This page contains notes that I took (by which I mean live-TeXed in almost all cases) during courses or seminars. If you find any mistakes in the notes, please email me or submit an issue (or pull request) on the git repository. Please note that the notes are not sorted in chronological order even within sections (instead, they are sorted roughly from less to more foundational).
Notes from the first two weeks of the program Recent developments in higher genus curve counting at the Simons Center for Geometry and Physics organized by Qile Chen, Felix Janda, Sheldon Katz, Melissa Liu, John Pardon, and Rachel Webb.
Topics: Topological B-model, Givental formalism, log GLSM, and MSP.
Abstract: In the 1990s, a remarkable correspondence was discovered between the geometry of algebraic curves and infinite-dimensional systems of differential equations. The correspondence has its origin in the study of two-dimensional quantum field theories and is related to many different areas of mathematics. After introducing the relevant objects, I will then state the first result in this story, which was conjectured by Witten and proved by Kontsevich.
Abstract: I will tell you why there are 27 lines on a cubic surface. Along the way, we will meet useful geometric notions such as Grassmannians, Schubert calculus, Chern classes, and intersection theory.
Notes from introductory lectures by Rachel Webb and Tudor Pǎdurariu about Bondal-Orlov reconstruction and Fourier-Mukai transforms at the Derived categories and moduli workshop at Cornell.
Lecture series at Columbia University by Nikita Nekrasov.
Description: Graduate level introduction to modern mathematical physics with the emphasis on the geometry and physics of quantum gauge theory and its connections to string theory. We shall zoom in on a corner of the theory especially suitable for exploring non-perturbative aspects of gauge and string theory: the instanton contributions. Using a combination of methods from algebraic geometry, topology, representation theory and probability theory we shall derive a series of identities obeyed by generating functions of integrals over instanton moduli spaces, and discuss their symplectic, quantum, isomonodromic, and, more generally, representation-theoretic significance. Quantum and classical integrable systems, both finite and infinite-dimensional ones, will be a recurring cast of characters, along with the other (qq-) characters.
Remark: The last lecture I attended was on March 22, 2024. Notes from lectures after this were taken by Davis Lazowski and are included in the file.
Final paper for Symplectic Topology course taught by R. Inanç Baykur. Explains the construction of a simply connected surface of general type using rational blowdowns and its realization as a smoothing of a related surface.
Final paper for Algebraic Topology 2 course taught by Francesco Lin. Constructs the spectrum MU and computes its value on a point using the Adams spectral sequence.
Topics: symplectic manifolds, almost complex structures, Kähler manifolds, geography of complex surfaces and symplectic 4-manifolds, isotopy, symplectic operations on 4-manifolds.