Here is a list of notes of courses I've taken
Algebraic DModules MATH731 Bhargav Bhatt Course Website 
The primary goal of this course is to develop the algebraic theory of Dmodules on an algebraic variety over a field of characteristic 0. Our perspective will be sheaftheoretic: our main aim is to explain why Dmodules 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 6functor formalism, and (aspects of) the RiemannHilbert correspondence. 
Berkovich Spaces MATH715 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 nonArchimedean valuation, e.g. padic 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. 
Introduction to Hodge Theory MATH731 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 Mel Hochster Course Website 
These lectures will deal with several advanced topics in commutative algebra, including the LipmanSathaye Jacobian theorem and its applications, including, especially, the BriançonSkoda 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 BriançonSkoda theorem. Moreover, intersection multiplicities, HilbertSamuel multiplicities of local rings, HilbertKunz multiplicities, and other notions will be studied, and interactions with the theory of integral closure of ideals will be developed. 
Combinatorial Matrix Theory MATH669 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 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 CohenMacaulay, as in the famous HochsterRoberts theorem for rings of invariants (over fields of arbitrary characteristic). In more geometric settings, we can analyze or quantify how singularthat is, how far from being smootha particular variety may be, or to establish that the singularities are suitably mild. 
Local Cohomology MATH615 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, MayerVietoris, HartshorneLichtenbaum, 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 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. 
Seminar in Etale Cohomology 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 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 FourierMukai transform, and give some geometric applications. 
Coxeter Groups MATH665 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 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. 
Lie Group MATH637 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 leftinvariant 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 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 3dimensional 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; Ktheoretic Schubert calculus; affine Schubert calculus; and equivariant Schubert calculus. 
Commutative Algebra II MATH615 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 CohenMacaulay rings, but with focus on techniques that show that all local rings are, in some sense, close to being CohenMacaulay. 
Combinatorial Representation Theory MATH669 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 representationtheoretic tools. Enumeration of plane partitions, unimodality theorems, and the RogersRamanujan 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.

Algebraic Geometry II MATH632 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. 
Symmetric Functions MATH665 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 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 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. 
Linear Algebraic Groups Shenghao Sun 
This is seminar on algebraic groups. 
Algebraic Topology Wenxuan Lu 
This is a course at Tsinghua University. 
Algebraic Geometry I 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 kalgebra without nilpotent elements, then A can be understood as an algebra of kvalued 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. 