course-notes

Course Notes

These are the (live-TeXed unless noted otherwise) notes for various graduate math courses and seminars that I attended as an undergrad or grad student. If you find any errors, do not hesitate to email me or submit a pull request.

If you treat this as an entire body of work, then anyone who has taken undergrad algebra and analysis can read and understand the material in these notes. However, the notes are most definitely not self-contained, even if you treat them together as a single text (for example, some definitions that were included as remarks during class and I was already comfortable with at the time are deliberately omitted).

Some notes:

• The overall exposition in the algebraic geometry course at Columbia was not very good. Individual topics were explained reasonably well though.
• As much as I like Andrei, the exposition in the Lie groups classes at Columbia is all over the place.
• The exposition in my representation theory class from UMass is notoriously horrible, as much as I like Ivan, and the content about Lie algebras should be considered superseded by the Lie algebras class at UMass and the Lie groups classes at Columbia.

Mathematical programs and workshops

• Integrable systems, enumerative geometry, and quantization program organized by Gaëtan Borot, Alexandr Buryak, Melissa Liu, Nikita Nekrasov, Paul Norbury, and Paolo Rossi (ISEG/)

Gromov-Witten/Pandharipande-Thomas correspondence for toric threefolds, Gromov-Witten/Hurwitz correspondence for curves, Donaldson-Thomas/Hilb correspondence, Nakajima quiver varieties and quantum integrable systems, double ramification cycles, Airy structures and topological recursion, and integrable systems.

Learning Seminars

These are learning seminars that I attended as a graduate student. Use the directory seminars/

• Informal enumerative geometry seminar organized by Melissa Liu (IEG/)

Virtual fundamental classes, Behrend function, Gromov-Witten and Donaldson-Thomas crepant resolution conjectures, and stable envelopes.

• Hyperbolicity seminar organized by Johan de Jong (HYP/)

Notions of hyperbolicity for algebraic varieties and applications to rational curves and arithmetic.

• Geometric Representation Theory seminar organized by Kevin Chang, Fan Zhou, and me (GRT/)

D-modules, the Riemann-Hilbert correspondence, and the Kazhdan-Lusztig conjectures.

• Hodge Theory seminar organized by Kevin Chang (HT/)

• Category O seminar organized by Kevin Chang, Fan Zhou, and me (CO/)

Basic theory of category O, applications to Kazhdan-Lusztig theory, and Koszul duality.

• DAHA and Knot Homology seminar organized by Sam Dehority, Zoe Himwich, and Davis Lazowski (DAHA/)

Knot invariants and their refinements, relation to enumerative geometry, and double affine Hecke algebras.

• Deformation Theory seminar organized by Johan de Jong (DEF/)

Deformation theory of schemes and sheaves, applications to the moduli of curves, and Artin's axioms for algebraic stacks.

• Intersection Theory seminar organized by Caleb Ji and me (INT/)

Intersection theory, moduli spaces, Grothendieck-Riemann-Roch, and motivic cohomology.

• Minimal Model Program seminar organized by Joaquin Moraga (MMP/)

Minimal model program for threefolds.

• Geometric Invariant Theory seminar organized by Anna Abasheva and me (GIT/)

GIT, applications to constructions of moduli spaces, and symplectic reduction.

• FGA Explained seminar organized by Caleb Ji (FGA/)

Grothendieck topology, stacks, and Jacobians of curves.

Courses

Columbia - Graduate School

These are courses that I took or sat in on as a PhD student in the math department at Columbia. Use the directory Columbia/. Folders are named descriptively.

Fall 2021

Use the subdirectory f2021/

• Moduli Spaces and Hyperkähler Manifolds with Giulia Sacca (HK/)

Geometry and topology of hyperkähler varieties and moduli spaces of sheaves on K3 surfaces.

Spring 2021

Use the subdirectory s2021/

• Algebraic Geometry with Giulia Sacca (AG/)

Schemes, sheaves, and cohomology.

• Algebraic Number Theory with Chao Li (NT/)

Local and global class field theory using group cohomology.

• Algebraic Topology with Francesco Lin (AT/)

Serre spectral sequence, K-theory, and the Atiyah-Singer index theorem.

• Lie Groups and Representations with Andrei Okounkov (RT/)

Invariant theory, structure of algebraic groups, Lie algebra cohomology, and Kac-Moody Lie algebras.

Fall 2020

Use the subdirectory f2020/

• Algebraic Topology with Mohammed Abouzaid (AT/)

Homotopy groups, homology, cohomology, and Poincaré duality.

• Commutative Algebra with Eric Urban (CA/)

Rings, flatness, dimension theory, homological aspects, and Cohen-Macaulay/normal/regular rings.

• Lie Groups and Representations with Andrei Okounkov (RT/)

Structure and classification of Lie groups and their finite-dimensional representation theory.

UMass Amherst

These are graduate courses that I took as an undergrad at UMass Amherst. Use the directory UMass/. Folders are named by the course number.

Spring 2020

Use the subdirectory s2020/

• Singular Spaces with Paul Gunnells (math797d/)

Singular spaces, intersection cohomology, local structure, and locally symmetric spaces.

• Symplectic Topology with R. Inanc Baykur (math705/)

Symplectic topology, Kähler manifolds, geography of complex surfaces and symplectic 4-manifolds, and surgery operations and isotopy on symplectic 4-manifolds.

Fall 2019

Use the subdirectory f2019/

• Lie Algebras with Eric Sommers (math718/)

Structure theory and finite-dimensional representation theory of semisimple Lie algebras.

• Topics in Geometry with Mike Sullivan (math703/)

(Unfortunately named) smooth manifolds, vector fields, differential forms, Lie derivatives, and de Rham cohomology.

• Missing: Complexity Theory with David Mix Barrington
  

See here for the course webpage and here for the lecture notes, written by Barrington and Maciel for the 2000 IAS/PCMI summer session. The course itself covered machine classes, circuit classes, first-order logic, monoid classes, and their relationship.

Note: This course was offered by the computer science department, but the material is sufficiently close to math for it to be listed.

Spring 2019

Use the subdirectory s2019/

• Analysis 2 with Robin Young (math624/)

Compactness and convergence in infinite dimensions, Hilbert and Banach spaces, distributions, and Fourier theory.

• Algebraic Geometry with Jenia Tevelev (math797W/)

The basic theory of quasiprojective varieties, local properties, divisors, and differentials.

• Algebra 2 with Paul Gunnells (math612/)

Fields, Galois theory, and a bit of commutative algebra.

Fall 2018

Use the subdirectory f2018/

• Analysis 1 with Robin Young (math623/) (Note: transcribed from handwritten notes)

Lebesgue measure, integration, differentiation, and some general measure theory.

• Representation Theory with Ivan Mirkovic (math797rt/) (Note: transcribed from handwritten notes)

Representation theory of finite groups, some Springer theory, and a bit of semisimple Lie algebras.

• Missing: Algebra 1 with Paul Gunnells

See here for notes for the Fall 2019 version of this course, by Jimmy Hwang. The course itself covers the basic theory of groups, rings, and modules.

Spring 2018

Use the subdirectory s2018/

• Complex Analysis with Paul Hacking (math621/) (Note: transcribed from handwritten notes)

Local theory of holomorphic functions, Riemann mapping theorem, and elliptic functions.