chicago math phd

Dept of Math, Stat, & Comp Sci

College of liberal arts and sciences, phd in mathematics.

Doctor of Philosophy in Mathematics

The PhD in Mathematics is designed to provide the highest level of training for independent research. Students may apply with or without a Master's degree. For those with a previous Master's degree in mathematics (or related field) the PhD is typically 5 years in duration, whereas for those without a previous Master's degree it is typically 6 years.

To earn the PhD, the student must fulfill the Graduate College requirements specified in the Graduate College Catalog as well as departmental requirements detailed in the MSCS Graduate Handbook , which includes:

• Provide proof of an equivalent MS degree or earn a high pass on the Department's written Master's Examination.
• Fulfill the doctoral preliminary examinations and minor sequence requirement.
• Produce and defend a thesis that makes a contribution to original research.
• Earn 96 semester hours of graduate credit including:
• 32 credit hours for a previously earned Master's degree (requires DGS approval), or earn a high pass on the Department's Master's Exam
• 40 credit hours of departmental 500-level courses which may include 500-level courses taken from the MS degree earned in residence but may NOT include thesis research (MATH 599 or MCS 599)
• 32 hours of thesis research (MATH 599 or MCS 599)

Committee on Computational and Applied Mathematics

How to apply, phd application requirements.

The deadline to submit admissions applications to the PhD program along with required accompanying documents for Fall 2025 is  January 9, 2025 . The application portal is open . If you have any questions regarding admission, please send your inquiry to Jonathan G. Rodriguez at [email protected] .

MS Application Requirements

For details regarding applying to the Masters program, please visit https://voices.uchicago.edu/cammasters/prospectivestudents/ .

The deadline to submit admissions applications to the Master's program along with required accompanying documents for Fall 2025 is  February 3, 2025 . The application portal  is open . If you have any questions regarding admission, please send your inquiry to Jonathan G. Rodriguez at [email protected] .

Department of Mathematics

First-year courses.

The courses described below are the core of the first year graduate program. The program undergoes regular reevaluation and change, so the list of topics is only approximate, and the content of the courses also varies from year to year at the discretion of the instructor.

Math 312: Analysis I

Measure theory, integration and \(L^p\) spaces, differentiation, basic functional analysis. Additional topics may be discussed depending on the instructor.

Prerequisite: Graduate student status or instructor consent

Math 313: Functional Analysis

Weak convergence, compact operators, spectral theory, Sobolev spaces, and some applications. Additional topics may be discussed depending on the instructor.

Prerequisite: Math 312

Math 314: Complex Analysis and Topics in Analysis

Basic complex analysis, Cauchy theorem in the homological formulation, residues, meromorphic functions, Mittag-Leffler theorem, Gamma and Zeta functions, analytic continuation, mondromy theorem, the concept of a Riemann surface, meromorphic differentials, divisors, Riemann-Roch theorem, compact Riemann surfaces, uniformization theorem, Green functions, hyperbolic surfaces, covering spaces, quotients.

Prerequisite: Math 313

Geometry and Topology

Math 317: algebraic topology.

Fundamental group, covering space theory and Van Kampen's theorem (with a discussion of free and amalgamated products of groups), homology theory (singular, simplicial, cellular), cohomology theory, Mayer-Vietoris, cup products, Poincare Duality, Lefschetz fixed-point theorem, some homological algebra (including the Kunneth and universal coefficient theorems), higher homotopy groups, Whitehead's theorem, exact sequence of a fibration, obstruction theory, Hurewicz isomorphism theorem.

Prerequisite: Undergraduate analysis, algebra, and (preferably) topology.

Math 318: Differential Topology

Definition of smooth manifolds, tangent and cotangent bundles, vector bundles. Inverse and implicit function theorems. Sard's theorem and the Whitney embedding theorem. Degree of maps. Vector fields and flows, transversality, and intersection theory. Frobenius' theorem, differential forms and the associated formalism of pullback, wedge product, integration, etc. Cohomology via differential forms, and the de Rham theorem. Further topics may include: compact Lie groups and their representations, Morse theory, cobordism, and differentiable structures on the sphere.

Prerequisite: Math 317

Math 319: Differential Geometry

Riemannian metrics, connections and curvature on vector bundles, the Levi-Civita connection, and the multiple interpretations of curvature. Geodesics and the associated variational formalism (formulas for the 1st and 2nd variation of length), the exponential map, completeness, and the influence of curvature on the topological structure of a manifold (positive versus negative curvature). Lie groups. The Chern-Weil description of characteristic classes, the Gauss-Bonnet theorem and possibly the Hodge Theorem.

Prerequisite: Math 318

Math 325: Representation Theory

Representation theory of finite groups, including symmetric groups and finite groups of Lie type; group rings; Schur functors; induced representations and Frobenius reciprocity; representation theory of Lie groups and Lie algebras, highest weight theory, Schur–Weyl duality; applications of representation theory in various parts of mathematics.

Math 326: Commutative Algebra and Algebraic Geometry

This course will explain the dictionary between commutative algebra and algebraic geometry. Topics will include the following. Commutative ring theory: Noetherian property; Hilbert Basis Theorem; localization and local rings; etc. Algebraic geometry: affine and projective varieties, ring of regular functions, local rings at points, function fields, dimension theory, curves, higher-dimensional varieties.

Prerequisite: Math 325

Math 327: Topics in Algebra

According to the inclinations of the instructor, this course may cover: algebraic number theory; homological algebra; further topics in algebraic geometry and/or representation theory.

Prerequisite: Math 326