Graduate Courses

Fundamentals of complex analysis, and further topics such asconformal mapping, hyperbolic geometry, canonical products, elliptic functions and modular forms.

Prerequisites::

Basic complex analysis, topology of covering spaces, differential forms.

:

Fundamentals of algebraic curves as complex manifolds of dimension one. Topics may include branched coverings, sheaves and cohomology, potential theory, uniformization and moduli.

Recommended Prep:

Knowledge of the material in Mathematics 213a.

:

This class is a graduate level course in Commutative Algebra, also aimed at undergraduates.Commutative Algebra lies at the foundation of many active research areas in mathematics like Algebraic Geometry, Number Theory and Algebraic Topology. It is also an important subject in itself. The different topics we will cover in this class will give a taste of these different directions.We will start by studying rings and ideals, associated prime ideals and primary decomposition in Noetherian rings. We will then study dimension theory and briefly touch upon Zariski topology. Then we will move to study integral extensions: going-up, going-down and Noether normalization theorems. The last part of the class will cover valuation rings, graded rings, Hilbert polynomials and homological methods in commutative algebra.

Recommended Prep:

Mathematics 123.

:

Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.

Recommended Prep:

Knowledge of the material in Mathematics 114, 123 and 132.

:

A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.

Recommended Prep:

Knowledge of the material in Mathematics 129.

:

Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate’s thesis or Euler systems.

Recommended Prep:

Knowledge of the material in Mathematics 223a.

:

Fundamental methods, results, and problems of analytic number theory. Riemann zeta function and the Prime Number Theorem; Dirichlet’s theorem on primes in arithmetic progressions; lower bounds on discriminants from functional equations; sieve methods, analytic estimates on exponential sums, and their applications.

Recommended Prep:

Knowledge of the material in Mathematics 113 and 123.

:

Basic properties and examples of smooth manifolds, Lie groups, and vector bundles; Riemannian geometry (metrics, geodesics, Levi-Civita connections, and Riemann curvature tensors); principal bundles and associated vector bundles with their connections and characteristic classes.

Recommended Prep:

Knowledge of the material in Mathematics 132 and 136.

:

Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.

Recommended Prep:

Knowledge of the material in Mathematics 131 and 132.

:

Continuation of Mathematics 231a. Topics will be chosen from: Cohomology products, homotopy theory, bundles, obstruction theory, characteristic classes, spectral sequences, Postnikov towers, and topological applications.

Recommended Prep:

Knowledge of the material in Mathematics 231a.

:

Introduction to complex algebraic curves, surfaces, and varieties.

Recommended Prep:

Knowledge of the material in Mathematics 123, 132, and 137.

:

This is a continuation of the material covered in the first semester, namely a complex analytic introduction to algebraic geometry. Among other things, we will discuss: higher dimensional manifolds, Hodge structures, polarizations, complex tori and abelian varieties, deformations of complex structures.

Recommended Prep:

Knowledge of the material in Mathematics 232a.

:

This course introduces basic concepts of mathematical biology and evolutionary dynamics: reproduction, selection, mutation, genetic drift, quasi-species, finite and infinite population dynamics, game dynamics, evolution of cooperation, language, spatial models, evolutionary graph theory, infection dynamics, virus dynamics, somatic evolution of cancer.

Recommended Prep:

Mathematics 19a,b or 21a,b or 22a,b or 23a,b or 25a,b or 55a,b; or an equivalent background in mathematics.

:

Research seminar on evolutionary dynamics, spanning mathematical and computational models of evolution in biological and social systems. Students attend a weekly lecture and conduct an original research project.

Recommended Prep:

Experience with mathematical biology at the level of Mathematics 153.

:

This course will cover the recent work of Burklund-Hahn-Levy-Schlank giving a counterexample to the Ravenel’s famous Telescope Conjecture. An understanding of chromatic homotopy theory will be assumed.

Additional Course Attributes:

This is the first semester of a year-long course centered aroundconcepts in quantum mechanics and quantum field theory. We treatthe subject from a mathematical perspective that emphasizesstructural aspects and applications to mathematics. Some lectureswill be devoted to background mathematics. Topics include: quantummechanics, relativistic quantum field theory, Segal’s axioms forWick-rotated theories, topological field theories, invertible fieldtheories, supersymmetry, supersymmetric field theories.

Recommended Prep:

The course assumes mathematical knowledge at the level of basic graduate courses; no physics is assumed.

:

This course will be an introduction to quantum topology, a branch of low-dimensional topology informed by Chern-Simons theory and its generalizations. Tentative topics include: quantum groups and their representation theory, semisimple and non-semisimple quantum link and 3-manifold invariants, skein modules, and knot homologies.

:

This is the second semester of a year-long course centered aroundconcepts in quantum mechanics and quantum field theory. We treatthe subject from a mathematical perspective that emphasizesstructural aspects and applications to mathematics. Some lectureswill be devoted to background mathematics. Topics include: quantum mechanics, relativistic quantum field theory, Segal’s axioms for Wick-rotated theories, topological field theories, invertible field theories, supersymmetry, supersymmetric field theories.

Recommended Prep:

The course assumes mathematical knowledge at the level of basic graduate courses; no physics is assumed.

:

This class will be an introduction to topological and geometriccombinatorics at the graduate level, covering several general areas: (1) Posets (2) Simplicial complexes (3) Matroids (4) PolytopesOne of the main themes of the class will be the question “To whatextent do combinatorial properties of an object determine its topologyor geometry?” For example, to what extent does the face lattice of asimplicial or cell complex determine its homotopy type? To what extent does the graph of a polytope determine the polytope? And what kinds of combinatorial techniques can we use to then understand the topology of the object in question? Along the way we will discuss interesting examples coming from posets, polytopes, matroids, Coxeter groups, the Grassmannian and flag varieties, etc.

:

Introduction to 3-manifolds with some sutured structures on their boundaries and various constructions of the Floer homology for sutured manifolds using Heegaard Floer theory, monopole (Seiberg-Witten) theory, and instanton (Donaldson-Floer) theory.

:

A survey of fundamental results and current research in several interacting areas. Topics may include: hyperbolic manifolds in dimensions 2 and 3; arithmetic, ergodic theory and rigidity; moduli spaces of Riemann surfaces and 1-forms; conformal dynamical systems.

:

This course is an introduction to higher-dimensional algebraic dynamics through the lens of entropy, the most fundamental dynamical invariant. We will introduce three flavors of entropy — algebraic, arithmetic, and topological — and prove the powerful comparison theorems of Gromov-Yomdin and Kawaguchi-Silverman that relate them. Along the way, we will compute the algebraic entropy, or dynamical degree, for some guiding examples: self-maps of algebraic tori, abelian varieties, the projective plane, and Markoff surfaces.

:

Arithmetic statistics can be thought of as the study sequences of arithmetic interest, such as the number of divisors of integers or the number of points on an elliptic curve over finite fields. In this course we’ll encode these sequences in “automorphic forms” and then extract statistical information using techniques from analytic number theory. We’ll focus primarily on explicit calculations involving the spectral decomposition of weight 0 GL2 forms to study shifted convolutions.

Recommended Prep:

Some basic number theory (Dirichlet characters) and basic complex analysis (Cauchy residue theorem).

:

In this course, we will develop the tools and techniques for study the of nonlinear elliptic PDEs and their applications to geometric problems. We will begin with the study of Laplace equation on R^n and discuss basic estimates via mean-value properties, maximum principles, and integral methods. Next we will see how each of these methods can be applied to the study of linear elliptic equations, and also discuss the classical estimates of Schauder. Then we will move on to the theory of DeGiorgi-Nash-Moser and how it can be applied to the study of semi-linear equations and the solution of Hilbert’s 19th problem. Lastly, I would like to discuss the modern theory of fully nonlinear equations. Throughout the course, we will also discuss various examples of how the theory can be applied to the study of specific problems arising from geometry and physics, such as the regularity of minimal surfaces, the Minkowski problem in differential geometry, and the Calabi-Yau problem in K\”ahler geometry.

:

Infinity categories are categories up to infinite coherence. Many questions and constructions in algebraic topology can be formulated in the language of infinity categories, where they are clarified by the presence of new structures and phenomena. This will be an introductory course on the theory of infinity categories, following the book ” Introduction to Infinity-Categories” by Markus Land, and lecture notes from Charles Rezk. It will be expected that participants have familiarity with basic algebraic topology and some fluency in the language of categories.

:

An important class of profinite groups often occuring in number theory is known as p-adic analytic groups, which in many ways behave like a p-adic analogue of Lie groups. This course will be an introduction to p-adic analytic groups, with an emphasis on how their group-theoretic properties are reflected in their cohomology ring. One example guiding our journey through this beautiful subject will be the Morava stabilizer group, whose cohomology forms the building blocks of the stable homotopy groups of spheres.

:

This course is a case study in the interconnectivity of math. We will begin with Kepler’s laws of planetary motion, pass through the Atiyah—Singer index theorem and elliptic cohomology, and end at the spectrum of topological modular forms. This is a beautiful story involving a wide range of topics including operator theory, modular forms, elliptic curves, string theory, and homotopy theory. The real goal of this course is to encounter and learn about various branches of math in an organic manner, with the historical development of topological modular forms serving as our leitfaden.

:

In this course, we will discuss methods for studying the distributions of arithmetic objects, such as polynomials, solutions to Diophantine equations, class groups of number fields, and more. We will focus on the parametrize- and-count approach developed in the work of Bhargava and his collaborators. This approach consists of two steps — an algebraic parametrization step in which one describes the desired arithmetic objects in terms of orbits of a representation — and an analytic counting step in which one determines asymptotic counts of these orbits. Thus, the course will feature techniques from diverse subfields of mathematics.

:

The course will focus on computations of different spectral sequences in motivic stable homotopy theory. Topics include: the motivic Adams spectral sequence over complex numbers, real numbers and finite fields; the $\rho$- Bockstein spectral sequence; the motivic slice spectral sequence.

:

We will study recent advances in number theory. One of the aims of the class is to help participants improve their expository skills by giving talks on recent number theory papers, or on their own work in this subject.

:

Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students.

: