# Graduate Courses

MSCS 500-Level Graduate Courses

Below is a list of courses we expect to be offering in the semesters ahead. For complete course information including location, please consult the UIC Schedule of Classes.

## Fall 2024 Planned 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 | Caroline Terry | 38305 |

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 Instructor. | MWF 10:00 - 10:50 | Joel (Ronnie) Nagloo | 34413 |

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 | Nathan Jones | 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 9:00 - 9:50 | Wenliang Zhang | 13724 |

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 | Ramin Takloo-Bighash | 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 in TH 215 | Mimi Dai | 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 11:00 - 11:50 | Osama Khalil | 37062 |

MATH 549: Differentiable Manifolds | 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 10:00 - 10:50 | Daniel Groves | 38306 |

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 | 31610 | |

MATH 555: Complex Manifolds | 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. | MWF 2:00 - 2:50 | Ben Bakker | 39215 |

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. | MWF 12:00 - 12:50 | Emily Dumas | 38308 |

MATH 577: Advanced Partial Differential Equations | Linear elliptic theory, maximum principles, fixed point methods, semigroups and nonlinear dynamics, systems of conservation laws, shocks and waves, parabolic equations, bifurcation, nonlinear elliptic theory. Course Information: Prerequisite(s): MATH 533 and MATH 576 or consent of the instructor. | MWF 12:00 - 12:50 | Christof Sparber | 40006 |

MATH 580: Mathematics of Fluid Mechanics | Development of concepts and techniques used in mathematical models of fluid motions. Euler and Navier Stokes equations. Vorticity and vortex motion. Waves and instabilities. Viscous fluids and boundary layers. Asymptotic methods. Course Information: Prerequisite(s): Grade of C or better in MATH 410 and grade of C or better in MATH 417 and grade of C or better in MATH 481. | MWF 2:00 - 2:50 | Roman Shvydkoy | 40613 |

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 3:00 - 3:50 | Brooke Shipley | 32552 |

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 10:00 - 10:50 | 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 | High dimensional and big data analysis is one of the most active research areas in statistics today given the unprecedented size and complexity of high-throughput data. We will study cutting-edge developments in the methods and theory of statistical inference including data from genetic, microarrays, proteomics, fMRI, cancer clinical trials and high frequency financial data. | MWF 1:00 - 1:50 | Yichao Wu | 37945 |

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. | TR 5:00 - 6:15 | Xiaorui Sun | 34849 |

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. | MWF 11:00 - 11:50 | Gyorgy Turan | 39221 |

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 9:00 - 9:50 | Lev Reyzin | 43424 |

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. | MWF 12:00 - 12:50 | Jan Verschelde | 39222 |

MCS 583: Extremal Combinatorics | Extremal combinatorics, including extremal graph and set theory, Ramsey theory, the linear algebra method, and applications to computer science. Prerequisite(s): MCS 421 and MCS 423, or consent of the instructor. | MWF 10:00 - 10:50 | Dhruv Mubayi | 44948 |

## Spring 2025 Planned Graduate Courses Heading link

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. | Matthew Harrison-Trainor | ||

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. | 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. | 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. | Frederick Saia | ||

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

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. | Kevin Whyte | ||

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. | 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. | Gwyneth Moreland | ||

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. | Ben Bakker | ||

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. | Christof Sparber | ||

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 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 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. | Vishesh Jain | ||

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 |