Derived Algebraic Geometry Student Seminar Fall 2024

Our current plan is to read 3d CoHA. We will begin by working through PTTV. We are currently meeting in Evans 732 on Tuesdays at 10 a.m. All are welcome; please e-mail me to be added to the mailing list.

If you’d like to join remotely, send me an e-mail so that I open Zoom. Here’s the Zoom link.

Date	Speaker	Title	Abstract	Notes
9/17	John Nolan	A brief overview of DAG and shifted symplectic structures	I’ll begin by giving a (very quick) summary of derived algebraic geometry and why new / prospective members might find it useful. After that, I’ll give a partial definition of shifted symplectic structures, and I’ll discuss some contexts in which shifted symplectic geometry plays a significant role. This talk will be relatively non-technical, and hopefully future talks will fill in the (important) details I omit.	PDF
9/24	Justin Wu	Shifted Symplectic Structures	I will go into detail about shifted symplectic structures, and give some examples. Then I will discuss lagrangian correspondences and a Darboux theorem for (-1)-shifted symplectic stacks.
10/1	Yuji Okitani	Examples of Lagrangians	I will be talking tomorrow (and asking questions) about some examples of Lagrangians from https://arxiv.org/abs/1306.3235, and also about the loop space interpretation of the spaces introduced last week.
10/8	Chris Li	CoHAs	I will be talking Cohomological hall algebras today. Reference is KS11.
10/15	Chris Li	CoHAs with Potential	We will continue talking about cohomological Hall algebras, focusing on those with potential. We also discussed rapid decay cohomology. Reference is KS11.	PDF; Board Photo 1; Board Photo 2
10/22	Elliot Kienzle	CoHAs and BPS Algebras	I’m going to tell u all why CoHA is an algebra of BPS states, and what that even means.	PDF
10/29	Elliot Kienzle	CoHAs and BPS Algebras	Continuation of previous talk
11/5	Elliot Kienzle	CoHAs and BPS Algebras	Finishing previous talks
11/12	Swapnil Garg	Hall Algebras and Stability Conditions	The general goal of this talk and the next is to give an overview of the circle of mathematical ideas connecting Hall algebras, stability conditions, and DT invariants. In this talk, I defined finitary Hall algebras and stability conditions. I then explained (first due to Reineke) how a multiplication in a Hall algebra can be factorized via a stability condition, and how “wall-crossing” shows the equality of different factorizations, allowing us to prove enumerative geometry identities (e.g. involving the quantum dilogarithm) via a homomorphism to a quantum torus. We mainly followed Bridgeland’s notes.
11/19	Swapnil Garg	Hall Algebras, Wall Crossing, and Motivic DT Invariants	Today I will explain how one can define DT invariants of 3-Calabi-Yau categories, and how wall-crossing formulas arise intrinsically, motivating factorization via Hall algebra multiplication. I will then define the motivic Hall algebra and relate the motivic DT series arising from this to the CoHA one. The main references are notes by Kontsevich–Soibelman, (https://arxiv.org/pdf/0910.4315v2), along with the associated papers, and some notes by Bridgeland.
11/26	Jacob Erlikhman	CoHAs and Vanishing Cycles	After giving a quick run-through of vanishing cycles by examples, I plan to first introduce the critical CoHA and explain its relation to the rapid decay CoHA for moduli of quiver representations (with potential). I will then explain why this is insufficient for more general 3CY categories, motivating the modern definition by Joyce et al: The CoHA is the cohomology of the moduli stack with coefficients in a certain perverse sheaf, which locally looks like a sheaf of vanishing cycles. Hopefully, I’ll have time to indicate some of the ingredients which go into the definition of this sheaf, as well as their significance. The references are (parts of) KS11 Section 7 and KPS24 Section 3, supplemented by the ~5 previous papers of Joyce and collaborators.