Skip to Content

Graduate Courses 2018/2019

Please note: the class schedule is subject to change at any time. Such changes are not always reflected immediately on this page.

All the other courses are described in the catalog.  To view graduate courses offered in previous years, please visit the archived pages at

FALL 2018 WINTER 2019 SPRING 2019
510 Differential Forms and De Rham Cohomology
P. Lu (13:00)

513 Intro to Analysis I

C. Phillips (11:00)

514 Intro to Analysis II

C. Phillips (11:00)

515 Intro to Analysis III

P. Gilkey (11:00)

531 Intro to Topology I

N. Addington (9:00)

532 Intro to Topology II

V. Ostrik (9:00)

533 Intro to Differential Geometry

W. He (9:00)

544 Intro to Algebra I

A. Polishchuk (13:00)

545 Intro to Algebra II

A. Polishchuk (13:00)

546 Intro to Algebra III

A. Polishchuk (13:00)

  607 Seminar: Applied Stochastic Processes I

P. Ralph (10:00)

 607 Seminar: Applied Stochastic Processes II

Y. Ahmadian (10:00-11:20 MW)

607 Seminar: Stable Homotopy Theory

D. Sinha (10:00)


  607 Seminar: Moduli Spaces of Curves

A. Vaintrob (12:00)

 607 Seminar: Number Theory I

E. Eischen (10:00-11:20 TR)

607 Seminar: Number Theory II

E. Eischen (10:00-11:20 TR)


  607 Seminar: Set Theory

C. Phillips (14:00)

 607 Seminar: Mirror Symmetry

Y. Shen (14:00)

616 Real Analysis I

M. Bownik (11:00)

617 Real Analysis II

M. Bownik (11:00)

618 Real Analysis III

M. Bownik (11:00)

634 Algebraic Topology I

R. Lipshitz (9:00)

635 Algebraic Topology II

R. Lipshitz (15:00)

636 Algebraic Topology III

R. Lipshitz (15:00)

647 Abstract Algebra I

J. Brundan (13:00)

648 Abstract Algebra II

J. Brundan (13:00)

649 Abstract Algebra III

J. Brundan (13:00)

672 Probability
D. Levin (14:00)
673 Probability
D. Levin (14:00)

681 Representation Theory

A. Kleshchev (13:00)

682 Representation Theory

A. Kleshchev (13:00)

683 Representation Theory

A. Kleshchev (15:00)

684 Harmonic Analysis

Y. Xu (12:00)

685 Harmonic Analysis

Y. Xu (12:00)

686: Four-manifolds

P. Lu (14:00)


690 Morse Theory

M. Gomez Lopez (11:00)

 691 Spectral Sequences

B. Botvinnik (9:00)

N. Addington (9:00)

Fall Seminars

 Peter Ralph/Yashar Ahmadian 607 Applied Stochastic Processes

This course will cover a wide range of stochastic processes that are commonly used in modeling, and appear in subjects from biology to physics to finance to combinatorics. The course will focus on building intuition and understanding common mathematical structures, techniques, and applications. Examples will be taken from a variety of applied fields, especially neurobiology and genetics. Topics to be covered include:

(Fall term) Gaussian processes, Poisson processes, Markov chains, generators and semigroups, Brownian motion, the Karhunen-Loeve (spectral) decomposition, Levy processes, and

(Winter term) dynamical systems, bifurcation theory, diffusions and stochastic differential equations, point processes, and random matrix theory.

We will assume basic familiarity with undergraduate probability or statistics (as for instance in Pitman, “Probability”, or Ross, “A first course in probability”), linear algebra, and differential equations.

 Arkady Vaintrob (12:00) 607 Moduli Spaces of Curves

A moduli space is, roughly speaking, a parameter space whose points represent (equivalence classes of) geometric objects of a certain kind. They are ubiquitous in different areas of mathematics and include some of the most important spaces: projective spaces, Grassmanians and flag varieties, Hilbert schemes and quiver varieties, Jacobians, etc.

In this course we will study the moduli spaces of complex algebraic curves which play a central role not only in classical geometry, but also in number theory, topology, complex analysis, dynamical systems, group theory, combinatorics, and theoretical physics. We will start with a brief review of algebraic curves, then describe various ways of constructing their moduli spaces and finally focus on their geometric, topological and combinatorial properties. In particular, we will define and study special (tautological) homology and cohomology classes on these and related moduli spaces. This course would be a good preparation for the winter course on mirror symmetry.

 Chris Phillips (14:00) 607 Set Theory

In the first several weeks, we intend to present a mostly complete proof of Goedel’s Incompleteness Theorem. The rest of the course will be devoted to forcing and some applications, particularly independence of the Continuum Hypothesis from Zermelo-Frankel set theory with the Axiom of Choice.

Formal logic is not a prerequisite, but will also not be presented in detail. Instead, an overview will be given of how it works, which, I hope, will be sufficient for understanding the ideas of the Incompleteness Theorem and forcing.

Winter Seminars

Yashar Ahmadian 607 Applied Stochastic Processes II See above

 Ellen Eischen (10:00-11:20) 607 Number Theory

This two-term algebraic number theory sequence will introduce standard topics essential for students considering working in number theory and also potentially useful for students working in related fields, including algebraic geometry, representation theory, and topology. These courses will focus on the structure of number fields (finite extensions of the rational numbers), including ideal class groups, unit groups, cyclotomic extensions, quadratic reciprocity, special cases of Fermat’s Last Theorem, local fields, global fields, adeles, and an introduction to class field theory. The main prerequisite is abstract algebra at the level of the 500-level algebra sequence (with an emphasis on the Galois theory covered in Math 546), as well as elementary knowledge of modules. Students who have seen some commutative algebra (at the level covered in the 600-level algebra sequence) will be better prepared. Students who have not seen commutative algebra will need to do a little extra work or accept a few facts as black boxes. Topics and pace may be adjusted according to the background and interests of the students enrolled in the course (e.g. since some of these topics have been covered in the student number theory seminar in 2017-2018).

 Yefeng Shen (14:00) 607 Mirror Symmetry

Mirror symmetry has been a driving force in geometry and physics for the last twenty years. In this course, we will give an introduction to the mirror symmetry for toric varieties, Calabi-Yau varieties, and Landau-Ginzburg models.

The majority part of the course will be mirror constructions and mirror symmetry at the level of cohomology. We will introduce some basics of toric geometry and then talk about various mirror constructions, including Batyrev-Borisov’s mirror construction between toric varieties, Hori-Vafa’s mirror construction, and Berglund-Hubsch-Krawitz’s mirror construction between Landau-Ginzburg models.

If time allowed, in the rest part of course, we will introduce genus zero Gromov-Witten invariants and quantum cohomology of projective spaces.

Spring Seminars

Dev Sinha (10:00) Stable Homotopy Theory

Stable homotopy theory, which is essentially the study of objects called spectra, arose from the study of generalized cohomology theories such as K-theory and cobordism. The theory is now realizing its promise to form a basis for “fully derived algebra.”

After preliminaries as to why cohomology theories and infinite loop spaces give rise to one another, we will take the point of view that an infinite loop space is like a topological abelian group, but with addition which is coherently homotopy commutative rather than just commutative. We first treat the special case of topological abelian groups, which are generalized Eilenberg-MacLane spaces. We then develop the notion of operad to make the notion of coherent homotopy precise, and discuss examples, starting with Stasheff’s recognition theorem for loop spaces. We will also just begin to discuss the considerable effort needed to go from just having an addition operation to also having a multiplication, that is the theory of ring spectra. Here we will mostly focus on the question of what properties a multiplication needs to have before realizing properties which classical cohomology enjoys (e.g. when would there be a Künneth theorem; when would there be a good theory of cohomology operations) rather than going into details as to how to develop such properties.

In the second half of the term, there are a few directions we could go, depending on the interests of students. These include:
– going more deeply into examples and/or basic applications (e.g. dualities and transfer maps; setting up the Adams spectral sequence).
– more fully developing basic constructions such as function spectra and smash products.
– investigating the interplay between stable and unstable through looking at the homology of infinite loop spaces, and in particular Dyer-Lashof operations.
– a survey (which is all we would have time for) of the deeper structure of the stable homotopy category, called chromatic homotopy theory, which is organized around ideas from algebraic number theory.
– models for ring spectra – most likely orthogonal spectra, with some mention of symmetric spectra in parallel – and then discussion of the higher algebraic approach (that is, seeing the story in coordinates before discussing the coordinate-free version).

I would be open to continuing any such topics as a reading course.

Ellen Eischen (TR 10:00-11:20) 607 Number Theory II See above

Peng Lu (14:00) 4-Manifolds

The world of differential 4-manifolds is fascinating. Simply connected closed topological 4-manifolds are classified by M. Freedman in early 1980’s via a difficult version $h$-cobordism theorem in 4d. Which of these topological manifolds admits smooth structure? There are the results of S. Donaldson which tells when there is not any. The Seiberg-Witten theory can be used to show the existence of exotic smooth structures. Given a topological
manifold there is some smooth structure on which the normalized Ricci flow exists for long time and some other smooth structure on which the normalized Ricci flow develops singularity in finite time. One conjecture says that every closed topological 4-manifold has either zero or (countably) infinitely many distinct smooth structures. The list of stories can go on much longer. The proposed course will be a sampler, we may dwell on some topic based on demands in case the course flies.