# Graduate Courses

MSCS 500-Level Graduate Courses

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

## Spring 2024 Planned Graduate Courses Heading link

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. | 12:00 - 12:50 | 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. | 9:00 - 9:50 | Ramin Takloo Bighash | |

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. | 1:00 - 1:50 | Izzet Coskun | |

Math 525: Advanced Topics in Number Theory | This course will be an introduction to arithmetic dynamics, a relatively new area which studies dynamical systems over number fields and other fields of arithmetic interest. We will focus especially on the interplay between the local geometry of a dynamical system and the arithmetic of its forward orbits. The class will develop the necessary complex dynamics and non-archimedean geometry in dimension 1, and will also explore connections/applications to higher-dimensional dynamical systems and abelian varieties, along with some open problems. The target audience covers a wide range of interests in the areas of number theory, algebraic geometry, and dynamics. | 11:00 - 11:50 | Nicole Looper | |

Math 535: Complex Analysis I | Analytic functions as mappings. Cauchy theory. Power Series. Partial fractions. Infinite products. Course Information: Prerequisite(s): MATH 411. | 2:00 - 2:50 | Alex Furman | |

Math 548: Algebraic Topology II | Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics. Course Information: Prerequisite(s): MATH 547. | 11:00 - 11:50 | Kevin Whyte | |

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. | 12L00 - 12:50 | Wouter Van Limbeek | |

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. | 10:00 - 10:50 | Nolan Schock | |

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. | 1:00 - 1:50 | Lawrence Ein | |

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. | 12:00 - 12:50 | Irina Nenciu | |

Math 586: Computational Finance | 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. Course Information: Prerequisite(s): Grade of C or better in MATH 220 and grade of C or better in STAT 381; or consent of the instructor. | 9:00 - 9:50 | David Nicholls | |

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. | 9:00 - 9:50 | 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. | 1:00 - 1:50 | 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. | 2:00 - 2:50 | Gerard Awanou | |

MCS 573: Topics in Numerical Analysis of Partial Differential Equations | The course will be an introduction to physics informed learning which integrates elements of machine learning into scientific computing. The focus will be on methods for the numerical solution of differential equations and the emerging paradigms for proving convergence results. | 11:00 - 11:50 | Gerard Awanou | |

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. | 12:00 - 12:50 | Dhruv Mubayi | |

MCS 591: Advanced Topics in Combinatorial Theory | We will cover tools in probability theory with an eye towards applications in combinatorial and high-dimensional problems. Classical theorems in probability theory such as laws of large numbers and limit theorems will be reviewed with the main goal being to provide students with the tools of probability theory as well as a wide number of examples that require going beyond the classic toolbox. | 10:00 - 10:50 | Marcus Michelen | |

Stat 502: Probability Theory II | Radon-Nikodym theorem, conditional expectations, martingales, stationary processes, ergodic theorem, stationary Gaussian processes, Markov chains, introduction to stochastic processes, Brownian motions. Course Information: Prerequisite(s): STAT 501. | 2:00 - 2:50 | Cheng Ouyang | |

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. | 10:00 - 10:50 | Kyunghee Han | |

Stat 535: Optimal Design Theory I | Gauss-Markov theorem,optimality criteria, optimal designs for 1-way, 2-way elimination of heterogeneity models,repeated measurements, treatment-control ; Equivalence theorem,approximate designs for polynomial regression. Course Information: Prerequisite(s): STAT 521. | 12:00 - 12:50 | Min Yang | |

Stat 591: Advanced Topics in Statistics, Probability and Operations Research | Likelihood estimation, testing hypothesis, and model selection in generalized linear models, Logistic and Poisson regression, Linear mixed model and generalized linear mixed model, generalized additive models via kernel and spline regression smoothing, generalized estimation equation, quasi-likelihood method. Prerequisite(s): STAT 521. | 11:00 - 11:50 | Jing Wang |