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2020

Algebraic D-Modules MATH731 Fall 2020 Bhargav Bhatt Course Website

The primary goal of this course is to develop the algebraic theory of D-modules on an algebraic variety over a field of characteristic 0. Our perspective will be sheaf-theoretic: our main aim is to explain why D-modules provide a coherent counterpart of constructible sheaves. In particular, we plan to discuss the notion of holonomicity (which is key finiteness condition in this theory), the 6-functor formalism, and (aspects of) the Riemann-Hilbert correspondence.

Berkovich Spaces MATH715 Winter 2020 Mattias Jonsson Course Website

Berkovich spaces are analogues of complex manifolds that appear when replacing complex numbers by the elements of a field equipped with a non-Archimedean valuation, e.g. p-adic numbers or formal Laurent series. They were introduced in the late 1980’s by Vladimir Berkovich as a topologically more satisfying alternative to the rigid spaces earlier used by Tate. In recent years, Berkovich spaces have seen a large and growing range of applications. The first part of the course will be devoted to the basic theory of local (affinoid) and global Berkovich spaces. In the second part, we will discuss various applications or specialized topics, partly depending on the interest of the audience.

2019

Introduction to Hodge Theory MATH731 Fall 2019 Mircea Mustaţă Course Website	The goal of the course is to give an introduction to the basic results in Hodge theory. Prerequisites: familiarity with algebraic varieties and sheaf cohomology (no familiarity with scheme theory is required) and with smooth manifolds (the tangent bundle, differential forms, integration).
Commutative Algebra II MATH615 Winter 2019 Mel Hochster Course Website	These lectures will deal with several advanced topics in commutative algebra, including the Lipman-Sathaye Jacobian theorem and its applications, including, especially, the Briançon-Skoda theorem, and the existence of test elements in tight closure theory: basic tight closure theory will be developed as needed. In turn, tight closure theory will be used to prove theorems related to the Briançon-Skoda theorem. Moreover, intersection multiplicities, Hilbert-Samuel multiplicities of local rings, Hilbert-Kunz multiplicities, and other notions will be studied, and interactions with the theory of integral closure of ideals will be developed.
Combinatorial Matrix Theory MATH669 Winter 2019 Sergey Fomin Course Website	This is an introductory graduate course in Combinatorial Matrix Theory, emphasizing its algebraic aspects. Topics included: Combinatorial techniques in linear algebra. Applications of linear algebra in enumerative combinatorics. Special classes of matrices. Determinantal identities. Matroids and projective geometry. Grassmannians and Schubert varieties. Polynomials with real roots. Total positivity.
Prime Characteristic Methods MATH732 Winter 2019 Karen Smith Course Website	The goal of this course is to introduce the Frobenius morphism and its uses in commutative algebra and algebraic geometry. These "characteristic p techniques" have been used in commutative algebra, for example, to establish that certain rings are Cohen-Macaulay, as in the famous Hochster-Roberts theorem for rings of invariants (over fields of arbitrary characteristic). In more geometric settings, we can analyze or quantify how singular---that is, how far from being smooth---a particular variety may be, or to establish that the singularities are suitably mild.

2018

Local Cohomology MATH615 Winter 2018 Jack Jeffries Course Website

The focus this semester will be on local cohomology. The first part of the class will be on homological Commutative Algebra material: depth, properties of free resolutions, Ext, and Tor, and the structure of injective modules. The second part of the class is on local cohomology and its basic applications: its various definitions, Grothendieck (non)vanishing, arithmetic rank, canonical modules, local duality, Mayer-Vietoris, Hartshorne-Lichtenbaum, and connectedness results. In the last part of the class, we will discuss some of the structural results on local cohomology using differential operators and the Frobenius map.

Cluster Algebras MATH669 Winter 2018 Sergey Fomin Course Website

Cluster algebras are a class of commutative rings constructed via a recursive combinatorial process of "seed mutations." They arise in a variety of algebraic and geometric contexts including representation theory of Lie groups, Teichmüller theory, discrete integrable systems, classical invariant theory, and quiver representations. This course will provide an elementary introduction to the basic notions and results of the theory of cluster algebras, and present some of its most accessible applications. Combinatorial aspects will be emphasized throughout. No special background in commutative algebra, representation theory, or combinatorics is required.

2017

Seminar in Etale Cohomology Fall 2017 Yiwang Chen, Gilyoung Cheong, Angus Chung, Yifeng Huang, Zhan Jiang, Patrick Lenning	This is a seminar organized by Gilyoung Cheong. We basically follow the book "Etale Cohomology" written by J. S. Milne.
Abelian Varieties MATH731 Fall 2017 Bhargav Bhatt Course Website	The goal of the first half of this class is to introduce and study the basic structure theory of abelian varieties, as covered in (say) Mumford’s book. In the second half of the course, we shall discuss derived categories and the Fourier-Mukai transform, and give some geometric applications.
Coxeter Groups MATH665 Fall 2017 David E Speyer Course Website	In the first month of the course, we will cover the structure theory and geometry of Coxeter groups in general. We will then study weak order on Coxeter groups. This is a lattice, with many applications in cluster algebras, representation theory of quivers, and enumerative combiantorics. Finally, we will study Bruhat order, also known as strong order. This is a different partial order on a Coxeter group, with key importance in the geometry of flag varieties and the representation theory of Lie groups.
Commutative Algebra II MATH615 Winter 2017 Mel Hochster Course Website	Topics include: Zariski’s main theorem, structure of smooth, unramified, and étale homomorphisms, henselian rings and henselization, artin approximation, and reduction to characteristic p.

2016

Lie Group MATH637 Fall 2016 Gopal Prasad	This is a basic introduction to Lie groups. We will begin with the definition of Lie groups, and give many examples. We will define the Lie algebra of a Lie group and connect them via left-invariant vector fields and the exponential map (prior knowledge of Lie algebras is not assumed). We will establish some structure theorems in Lie theory. Toward the end of the course we will discuss representation theory, with an emphasis towards representations of compact groups.
Schubert Calculus MATH665 Fall 2016 Thomas Lam Course Website	This class is an introduction to combinatorial aspects of Schubert Calculus. A typical Schubert calculus problem is the following: given four lines in a complex 3-dimensional space, how many other lines intersect all four of these four lines? In modern language, problems of this kind translate into calculations in the cohomology ring of the Grassmannian. We will discuss the related combinatorics of Young tableaux, symmetric functions, Bruhat order, and Schubert polynomials. If time permits, we will also discuss a range of generalizations such as: Schubert Calculus in other Lie types; quantum Schubert calculus; K-theoretic Schubert calculus; affine Schubert calculus; and equivariant Schubert calculus.
Commutative Algebra II MATH615 Winter 2016 Mel Hochster Course Website	This course will deal with several topics including: (1) The theory of Gröbner bases and applications: a lot more about this momentarily. (2) The structure theory of complete local rings. (3) The theory of Cohen-Macaulay rings, but with focus on techniques that show that all local rings are, in some sense, close to being Cohen-Macaulay.
Combinatorial Representation Theory MATH669 Winter 2016 John Stembridge	Our goal is to survey some of the many connections between representation theory and combinatorics. Some of the most interesting results in combinatorics have been derived by means of representation-theoretic tools. Enumeration of plane partitions, unimodality theorems, and the Rogers-Ramanujan identities are all examples of this. In the opposite direction, the symmetry groups that occur most frequently in nature (the symmetric groups, the classical groups) have representations with intrinsic combinatorial structure. The course will be divided into three unequal parts. Part 0 will consist of a self-contained development of the representation theory of finite groups in characteristic 0, and a less self-contained discussion of compact groups. Part 1 will be a detailed study of the representation theory of the symmetric groups and closely related groups, and the applications thereof. If there is time, we hope to discuss the W-graphs of Kazhdan-Lusztig theory. Part 2 will be concerned with a combinatorial approach to the representations of GL(n) and related groups (SL(n), U(n), etc).
Algebraic Geometry II MATH632 Winter 2016 Tyler Foster	This course will cover the foundations of modern algebraic geometry, with a focus on schemes and their morphisms, relative properties of schemes, and coherent sheaves and their cohomology. Here is the textbook written by Ravi Vakil.

Fall 2015

Symmetric Functions MATH665 Fall 2015 Sergey Fomin Course Website	This is an introduction to the foundations of the classical theory of symmetric functions from a combinatorial perspective. Core topics include Young tableaux, Schur functions, and related combinatorial algorithms and enumeration problems. The course will conclude by a survey of applications of symmetric functions to various areas of mathematics such as linear algebra, representation theory, and enumerative geometry.
Commutative Algebra MATH614 Fall 2015 Mel Hochster Course Website	This course is an introduction to commutative algebra with emphasis on the theory of commutative Noetherian rings. One theme in the course will be to explain why every commutative ring is a geometric object. Specific topics covered will include localization, uses of the prime spectrum, integral extensions, structure of finitely generated algebras over a field (Noether normalization), Hilbert's Nullstellensatz, an introduction to affine algebraic geometry, primary decomposition, discrete valuation rings, Dedekind domains, Artin rings, dimension theory, completion, and Hilbert functions. Some category theory will be discussed. This material is particularly useful to students with interests in commutative or non‐ commutative algebra, algebraic geometry, several complex variables, algebraic groups or Lie theory, number theory, or algebraic combinatorics.
Algebraic Topology MATH695 Fall 2015 Igor Kriz Course Website	This course is a continuation of the first year algebraic topology course 592, but is self‐contained and can be used as a first algebraic topology course by those students who have seen the basics of the fundamental group and homology. We begin with a review of homology and cohomology with coefficients, focusing on further related topics such as products, homological algebra, Tor and Ext, group (co)homology and duality theory. We then explore how these concepts lead to derived categories of spaces and modules, basic homotopy theory and also stable homotopy theory.

Before Fall 2015

Linear Algebraic Groups Spring 2015 Shenghao Sun	This is seminar on algebraic groups.
Algebraic Topology Spring 2014 Wenxuan Lu	This is a course at Tsinghua University.
Algebraic Geometry I Fall 2013 Eduard Looijenga Course Website	We begin with developing the dictionary between geometry and algebra in the most basic and elementary way: if k is an algebraically closed field, and A is a finitely generated k-algebra without nilpotent elements, then A can be understood as an algebra of k-valued functions on a topological space, the maximal ideal spectrum of A. This leads us into algebraic geometry over the field k. Soon enough we will recognize the need to replace the maximal ideal spectrum by somewhat larger space, the prime ideal spectrum (=ordinary) spectrum. This may look strange at first, but turns out to do a much better job in geometrically representing algebraic properties. This then will lead us naturally to the notion of a scheme. Along the way we consider some examples in a fair amount of detail.