chicago math phd

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DEPARTMENT OF MATHEMATICS

Graduate Program: Welcome

Questions about the graduate program, dear prospective graduate student,.

Thank you for your interest in the Northwestern graduate program in mathematics. Our program aims to develop graduate students into productive research mathematicians. We offer the benefits of an internationally-renowned faculty together with the close-knit collegiality of a small department. Our extremely low student-faculty ratio, small class size, and close interactions among students and faculty allow us to give you individualized, expert attention that would be hard to find at a larger school.

Our department has research strength in many areas of mathematics, including (but not limited to!) algebra, algebraic topology, differential equations, probability, dynamical systems and ergodic theory, homotopy theory, mathematical physics, number theory and representation theory. Students with a particular interest in one of these fields, or with a general interest in mathematics, are likely to have their intellectual needs satisfied by our program. Students with a very specific mathematical interest not mentioned above should closely review the  research interests of our faculty .

A large majority of students who matriculate at Northwestern graduate with a Ph.D. in mathematics, and most graduates go on to postdoctorate research positions in academia. These students typically go on to careers in academia. As long as they remain in good standing, students receive stipend support and full health coverage from the department for this whole period, with fellowship support in the first and fifth year, and as a teaching assistant otherwise. The first year of graduate study is usually spent taking courses and preparing for the preliminary examination at the end of the year.

While there is no "typical" graduate-school application, students with excellent grades and recommendations from mathematicians who know them well have a clear advantage in the admissions process.

If you are thinking of applying, please review the  deadlines and other application policies.

Northwestern is set in a beautiful, tree-lined campus along the lake in Evanston, Illinois. The suburb of Evanston is a pleasant college town with nice walks, lakefront beaches, and plenty of cafes. At the same time, it is adjacent to the city of Chicago, and is connected to the city's cultural attractions by both subway (or "el") and commuter trains.

We look forward to your continued interest in our program.

The Director of Graduate Studies

• Applied Mathematics (Ph.D.)

Applied mathematics addresses problems in science, engineering, and society. Find new ways to solve real-world problems through original, creative research in Illinois Tech’s applied mathematics Ph.D. program.

• Academic Programs

Illinois Tech’s Ph.D. program in Applied Mathematics is a flagship graduate program that prepares talented mathematicians and statisticians for careers in research or academia through a rigorous education, which includes advanced coursework, independent study, and original research. With almost 100 percent job placement at graduation, our alumni work at Goldman Sacks, UBS, Amazon, University of Michigan, DePaul University, United Airlines, Grant Thornton, as well as start-ups and early stage companies.

The Department of Applied Mathematics and the College of Computing offer generous scholarships in the form of teaching or research assistantships that cover tuition and provide a competitive monthly stipend. 

Courses cover a wide range of topics in applied mathematics and statistics including mathematical courses offered in popular graduate programs such as Data Science , Financial Technology , and Computational Decision Science and Operations Research . 

The Department of Applied Mathematics is a vibrant research hub with internationally recognized faculty working in a variety of applied research areas such as computational mathematics, stochastic analysis, statistics, data science, applied discrete mathematics, and optimal control. 

In addition to numerous academic activities, the Department of Applied Mathematics is home to several student organizations such as Illinois Tech SIAM Student Chapter, American Statistical Association, Association for Women in Mathematics, Machine Learning @IIT, and Fun Math Problems.

Program Overview

Prepare for a career in industrial research or academia through a rigorous education that includes advanced coursework, independent study, and original research to make a significant contribution to the field of applied mathematics.

Career Opportunities

Career opportunities exist across industries, as so many need mathematics experts.

• Operations researcher/analyst
• Mathematician/statistician
• Post-secondary mathematics/science teacher
• Post-secondary mathematics/science administrator

View Details

Admission Requirements

The program typically requires a bachelor’s degree in mathematics or applied mathematics. Candidates whose degree is in another field and whose background in mathematics is strong are also eligible for admission and are encouraged to apply.

Applicants should have a bachelor’s degree from an accredited university. A cumulative GPA of 3.5/4.0 is usually required.

TOEFL scores, if required, should be a minimum of 80/550 (internet-based/paper-based test scores).

A two-page professional statement of goals/objectives and a curriculum vitae must be submitted.

Three letters of recommendation are required.

All applications are automatically considered for full scholarship in from of Teaching or Research Assistantship, with no additional application process for such funds. The scholarships are awarded to top candidates based on the strength of the entire portfolio, the departmental needs and the availability of funds. Full consideration are given to applications for Fall semesters received before the priority deadline of January 31.

Ask a Professor

What do climate change, finance, data science, sports analytics, engineering, and software development all have in common? They all have foundations in mathematics. Discover how a degree in applied mathematics can open doors to these careers, and many more, by speaking with Professor Igor Cialenco, director of graduate studies at Illinois Tech’s Department of Applied Mathematics. These virtual visits occur on Wednesdays from 3 p.m. to 4 p.m. CST.

Featured Faculty

Igor Cialenco

Jinqiao (Jeffrey) Duan

Sonja Petrović

Fred J. Hickernell

Maggie Cheng

Our alumni hit the ground running

By planning together, the Department of Applied Mathematics and Jeffery Mudrock forged a plan allowing him to work full-time while pursuing a Ph.D. and his career goals.

Learning experiences from a network of collaborators helped Lluís Antoni Jiménez Rugama evaluate mathematics from a variety of perspectives.

After emigrating to the United States, Kabre ran her own business before pursuing a Ph.D. in applied mathematics and a career in academia.

Yicong Huang finds himself incorporating his research experience into his daily work.

With a combination of theoretical and practical research experience, Xiao Huang can’t find a work problem he is unable to solve.

Learn more...

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Department of Mathematics

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• Bao Châu Ngô
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• Guillaume Bal, Statistics and Mathematics
• Alexander A. Beilinson
• Danny Calegari
• Francesco Calegari
• Kevin Corlette
• Marianna Csörnyei
• Vladimir Drinfeld
• Matthew Emerton
• Benson Farb
• Robert A. Fefferman
• Simion Filip
• Victor Ginzburg
• Ewain Gwynne
• Denis Hirschfeldt
• Kazuya Kato
• Carlos E. Kenig
• Gregory Lawler, Mathematics and Statistics
• Maryanthe Malliaris
• J. Peter May
• Andre Neves
• Madhav Vithal Nori
• Alexander Razborov, Mathematics and Computer Science
• Luis Silvestre
• Panagiotis Souganidis
• Sidney Webster
• Shmuel Weinberger
• Amie Wilkinson
• Robert Zimmer

Associate Professors

• Akhil Mathew

Assistant Professors

• Daniil Rudenko

Instructors

• Aaron Calderon
• Justin Campbell
• Leonardo Coregliano
• Haoyang Guo
• Joe Jackson
• Michael Klug
• Yangyang Li
• Zhimeng Ouyang
• Minjae Park
• Chengyang Shao
• Tobias Shin
• Andreas Stavrou
• Donald Stull
• Tina Torkaman
• Zhihan Wang
• Nick Wawrykow
• Jincheng Yang

Senior Instructional Professors

• John Alongi
• John Boller
• Lucas Culler
• Jitka Stehnova
• Sarah Ziesler

Assistant Instructional Professors

• Subhadip Chowdhury
• Charles Cunningham
• Kale Davies
• Nicole Pitcher
• Beniada Shabani
• Stephen Yearwood
• Selma Yildirim
• Seyed Zoalroshd

Emeritus Faculty

• Jonathan Alperin
• Spencer Bloch
• Jack D. Cowan
• Todd Dupont
• George Glauberman
• Robert Kottwitz
• Steven Lalley
• Matam P. Murthy
• Niels Nygaard
• L. Ridgway Scott, Computer Science and Mathematics
• Robert I. Soare, Computer Science and Mathematics

The Department of Mathematics provides a comprehensive education in mathematics which takes place in a stimulating environment of intensive research activity. The graduate program includes both pure and applied areas of mathematics. Ten to fifteen graduate courses are offered every quarter. Several seminars take place every afternoon. There is an active visitors program with mathematicians from around the world coming for periods from a few days to a few months. There are four major lecture series each year: the Adrian Albert Lectures in Algebra, the Antoni Zygmund and Alberto Calderón Lectures in Analysis, the Unni Namboodiri Lectures in Topology, and the Charles Amick Lectures in Applied Mathematics. The activities of the department take place in Eckhart and Ryerson Halls. The Departments of Mathematics, Computer Science and Statistics have several joint appointments, and they coordinate their activities. 

Graduate Degrees in Mathematics

The graduate program of the Department of Mathematics is oriented towards students who intend to earn a Ph.D. in mathematics on the basis of work done in mathematics. The Department also offers the degree of Master of Science in mathematics, which is acquired as the student proceeds on to the Ph.D. degree. Students are not admitted with the Master of Science degree as their final objective. In addition, the department offers a separate Master of Science in Financial Mathematics degree program which is taught in the evenings. See the program listing for Financial Mathematics for more information.

The divisional requirements for these degrees can be found in the section on the Physical Sciences Division in these Announcements. Otherwise, the requirements are as follows.

The Degree of Master of Science

The candidate must pass, the nine basic first year graduate courses in the areas of

At the beginning of each quarter a placement exam is offered for each of the courses above. Students who pass the exam can place out of the course, but must take another course in a related area.

Grading Policy:

A passing grade for graduate student's in Math 59900:   Reading/Research Mathematics courses is the grade of "P".

The grading policy for first year graduate courses in mathematics. P = Satisfactory progress. C = Student's performance raised some concerns but these can be resolved by asking the student to do some additional work. D = Student should retake course.  

The Degree of Doctor of Philosophy

For admission to candidacy for the Doctor of Philosophy, an applicant must demonstrate the ability to meet both the divisional requirements and the departmental requirements for admission.

The applicant must pass at least six of the core courses required for the Master in Science in Mathematics, either by taking and passing the course or by passing a placement exam. The applicant must pass at least one core course from each of the three sequences in Algebra, Analysis, and Geometry and Topology.

The applicant must satisfactorily complete a topic exam. This exam covers material that is chosen by the student in consultation with members of the department and is studied independently. The topic presentation is normally made by the end of the student’s second year of graduate study, and includes both a written proposal and an oral presentation and exam.

The applicant must also successfully complete the department’s program of preparatory training in the effective teaching of mathematics in the English language at a level commensurate with the level of instruction at the University of Chicago.

After successful completion of the topic presentations, the student is expected to begin research towards the dissertation under the guidance of a member of the department. The remaining requirements are to:

• Complete a dissertation containing original, substantial, and publishable mathematical results
• Present the contents of the dissertation in an open lecture
• Pass an oral examination based both on the dissertation and the field of mathematics in which it lies

A joint Ph.D. in Mathematics and Computer Science is also offered. To be admitted to the joint program, students must be admitted by both departments as follows. Each student in this program will have a primary program (either Math or CS). Students apply to their primary program. Once admitted, they can apply to the secondary program for admission to the joint program. This secondary application can occur either before they enter the program or any time during their first four years in their primary program. Simultaneous applications to both programs will also be considered (one of the programs being designated as primary).

Students enrolling in this program need to satisfy the course requirements of both departments. They have to satisfy the course requirements of their primary program on the schedule of that program, and satisfy the course requirements of their secondary program by the end of their fifth year. They also need to satisfy the examination requirements of their primary program, and are expected to write a dissertation in an area relevant to both fields.

Mathematics Courses

MATH 30200-30300-30400. Computability Theory-1; Computability Theory-2; Computability Theory-3.

The courses in this sequence are offered in alternate years.

MATH 30200. Computability Theory I. 100 Units.

We investigate the computability and relative computability of functions and sets. Topics include mathematical models for computations, basic results such as the recursion theorem, computably enumerable sets, and priority methods.

Instructor(s): D. Hirschfeldt     Terms Offered: Spring Prerequisite(s): Consent of department counselor. MATH 25500 or consent of instructor. Equivalent Course(s): CMSC 38000

MATH 30300. Computability Theory II. 100 Units.

CMSC 38100 treats classification of sets by the degree of information they encode, algebraic structure and degrees of recursively enumerable sets, advanced priority methods, and generalized recursion theory.

Instructor(s): D. Hirschfeldt     Terms Offered: Spring Prerequisite(s): Consent of department counselor. MATH 25500 or consent of instructor. Equivalent Course(s): CMSC 38100

MATH 30400. Computability Theory-III. 100 Units.

MATH 30708. Simple Theories. 100 Units.

Simple theories (so called), introduced almost forty years ago, provide a model theoretic framework for studying certain familes of 'random" objects, such as the theories of random graphs and hypergraphs. Very recent work has shown the class to contain a much greater range of complexity than previously thought. This course will cover the fundamental theorems of simple theories along with some of the new developments.

Instructor(s): Maryanthe Malliaris     Terms Offered: Autumn

MATH 30813. Some Logic and Geometry for Mathematicians. 100 Units.

This quarter course will cover three major applications of logic to other branches of mathematics. The first is Tarski's theorem about quantifier elimination and decidability of the first order theory of the reals. This is a cornerstone of real algebraic geometry with major implications in analysis and geometry. The second is the existence of groups with unsolvable word problem and Higman's characterization of finitely generated subgroups of finitely generated groups. This implies that some geometric problems, such as the homeomorphism problem for manifolds, or deciding whether a simplicial complex is a manifold are undecidable (as we hope to explain). And, finally, we will explain why there is no algorithm to decide whether Diophantine equations with integer coefficients have integral solutions, along with some interesting definability questions in this context.

Instructor(s): Shmuel Weinberger, Maryanthe Malliaris     Terms Offered: Spring

MATH 30900-31000. Model Theory I-II.

MATH 30900 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In MATH 31000, we study saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero.

MATH 30900. Model Theory I. 100 Units.

First graduate course in model theory, covering the basics of the modern field, through stability.

Prerequisite(s): MATH 25500 or 25800 Note(s): This course is offered in alternate years.

MATH 31000. Model Theory II. 100 Units.

Second graduate course in model theory, focusing on the fundamentals of classification theory.

Terms Offered: Spring Prerequisite(s): MATH 30900 Note(s): This course is offered in alternate years.

MATH 30905. Decidability. 100 Units.

Decision problems in model theoretic algebra, learning and connections to classification theory, and other topics as time permits.

Instructor(s): Maryanthe Malliaris     Terms Offered: Winter

MATH 31200-31300-31400. Analysis I-II-III.

Analysis I-II-III

MATH 31200. Analysis I. 100 Units.

Topics include: Lebesgue integration, Lp spaces and Banach spaces, differentiation theory, Hilbert spaces and Fourier series, Fourier transform, probability spaces and random variables, strong law of large numbers, central limit theorem, conditional expectation and martingales, Brownian motion.

Terms Offered: Autumn Prerequisite(s): MATH 26200, 27000, 27200, and 27400; and consent of director or co-director of undergraduate studies

MATH 31300. Analysis II. 100 Units.

Topics include: Hilbert spaces, projections, bounded and compact operators, spectral theorem for compact selfadjoint operators, unbounded selfadjoint operators, Cayley transform, Banach spaces, Schauder bases, Hahn-Banach theorem and its geometric meaning, uniform boundedness principle, open mapping theorem, Frechet spaces, applications to elliptic partial differential equations, Fredholm alternative.

Terms Offered: Winter Prerequisite(s): MATH 31200

MATH 31400. Analysis III. 100 Units.

Topics include: 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.

Terms Offered: Spring Prerequisite(s): MATH 31300

MATH 31700-31800-31900. Topology and Geometry I-II-III.

Topology and Geometry I-II-III

MATH 31700. Topology and Geometry I. 100 Units.

Topics include: 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.

MATH 31800. Topology and Geometry II. 100 Units.

Topics include: Definition of 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.

Terms Offered: Winter Prerequisite(s): MATH 31700

MATH 31900. Topology and Geometry III. 100 Units.

Topics include: 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.

Terms Offered: Winter Prerequisite(s): MATH 31800

MATH 32500-32600-32700. Algebra I-II-III.

Algebra I-II-III

MATH 32500. Algebra I. 100 Units.

Topics include: 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.

Terms Offered: Autumn Prerequisite(s): MATH 25700-25800-25900, and consent of director or co-director of undergraduate studies

MATH 32600. Algebra II. 100 Units.

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.

Terms Offered: Winter Prerequisite(s): MATH 32500

MATH 32700. Algebra III. 100 Units.

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.

Terms Offered: Spring Prerequisite(s): MATH 32600

MATH 34200. Geometric Literacy-2. 100 Units.

This ongoing course might be subtitled: "what every good geometer should know". The topics will intersperse more elementary background with topics close to current research, and should be understandable to second year students. The individual modules (2-5 weeks each) might be logically interrelated, but we will try to maintain a "modular structure" so that people who are willing to assume certain results as "black boxes" will be able to follow more advanced modules before formally learning all the prerequisites. This years topics might include: basics of symplectic geometry, harmonic maps in geometry, pseudo-Anosov homeomorphisms and Thurston's compactification of Teichmuller space, algebraic geometry for non-algebraic geometers. Prereq: First year graduate sequence.

MATH 34300. Geometric Literacy - 3. 100 Units.

Instructor(s): Benson Farb     Terms Offered: Spring Prerequisite(s): First year graduate sequence.

MATH 34500. Topics in Geometry and Topology. 100 Units.

This course will cover various topics ranging from algebraic and differential geometry to algebraic and geometric topology, often with connections to representation theory and number theory. Recent topics have included Hodge theory, Mostow Rigidity, Topology and geometry of K3 surfaces (joint with Eduard Looijenga), "What all the 3-manifolds are", 4-manifold theory: from Seiberg-Witten to the classification of albebraic surfaces (joint with Danny Calegari), and the cohomology of arithmetic groups (joint with Matt Emerton).

Instructor(s): Benson Farb     Terms Offered: Winter Prerequisite(s): The first year math graduate courses or permission of instructor.

MATH 34600. Topics in Geometry and Topology-2. 100 Units.

Instructor(s): Benson Farb     Terms Offered: Spring Prerequisite(s): The first year math graduate courses or permission of instructor.

MATH 35600. Topics in Dynamical Systems. 100 Units.

This course covers selected topics in dynamical systems. Topics vary and may include: ergodic theory, smooth dynamical systems, statistical properties of dynamical systems, and geometry and dynamics.

Instructor(s): Anne Wilkinson     Terms Offered: Autumn

MATH 36000. Proseminar: Topology. 100 Units.

This informal proseminar is devoted to topics in algebraic topology and neighboring fields. Talks are given by graduate students, postdocs, and senior faculty. They range from basic background through current research.

Instructor(s): Staff

MATH 36100. Topology Proseminar. 100 Units.

This informal "proseminar" is devoted to topics in algebraic topology and neighboring fields. Talks are given by graduate students, postdocs, and senior faculty. They range from basic background through current research.

Instructor(s): J. Peter May     Terms Offered: Winter

MATH 36200. Topology Proseminar. 100 Units.

The Spring proseminar is a more formal version of the Fall and Winter topology proseminar. It will be taught primarily or completely by May, on topics of interest to the participants.

Instructor(s): J. Peter May     Terms Offered: Spring

MATH 36206. Algebraic Number Theory. 100 Units.

This is a course in basic algebraic number theory. The only prerequisites will be the material covered in the first year algebra graduate sequence.

Instructor(s): Francesco Calegari     Terms Offered: Spring

MATH 36507. Condensed Mathematics. 100 Units.

This course is an introduction to the new subject of condensed mathematics introduced by Clausen and Scholze. We will discuss some of the foundational results in the theory and study some of the growing applications to analytic geometry.

Instructor(s): Akhil Mathew and Matthew Emerton     Terms Offered: Autumn

MATH 36558. Algebra/Topology. 100 Units.

This will be an advanced course on topics in algebra and topology.

Instructor(s): Akhil Mathew     Terms Offered: Spring

MATH 36611. Topics in Analytic Geometry. 100 Units.

This course will cover some recent developments in analytic geometry arising from the new theory of condensed mathematics developed by Clausen and Scholze.

Instructor(s): Akhil Mathew     Terms Offered: Autumn Prerequisite(s): Algebra and Geometry sequences. (1st year), some category theory.

MATH 36704. Dynamics and Applications. 100 Units.

The course will provide an introduction to basic results and techniques in dynamical systems and then discuss selected applications.

Instructor(s): Simion Filip     Terms Offered: Winter

MATH 36888. Pseudodifferential Operators with Applications. 100 Units.

In this course I will introduce classical pseudodifferential operators and their calculus and give some applications, including to variable coefficient Schrodinger equations.

Instructor(s): Carlos Kenig     Terms Offered: Autumn Prerequisite(s): The first year graduate sequence in analysis.

MATH 36918. Min-max Methods in Minimal Surfaces. 100 Units.

Min-max methods in minimal surfaces have produced a series of spectacular results lately and settle old questions. I will develop the Algren-Pitts min-max theory from the beginning and explain how that can be used to prove existence of minimal surfaces.

Instructor(s): Andre Neves     Terms Offered: Spring

MATH 37001. Bernstein center and cocenter. 100 Units.

We discuss the Bernstein center of a p-adic reductive group, the cocenter of the Hecke algebra as a module over the Bernstein center and its completion. We present Langlands' theory of endoscopy in this language. We discussion the stable Bernstein center, the stable cocenter and their relation to Langlands' functoriality.

Instructor(s): Bao Chau Ngo     Terms Offered: Autumn

MATH 37104. Parabolic Equations with Irregular Data and Related Issues. 100 Units.

We present a general theory of existence and uniqueness of linear parabolic equations with Lebesgue/Sobolev regular coefficients and initial conditions. Applications to the theory of stochastic differential equations are also discussed. The course is based on the recent book C. Le Bris/ P.L. Lions, Parabolic Equations with irregular data and related issues. Application to Stochastic Differential Equations, De Gruyter Series in Applied and Numerical Mathematics, Vol. 4, 2019, 165 pages, ISBN 978-3-11-063313-9. For more information on the course: [email protected]

Instructor(s): Claude Le Bris     Terms Offered: Winter

MATH 37105. Topics in Geometric Measure Theory I. 100 Units.

A measure is a way to assign a size to collections of points. Lebesgue measure is the most important example but, depending upon the application, the 'size' of a set may be measured in many different, very interesting ways. The interplay between measure and geometry can be extremely subtle and has given rise to powerful ideas that are used in energy minimisation problems, the theory of partial differential equations and the study of fractal geometry. This is an advanced course on geometric measure theory and its applications.

Instructor(s): Marianna Csornyei     Terms Offered: Autumn

MATH 37111. Quiver Varieties. 100 Units.

Study of quiver varieties.

Instructor(s): Victor Ginzburg     Terms Offered: Spring

MATH 37214. Hecke algebras, Deligne-Lusztig characters, and character sheaves. 100 Units.

We will discuss Hecke algebras, Deligne-Lusztig characters, and character sheaves.

MATH 37219. Crystalline Differential Operators. 100 Units.

Introduction to crystalline differential operators.

Instructor(s): Victor Ginzburg     Terms Offered: Winter

MATH 37304. Theory of Elliptic PDES. 100 Units.

We will study the theory for existence and regularity of second order elliptic PDE's. After presenting the basic results (energy estimates, Schauder theory, spectral theory), we will do Di Giorgi-Nash estimates and present several methods to find solutions to quasilinear and fully non-linear second order elliptic PDE's.

Instructor(s): Andre Neves     Terms Offered: Winter Prerequisite(s): Analysis I and II

MATH 37392. Arithmetic Geometry. 100 Units.

I will explain important aspects in arithmetic geometry.

Instructor(s): Kazuya Kato     Terms Offered: Winter Prerequisite(s): Algebra 1-Algebra 3 of the first year graduate courses.

MATH 37410. Topics in low-dimensional topology. 100 Units.

We will discuss topics in low-dimensional topology.

Instructor(s): Danny Calegari     Terms Offered: Autumn

MATH 37411. 3-manifolds. 100 Units.

The topic will be foliations, order ability, and the L-space conjecture.

MATH 37801. Configuration spaces in topology, algebraic geometry and topological field theory. 100 Units.

Configuration spaces seem ubiquitous nowadays. In this course we discuss their role in various settings in which they occur: topology (involving among other things the little disk operad and a certain spectral sequence), algebraic geometry and associated mixed Hodge structures (the case of algebraic curves being the most interesting case) and their role in the theory of conformal blocks.

Instructor(s): Eduard Looijenga     Terms Offered: Autumn

MATH 37902. Topics in unique continuation. 100 Units.

The course will deal with selected topics in unique continuation and boundary unique continuation for elliptic equations, with applications.

Instructor(s): Carlos Kenig     Terms Offered: Autumn Prerequisite(s): First year analysis sequence, undergraduate pde

MATH 37904. Linear and semilinear Schrodinger evolutions, I. 100 Units.

We will develop harmonic analysis tools for the linear Schrodinger evolution that will then be used for the study of semilinear Schrodinger evolutions. In the first quarter we will treat for the semilinear case small data/short time results, while in the second quarter we will study large data for long times, in critical semilinear problems.

Instructor(s): Carlos Kenig     Terms Offered: Autumn Prerequisite(s): The first year graduate analysis sequence and familiarity with the Fourier transform and introductory partial differential equations.

MATH 37905. Linear and semilinear Schrodinger evolutions, II. 100 Units.

In the second quarter we will study large data for long times, in critical semilinear problems.

Instructor(s): Carlos Kenig     Terms Offered: Winter Prerequisite(s): The first year graduate analysis sequence and familiarity with the Fourier transform and introductory partial differential equations.

MATH 37907. Hodge Theory and Moduli. 100 Units.

Perhaps the most important tool for the study of moduli of complex algebraic varieties is the period map, which assigns to a variety its Hodge structure. In this course we shall develop some of the general theory of the notions involved here (including mixed Hodge theory), but examples of interest (some classical and others less so) will be at the center, such as hyperkaehler manifolds---this includes K3 surfaces---and hypersurfaces. This will lead us to see how locally symmetric varieties (such as ball quotients) parametrize Hodge structures. We will also touch on the mixed Hodge theory of the fundamental group and the interesting extensions of Hodge structures that it can give rise to.

Instructor(s): Eduard Looijenga     Terms Offered: Autumn Prerequisite(s): Prerequisites are basic knowledge of manifolds, (co)homology and De Rham theory (as taught in the courses Algebraic Topology and Topology and Geometry II). We shall treat the classical Hodge theorem as a given (but of course state it), which means that we will not get into its proof.

MATH 37908. Topics in Algebraic Geometry-1. 100 Units.

This is in order to develop some basic algebro-geometric literacy. Topics might include moduli spaces, Deligne-Mostow theory, period maps.

Instructor(s): Eduard Looijenga     Terms Offered: Autumn Prerequisite(s): Introductory course in algebraic geometry.

MATH 38002. Representation theory of p-adic groups. 100 Units.

Discussing representation theory of p-adic groups

MATH 38005. Decomposition theorem for perverse sheaves and Hodge theory. 100 Units.

Discussing decomposition theorem for perverse sheaves and Hodge theory

MATH 38010. The Hitchin Morphism. 100 Units.

The Hitchin morphism will be discussed.

MATH 38420. Mathematics of Quantum Computing. 100 Units.

This course is a gentle introduction to mathematical foundations of quantum computing taught in completely rigorous format: we will completely disregard physical aspects and specific questions pertaining to particular implementations. An (approximate) list of topics: reversible, probabilistic and quantum computation. Quantum complexity classes and relations to their classical counterparts. Fundamental quantum algorithms, notably Grover's search and Shor's factoring algorithm. Quantum (query) complexity theory and quantum communication complexity. Quantum probability, super-operators and non-unitary quantum computation. Basics of quantum information theory and quantum error-correction.

Equivalent Course(s): CMSC 38420

MATH 38515. Symplectic Topology. 100 Units.

This is an introduction to symplectic topology. The purpose is to provide background and details for some of the material covered in Leonid Polterovich's course.

Instructor(s): Danny Calegari     Terms Offered: Winter

MATH 38595. Topics in Complex Dynamics. 100 Units.

An introduction to the theory of complex dynamics in 1 dimension, especially the theory of rational maps and rational correspondences. Foundations of the theory, quasiconformal analysis, no wandering domains, Mandelbrot set and variations.

Instructor(s): Danny Calegari     Terms Offered: Spring

MATH 38599. Introduction to Floer Theories. 100 Units.

An introduction to the use of gauge theoretic methods in 3-manifold topology, including Seiberg-Witten and Heegaard Floer Homology, connections to taut foliations and sutured manifolds. Thurston norm, contact structures, etc.

MATH 39013. Crystalline Differential Operators. 100 Units.

Crystalline differential operators will be discussed.

MATH 39902. Topics in Invariant Theory. 100 Units.

We will discuss Topics in Invariant Theory.

MATH 42002. P-adic Hodge Theory. 100 Units.

Basic things in p-adic Hodge theory are explained.

Instructor(s): Kazuya Kato     Terms Offered: Winter Prerequisite(s): Algebra 1, 2, 3

MATH 47000. Geometric Langlands Seminar. 100 Units.

This seminar is devoted not only to the Geometric Langlands theory but also to related subjects (including topics in algebraic geometry, algebra and representation theory). We will try to learn some modern homological algebra (Kontsevich's A- infinity categories) and some "forgotten" parts of D- module theory (e.g. the microlocal approach).

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Autumn

MATH 47100. Geometric Langlands Seminar. 100 Units.

The seminar is devoted to the Geometric Langlands theory and related subjects, which covers topics in algebraic geometry, algebra, and representation theory.

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Winter

MATH 47200. Geometric Langlands Seminar. 100 Units.

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Spring

MATH 59900. Reading/Research: Mathematics. 300.00 Units.

Readings and Research for working on their PhD

MATH 70000. Advanced Study: Mathematics. 300.00 Units.

Advanced Study: Mathematics

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Admission Requirements

Transcripts of all undergraduate and any graduate work must be submitted. In addition to the Graduate College minimum requirements, applicants must meet the following program requirements:

Baccalaureate Field  Mathematics or a related field.

Grade Point Average   At least 3.00/4.00 for the final 60 semester hours (90 quarter hours) of undergraduate study, and an average of 3.00 in all mathematics courses beyond calculus.

Tests Required  Neither the GRE General Exam nor the subject exams are required. Applicants may still submit GRE scores; however, an absence of GRE scores will not negatively impact their application.

Minimum English Competency Test Score ( if native language is not English)

TOEFL iBT  100, with subscores of Reading 19, Listening 17, Speaking 23, and Writing 21, OR ,

IELTS  7.0, with 7.0 in each of the four subscores, OR ,

• PTE-Academic 54, with subscores of Reading 51, Listening 47, Speaking 53, and Writing 56.

Letters of Recommendation  Three required from persons familiar with the applicant’s academic work.

Personal Statement  Required.

Degree Requirements

In addition to the Graduate College minimum requirements, students must meet the following program requirements:

• Minimum Semester Hours Required  96 from the baccalaureate.
• Coursework  At least 40 hours must be in 500-level mathematics courses, excluding thesis research ( MATH 599 ,  MCS 599 , or  STAT 599 ).
• Preliminary Examination  Required.
• Dissertation  Required. Students earn at least 32 hours in thesis research ( MATH 599 , MCS 599 , or STAT 599 ).

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Graduate Program in Mathematics

The Department of Mathematics offers several graduate degrees, including

• a PhD in Mathematics
• a PhD in Mathematics with a Concentration in Actuarial Science and Risk Analytics
• MS degrees in Mathematics, Applied Mathematics, Actuarial Science, and the Teaching of Mathematics

The department won the Exemplary Program Award (2017) from the American Mathematical Society.

The department is housed largely in Altgeld Hall, a beautiful stone Romanesque building in the heart of the Illinois campus, and contains one of the world's best mathematics libraries. Doctoral students enjoy a collaborative atmosphere and a lively schedule of about twenty-five research seminars and lectures each week. A large share of our PhD students are women, and we support an active chapter of the Association for Women in Mathematics (AWM). Graduate students explore multiple career paths through an innovative program of internships in science and industry.

Graduate Program Staff

The Mathematics Graduate Studies office is located in 267 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801

Jared Bronski , Director of Graduate Studies 259 Altgeld Hall | 217-244-8218 | [email protected]

Marci Blocher , assistant to the director of Graduate Studies 267 Altgeld Hall | 217-333-5749 | [email protected]

Karen Mortensen , associate director of Graduate Studies 265 Altgeld Hall | 217-333-5749 | [email protected]  

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Dept of Math, Stat, & Comp Sci

College of liberal arts and sciences, phd and ms programs.

The Department of Mathematics, Statistics, and Computer Science offers the Doctor of Philosophy in Mathematics in pure mathematics, applied mathematics, statistics, and mathematical computer science.  We provide the Master of Science in Mathematics with concentrations in pure mathematics, applied mathematics, and mathematical computer science, and a Master of Science in Statistics. In mathematics education, we offer a Doctor of Arts in Mathematics and the Master of Science in Teaching degree, with certificates in elementary and secondary education.

Committee on Computational and Applied Mathematics

George Herbert Jones Laboratory

Home of the Committee on Computational and Applied Mathematics

Welcome to CCAM

In response to the critical need to train a new generation of computational and applied mathematicians who can confront data-centric problems in the natural and social sciences, the University of Chicago created the Committee on Computational and Applied Mathematics (CCAM) in 2016.  Read more ...

Committee on Computational and Applied Mathematics Spotlight

Congratulations to prof. bernd sturmfels, who received cam’s first honorary degree, congratulations to cam phd student, matthew oline for being awarded the physical sciences teaching prize, congratulations to cam phd student, andrew dennehy for being awarded a 2024 nsf graduate research fellowship, committee on computational and applied mathematics news & events, cam phd & mcam student dinner, an interview with the new director of the committee on computational and applied mathematics., peter nekrasov (cam phd student) presented research at the agu23 meeting., cam colloquium: yi sun.

4:00–5:00 pm Jones 303

CAM Colloquium: BIANCA M. DUMITRASCU

Cam colloquium: daniel sanz-alonso.

4:00–5:00 pm

The University of Chicago: Graduate Studies

Application management.

Applications for 2024 are closed. 2025 applications will be available here after the application opens. Welcome! Thank you for your interest in applying to a graduate program in the Physical Sciences Division of the University of Chicago. Through this site, you can apply to the following degree programs:

• Astronomy and Astrophysics (PhD)
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You may begin your application using the “Start New Application” link below . Please note that if you plan to apply to more than one program, you should begin a new application for each program. If you have any questions about your application, please contact the program to which you are applying. Contact information can be found under the listings of graduate programs on the Graduate Admissions website .

Non-Degree Visiting Student Status (NDVS) :   NDVS is available to advanced graduate and undergraduate students pursuing a degree at another institution who have been invited by a University of Chicago faculty member to temporarily work on their research project. This status also applies to undergraduate students participating in summer research opportunities such as the PKU-CCME exchange program or other research experiences for undergraduates (REUs).  Questions about NDVS status may be directed to Aleksandra Ninova-Parris at  [email protected]   Toyota Technological Institute at Chicago (TTIC):  TTIC students participating in research or University of Chicago courses should complete a non-degree application on this site to establish a student record with the University of Chicago.

We use data collected in the application to render admissions decisions, for internal and external reporting, institutional research, and other purposes. To learn more about how we use the information collected here please review our Privacy Notice .

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Math PhD After Math Undergrad at UChicago

Hi Everyone,

I recently noticed that it seems like very few of the math PhD students here went to UChicago for undergrad. I know that many institutions will favor their undergrad students in the grad admissions process, so it's a little weird to me that there are not that many UChicago undergrads continuing onto grad school here. Does UChicago not favor its undergrad students, or do students just want to leave after 4 years of "the life of the mind"?

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Department of Mathematics

Placement data for graduate students.

This page is being updated.

Undergraduate Programs

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College Admissions | The College

• I Interdisciplinary
• C UChicago Careers In Program
• J Joint Degree and Professional Option

Department of Mathematics

Joint math/cs phd program.

In Winter 2018, the Department of Mathematics and the Department of Computer Science launched a joint program through which participating students can earn the degree “Ph. D. in Mathematics and Computer Science.”

The basic structure is that students must gain admission to both PhD programs and satisfy both sets of course requirements. They write a single dissertation that satisfies both programs.

While the program is open to all eligible students, we expect that at least initially it will be most popular among students working in CS Theory, Discrete Mathematics, and Mathematical Logic.

Each student in this program will have a primary program (either Mathematics or Computer Science). Throughout the course of studies, the primary program will provide administrative support to the student, including in matters regarding financial support.

To be admitted to the joint program, students will have to be admitted by both departments as follows.

Application after entering the PhD program

Students enrolled in either the Mathematics or the Computer Science PhD program may apply to the joint program during the first four years in their current program. If admitted to the joint program, their current program will be primary.

Such an applicant must submit the following material to the Director of Graduate Studies/Graduate Committee Chair of the intended secondary program, while notifying the Director of Graduate Studies/Graduate Committee Chair of the primary program:

• statement of purpose, explaining why the joint program is the right program for the applicant
• statement of coursework and research done so far
• statement of a schedule how the applicant proposes to satisfy the secondary program's requirements
• advisor's recommendation, including endorsement of the applicant's statements (items 2, 3, 4 above)
• if the application occurs during the first year of the primary program, the applicant needs to provide two additional letters of recommendation and his/her undergraduate transcript. It is permitted to reuse material from the application to the primary program.

Requirements, monitoring

Course requirements.

Students enrolling in the joint program will need to satisfy the course requirements of both departments. They will have to satisfy the course requirements of their primary program on the schedule of that program, and satisfy the course requirements of their secondary program by the end of their fifth year in the primary program.

According to current rules, two of the CS electives can be courses offered by the Mathematics department. These courses are permitted to overlap with the Mathematics course requirements.

Exam Requirements

Students in the joint program shall fulfill the examination requirements of the primary program; the current list of requirements can be found at

• Computer Science Requirements
• Mathematics Requirements

For students participating in the joint program, the deadlines for these exams can be relaxed by petitioning the Director of Graduate Studies/Graduate Committee Chair of the primary program.

Monitoring student progress

Students' annual progress reports go to both departments' Director of Graduate Studies/Graduate Committee Chair in accordance with each department's format.

PhD dissertation and defense

Subject of the dissertation.

The dissertation is expected to be in an area relevant to both fields.

PhD thesis defense

The scheduling of the PhD Thesis defense follows the Mathematics Department's custom as follows.

• A nearly final draft of the thesis is made accessible to faculty at least two weeks prior to the defense, either in hard copy in the departmental office or, preferably, by posting on the internet.
• The dissertation is reviewed in writing by two readers, one of whom is typically the thesis advisor.
• The reports by the first and the second readers are circulated among faculty of both departments, along with the Thesis abstract and the following information: the location (physical or virtual) where the thesis can be viewed, the planned time and location of the defense, and the names and affiliations of the thesis committee members.
• There is a two-week period for comments by faculty before the admission of the thesis for defense.

The thesis defense itself consist of a 50-minute public presentation of the main results and methods of the dissertation, followed by a public question-answer period, followed by a closed-session question-answer period.

Oversight, committees

The program proceeds under joint Math-CS oversight, exercised by the Director of Graduate Studies/Graduate Committee Chair of each department.

Examination committees

The following rules apply to all examination committees (Qualifying/Topic Exam, Master's, Candidacy, and PhD). The committee will consist of at least three members, including the student's advisor(s). It will include at least one member of each department, and will either be chaired by a joint appointee of the two departments or co-chaired by a member of each department. Each department shall publicize these exams in accordance with its established customs.