# Graduate Courses

MSCS 500-Level Graduate Courses

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

## Fall 2022 Graduate Courses Heading link

Course | Description | Time and Location | Instructor | CRN |
---|---|---|---|---|

Math 502: Mathematical Logic | First order logic, completeness and incompleteness theorems, introduction to model theory and computability theory. Course Information: Same as PHIL 562. Prerequisite(s): MATH 430 or consent of the instructor. | MWF 1:00 - 1:50 | Ronnie (Joel) Najloo | 38305 |

MATH 514: Number Theory I | Introduction to classical, algebraic, and analytic, number theory. Euclid's algorithm, unique factorization, quadratic reciprocity, and Gauss sums, quadratic forms, real approximations, arithmetic functions, Diophantine equations. | MWF 11:00 - 11:50 | Alina Cojocaru | 42674 |

MATH 516: Second Course in Abstract Algebra I | Structure of groups, Sylow theorems, solvable groups; structure of rings, polynomial rings, projective and injective modules, finitely generated modules over a PID. Course Information: Prerequisite(s): MATH 330 and MATH 425. | MWF 10:00 - 10:50 | Kevin Tucker | 13724 |

Math 520: Commutative and Homological Algebra | Commutative rings; primary decomposition; integral closure; valuations; dimension theory; regular sequences; projective and injective dimension; chain complexes and homology; Ext and Tor; Koszul complex; homological study of regular rings. Course Information: Prerequisite(s): MATH 516 and MATH 517; or consent of the instructor. | MWF 1:00 - 1:50 | Wenliang Zhang | 39213 |

Math 525: Advanced Topics in Number Theory | Introduction to topics at the forefront of research in number theory. Topics will vary and may include elliptic curves, automorphic forms, diophantine geometry or sieve methods. Course Information: May be repeated. Prerequisite(s): MATH 515; or consent of the instructor. | MWF 2:00 - 2:50 | Nathan Jones | 43414 |

MATH 533: Real Analysis I | Introduction to real analysis. Lebesgue measure and integration, differ entiation, L-p classes, abstract integration. Course Information: Prerequisite(s): MATH 411 or MATH 414 or the equivalent. | MWF 1:00 - 1:50 | Alexander Furman | 42669 |

MATH 539: Functional Analysis I | Topological vector spaces, Hilbert spaces, Hahn-Banach theorem, open mapping, uniform boundedness principle, linear operators in a Banach space, compact operators. Course Information: Prerequisite(s): MATH 533. | MWF 12:00 - 12:50 | Roman Shvydkoy | 37062 |

MATH 549: Differentiable Manifolds I | Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes's theorem, degree theory, vector fields. Course Information: Prerequisite(s): MATH 445; and MATH 310 or MATH 320 or the equivalent. | MWF 2:00 - 2:50 | David Dumas | 40616 |

MATH 552: Algebraic Geometry I | Basic commutative algebra, affine and projective varieties, regular and rational maps, function fields, dimension and smoothness, projective curves, schemes, sheaves, and cohomology, posiive characteristic. | MWF 12:00 - 12:50 | Geoffrey Smith | 31610 |

MATH 554: Complex Manifolds I | Holomorphic functions in several variables, Riemann surfaces, Sheaf theory, vector bundles, Stein manifolds, Cartan theorem A and B, Grauert direct image theorem. Course Information: Prerequisite(s): MATH 517 and MATH 535. | MWF 10:00 - 10:50 | Izzet Coskun | 41072 |

MATH 585: Ordinary Differential Equations | Introduction to ordinary differential equations, existence, uniqueness of solutions, dependence on parameters, autonomous and non-autonomous systems, linear systems, nonlinear systems, periodic solutions, bifurcations, conservative systems. Course Information: Prerequisite(s): MATH 313 or MATH 480 or approval of the department. | MWF 10:00 - 10:50 | Rafail Abramov | 37428 |

MATH 589: Teaching and Presentation of Mathematics | Strategies and techniques for effective teaching in college and for mathematical consulting. Observation and evaluation, classroom management, presenting mathematics in multidisciplinary research teams. Required for teaching assistants in MSCS. Course Information: No graduation credit awarded for students enrolled in the Master of Science in the Teaching of Mathematics degree program. | MW | TBD | |

MCS 549: Mathematical Foundations of Data Science | Topics will include random graphs, small world phenomena, random walks, Markov chains, streaming algorithms, clustering, graphical models, singular value decomposition, and random projections. Course Information: Prerequisite(s): MCS 401 and MCS 441; or consent of the instructor. | MWF 12:00 - 12:50 | Lev Reyzin | 43424 |

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. | MWF 10:00 - 10:50 | Dhruv Mubayi | 43425 |

MCS 590: Advanced Topics in Computer Science | Topics in areas such as: mathematical aspects of artificial intelligence, symbolic methods in mathematics, mathematical cryptography, automated reasoning. Topics may vary from term to term. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. | MWF 2:00 - 2:50 | Yu Cheng | 38318 |

STAT 501: Probability Theory I | Abstract measure theory, probability measures, Kolmogorov extension theorem, sums of independent random variables, the strong and weak laws of large numbers, the central limit theorem, characteristic functions, law of iterated logarithm, infinitely divisible laws. Course Information: Prerequisite(s): MATH 534 or consent of the instructor. | MWF 1:00 - 1:50 | Cheng Ouyang | 22762 |

STAT 521: Linear Statistical Inference | Estimation and testing in linear models, generalized inverses of matrices, n-dimensional normal distribution, quadratic forms, likelihood ratio tests, best invariant tests, analysis of variance. Course Information: Prerequisite(s): STAT 411. | MWF 9:00 - 9:50 | Jing Wang | 37071 |

STAT 591: Advanced Topics in Statistics, Probability and Operations Research | Special topics. Topics drawn from areas such as: Data analysis; Bayesion inference; Nonlinear models; Time series; Computer aided design; reliability models; game theory. Course Information: May be repeated. Prerequisite(s): Approval of the department. | MWF 1:00 - 1:50 | Yichao Wu | 37945 |

## Spring 2023 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. | TBD | ||
---|---|---|---|---|

Math 506: Model Theory I | Elementary embeddings, quantifier elimination, types, saturated and prime models, indiscernibles, Morley's Categoricity Theorem. Course Information: Same as PHIL 567. Prerequisite(s): MATH 502 or PHIL 562. | TBD | ||

Math 507: Model Theory II | Stability theory: forking and indpendence, stable groups, geometric stability. Course Information: Same as PHIL 568. Prerequisite(s): MATH 506 or PHIL 567. | TBD | ||

Math 511: Descriptive Set Theory | Polish spaces and Baire category; Borel, analytic and coanalytic sets; infinite games and determinacy; coanalytic ranks and scales; dichotomy theorems. Course Information: Recommended background: MATH 445 or MATH 504 or MATH 533 or MATH 539. | TBD | ||

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. | Filippo Calderoni | ||

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 | Introduction to topics at the forefront of research in number theory. Topics will vary and may include elliptic curves, automorphic forms, diophantine geometry or sieve methods. Course Information: May be repeated. Prerequisite(s): MATH 515; or consent of the instructor. | Alina 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 | Various topics such as algebraic curves, surfaces, higher dimensional geometry, singularities theory, moduli problems, vector bundles, intersection theory, arithematical algebraic geometry, and topologies of algebraic varieties. Course Information: May be repeated. Students may register in more than one section per term. Prerequisite(s): Approval of the department. | 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. | Alexey Cheskidov | ||

Math 582: Linear and Nonlinear Waves | Analysis of partial differential equations describing (non-) linear wave phenomena. In particular, dispersive and hyperbolic equations. Analytical techniques include Fourier transformation and fixed point theorems. Course Information: Prerequisite(s): Graduate standing and MATH 533 and MATH 576 OR MATH 539 or consent of the instructor. | TBD | ||

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 582: The Probabilistic Method | Introduction to the probabilistic method, which includes a range of applications to address various problems that arise in combinatorics. Prerequisite(s): MCS 421 and 423, or consent of the instructor. | TBD | ||

MCS 591: Advanced Topics in Combinatorial Theory | Some of the following topics: combinatorial enumeration, designs, graph theory, matroid theory, combinatorial matrix theory, Ramsey theory. Contents vary from year to year. Course Information: May be repeated. Prerequisite(s): MCS 423. | 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 |