569. May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. | James Freitag | ||

Math 515: Number Theory II | Introduction to classical, algebraic, and analytic number theory. Algebraic number fields, units, ideals, and P-adic theory. Riemann Zeta-function, Dirichlet's theorem, prime number theorem. Course Information: Prerequisite(s): MATH 514. | Nathan Jones | ||

Math 517: Second Course in Abstract Algebra II | Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems. Course Information: Prerequisite(s): MATH 516. | Lawrence Ein | ||

Math 525: Advanced Topics in Number Theory | Function Field Arithmetic: We will focus on function field arithmetic, with a particular emphasis placed on the theory of Drinfeld modules as the function field counterpart to the theory of elliptic curves. The ring of polynomials in one indeterminate over a finite field exhibits strong similarities with the ring of integers. This similarity is an illustration of the broader analogy between function fields and number fields, whose exploration has had profound consequences on major branches of mathematics such as number theory, geometry, and topology. | Alina Carmen Cojocaru | ||

Math 535: Complex Analysis I | Analytic functions as mappings. Cauchy theory. Power Series. Partial fractions. Infinite products. Course Information: Prerequisite(s): MATH 411. | Jerry Bona | ||

Math 547: Algebraic Topology I | The fundamental group and its applications, covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology. Course Information: Prerequisite(s): MATH 330 and MATH 445. | Daniel Groves | ||

Math 550: Differentiable Manifolds II | Vector bundles and classifying spaces, lie groups and lie algbras, tensors, Hodge theory, Poincare duality. Topics from elliptic operators, Morse theory, cobordism theory, deRahm theory, characteristic classes. Course Information: Prerequisite(s): MATH 549. | Kevin Whyte | ||

Math 553: Algebraic Geometry II | Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces. Course Information: Prerequisite(s): MATH 552. | Kevin Tucker | ||

Math 555: Complex Manifolds II | Dolbeault Cohomology, Serre duality, Hodge theory, Kadaira vanishing and embedding theorem, Lefschitz theorem, Complex Tori, Kahler manifolds. Course Information: Prerequisite(s): MATH 517 and MATH 535. | Julius Ross | ||

Math 569: Advanced Topics in Geometric and Differential Topology | Topics from areas such as index theory, Lefschetz theory, cyclic theory, KK theory, non-commutative geometry, 3-manifold topology, hyperbolic manifolds, geometric group theory, and knot theory. Course Information: Prerequisite(s): Approval of the department. | Alexander Furman | ||

Math 571: Advanced Topics in Algebraic Geometry | Rational, Unirational and Rationally Connected Varieties. In this course, we will discuss classical and modern examples of rational and non-rational varieties. We will study spaces of rational curves on varieties with applications to unirationality and rational connectedness in mind. We will end the course by introducing recent developments due to Voisin, Colliot-Thélène, Pirutka, Schreieder and others. Prerequisites: Math 552 and Math 553. | Izzet Coskun | ||

Math 576: Classical Methods of Partial Differential Equations | First and second order equations, method of characteristics, weak solutions, distributions, wave, Laplace, Poisson, heat equations, energy methods, regularity problems, Green functions, maximum principles, Sobolev spaces, imbedding theorems. Course Information: Prerequisite(s): MATH 410 and MATH 481 and MATH 533; or consent of instructor. | Ian Tobasco | ||

Math 590: Advanced Topics in Applied Mathematics | One of the major problems in the theory of swarming is to understand formation of collective outcomes in large systems governed by local laws of interactions. This phenomenon, called emergence, occurs in a variety of applications -- flock of birds, school of fish, synchronization of UAVs, clustering in social networks, consensus of opinions. This course provides an introduction into mathematics of collective behavior and focuses on systems with alignment and other forces that model laws of self-organization. Prerequisite: Basic familiarity with ODEs and PDEs will be assumed and some functional analysis will be useful. | Roman Shvydkoy | ||

MCS 501: Computer Algorithms II | Continuation of MCS 401 (same as CS 401). Advanced topics in algorithms. Lower bounds. Union-find problems. Fast Fourier transform. Complexity of arithmetic, polynomial, and matrix calculations. Approximation algorithms. Parallel algorithms. Course Information: Same as CS 501. Prerequisite(s): MCS 401 or CS 401. | Gyorgy Turan | ||

MCS 541: Computational Complexity | Time and space complexity of computations, classification of mathproblems according to their computational complexity, P not equal NP problem. Course Information: Prerequisite(s): Consent of the instructor. | Lev Reyzin | ||

MCS 548: Mathematical Theory of Artificial Intelligence | Valiant's learning model, positive and negative results in learnability, automation inference, perceptrons, Rosenblatt's theorem, convergence theorem, threshold circuits, inductive inference of programs, grammars and automata. Course Information: Prerequisite(s): MCS 541. | Gyorgy Turan | ||

MCS 571: Numerical Analysis of Partial Differential Equations | Numerical analysis of Finite Difference methods for PDE of mathematical physics: Wave, heat, and Laplace equations. Introduction to numerical analysis of the Finite Element method. Course Information: Prerequisite(s): MATH 481 and MCS 471 or consent of the instructor. | David Nicholls | ||

MCS 572: Introduction to Supercomputing | Introduction to supercomputing on vector and parallel processors; architectural comparisons, parallel algorithms, vectorization techniques, parallelization techniques, actual implementation on real machines. Course Information: Prerequisite(s): MCS 471 or MCS 571 or consent of the instructor. | Jan Verschelde | ||

MCS 584: Enumerative Combinatorics | Enumerative methods in combinatorics, including inclusion/exclusion, recursion, partitions, Latin squares and other combinatorial structures. Prerequisite(s): MCS 421 and MCS 423, or consent of the instructor. | Marcus Michelen | ||

Stat 511: Advanced Statistical Theory I | Statistical models, criteria of optimum estimation, large sample theory, optimum tests and confidence intervals, best unbiased tests in exponential families, invariance principle, likelihood ratio tests. Course Information: Prerequisite(s): STAT 411. | Kyunghee Han | ||

Stat 522: Multivariate Statistical Analysis | Multivariate normal distribution, estimation of mean vector and covariance matrix, T-square statistic, discriminant analysis, general linear hypothesis, principal components, canonical correlations, factor analysis. Course Information: Prerequisite(s): STAT 521. | Jie Yang | ||

Stat 536: Optimal Design Theory II | Construction of optimal designs: BIB , Latin square and generalized Youden , repeated measurements , treatment-control studies; construction of factorial designs including orthogonal arrays Course Information: Prerequisite(s): STAT 535 or consent of the instructor. | Min Yang |