Cohomology Seminar

2024

Jan. 19, 11am, R638: Adeel

Derived Weil restriction

I'll talk about Weil restriction in derived algebraic geometry, and what this has to do with blow-ups and deformation to the normal cone. Based on forthcoming joint work with Jeroen Hekking and David Rydh.

Jan. 26, 11am, R638: Nawaz

Quantum K-Theory of a Point

I plan to review the basics of (permutation-equivariant) quantum K-theory and discuss computations related to the quantum K-theory of a point. In particular, I'll try to explain how the quantum K-theory of a point is important in many localization computations, and why the high genus theory is particularly difficult compared to the cohomological theory.

Feb. 16, 2pm, R638: Adeel

Virtual orbifold GRR

I will talk about a virtual, relative, and orbifold version of Riemann-Roch. Based on work in progress with Charanya Ravi.

Mar. 1, 2pm, R638: Wille

Braden's hyperbolic restriction theorem

Braden's hyperbolic restriction theorem is a powerful tool for proving functorial properties of D-modules and constructible sheaves. It has a vast amount of applications in the geometric representation theory. In this talk, I will explain a proof of it following the paper of Drinfeld and Gaitsgory and illustrate its significance via examples.

April 30, 2pm, R638: Ionut

Is there a reasonable abelian/non-abelian correspondence for higher genus GW invariants?

The abelian/non-abelian correspondence refers to expressing invariants of a GIT quotient X//G (with X quasiprojective and G reductive) via invariants of a corresponding “abelian” quotient X//T, with T a maximal torus in G. This is well-understood at the topological level (e.g, for cohomology rings), and has also been extended to genus zero Gromov-Witten theory (i.e., to quantum cohomology). I will explain the above story in the first part of the talk, while in the second I will ruminate about my unsuccessful efforts to uncover some version of the correspondence (which I believe to exist) in higher genus GW theory.

May 14, 2pm, R638: Ionut

Continuation.

May 16, 2pm, R638: Nawaz

Pardon's Grothendieck Group of 1-cycles

I'll attempt to explain the construction of John Pardon's Grothendieck group of 1-cycles and how he uses it to give a simple proof of the MNOP conjecture.

May 21, 2pm, R638: Ryota

Tropical cohomology and algebraic cycles

Tropical geometry is related to compactifications and degenerations of algebraic varieties. Tropical cohomology is a tropical analog of singular cohomology, and is much simpler than singular cohomology. I believe that tropical cohomology would be helpful to study cycle class maps. I will explain this story, including an algebraicity type result and a new cohomology theory (in progress), which is much better for our purpose.

2023

Nov. 27, 2pm, R638: Adeel Khan

Derived Fourier transforms

Various sheaf-theoretic incarnations of the Fourier transform have been introduced by Deligne, Sato, Brylinski-Malgrange-Verdier and others. I will discuss how these can be generalized to derived vector bundles (or perfect complexes). After briefly indicating why one might want such a thing, I will sketch a proof that the derived Fourier transform is involutive.

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