# TBA

Yu-Shen Lin

## Institution:

Boston University

## Time:

Tuesday, May 30, 2023 - 4:00pm

## Location:

ISEB 1200

On Tuesday due to Memorial Day holiday on Monday, 5/29.  Joint seminar with Differential Geometry Seminar.

# TBA

## Speaker:

Catherine Cannizzo

UC Riverside

## Time:

Tuesday, April 11, 2023 - 4:00pm

## Location:

ISEB 1200

Joint with Differential Geometry Seminar.

# TBA

Josh Lieber

UCI

## Time:

Monday, April 17, 2023 - 4:00pm to 5:00pm

RH 340N

# Towards knot homology for 3-manifolds

Aaron Mazel-Gee

Caltech

## Time:

Monday, March 13, 2023 - 4:00pm to 5:00pm

## Location:

RH 340N

The Jones polynomial is an invariant of knots in $\mathbb{R}^3$. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups.

Khovanov homology is a categorification of the Jones polynomial of a knot in $\mathbb{R}^3$, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds.

In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided $(\infty,2)$-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

# Massey Products in Galois cohomology

Federico Scavia

UCLA

## Time:

Monday, February 13, 2023 - 4:00pm to 5:00pm

## Location:

RH 340N

The Borromean rings are three interlinked circles such that no two circles are linked: if we cut or remove one of the circles, the other two fall apart. Massey products are an algebraic manifestation of this phenomenon. Born as part of Algebraic Topology, they have now made a surprising appearance in Number Theory and Galois Cohomology. The Massey Vanishing Conjecture of Minac and Tan predicts that all Masseyproducts in the Galois cohomology of a field vanish as soon as they are defined. In this talk, I will give an informal introduction to Massey products in Topology and Galois Theory, and then describe recent progress on the Massey Vanishing Conjecture, joint with Alexander Merkurjev

# A classification of rank 3 vector bundles on complex projective 5-space

Morgan Opie

UCLA

## Time:

Monday, April 3, 2023 - 4:00pm to 5:00pm

RH 340N

# Normalization in the integral models of Shimura varieties of abelian type

Yujie Xu

MIT

## Time:

Monday, February 27, 2023 - 4:00pm to 4:50pm

## Location:

RH 340N

Shimura varieties are moduli spaces of abelian varieties with extra structures (e.g. algebraic cycles, or more generally Hodge cycles). Over the decades, various mathematicians (e.g. Mumford, Deligne, Rapoport, Kottwitz, etc.) have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type (or more generally abelian type) constructed by Kisin and Kisin-Pappas. I will talk about my recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Ag. I will also mention an application to toroidal compactifications of such integral models.

If time permits, I will also mention a new result on connected components of affine Deligne–Lusztig varieties, which gives us a new CM (i.e. complex multiplication) lifting result for integral models of Shimura varieties at parahoric levels and serves as an ingredient for my main theorem at parahoric levels.

# Resolvent Degree for Arithmetic Groups and Variations of Hodge Structure

Jesse Wolfson

UCI

## Time:

Monday, January 23, 2023 - 4:00pm to 5:00pm

## Location:

RH 340N

In the 13th of his list of mathematical problems, Hilbert conjectured that the general degree 7 polynomial cannot be solved using only arithmeticoperations and algebraic functions of 2 or fewer variables. In the language of resolvent degree, Hilbert conjectures that RD(S_7) = 3. Reichstein has recently extended the notion of resolvent degree to general algebraic groups G. In this context, a conjecture of Tits asserts that RD(G) = 1 for any connected complex linear algebraic group. Reichstein proves unconditionally that RD(G)\le 5 for such G, and he offers this as possible evidence against Hilbert's conjecture. The goal of this talk is to offer analogous evidence *for* Hilbert's conjecture by extending Reichstein's definition to a notion of resolvent degree for arithmetic groups, variations of Hodge structure, and related moduli problems. We then use geometric techniques to give examples of problems F with RD(F) arbitrarily large. From this perspective, one can paraphrase Hilbert's 13th as asking which is a finite group more like: a connected complex linear algebraic group or an arithmetic lattice? This is joint work with Benson Farb and Mark Kisin.

# Exotic four-manifolds and TQFT

Tye Lidman

## Institution:

North Carolina State University

## Time:

Monday, November 21, 2022 - 4:00pm to 5:00pm

## Location:

RH 340N

A major problem in four-dimensional topology is to understand the difference between topological and smooth four-manifolds, e.g. four-manifolds which are homeomorphic but not diffeomorphic. Smooth manifolds are usually studied by considering invariants which count solutions to a PDE on the four-manifold, like the instanton or Seiberg-Witten equations. These invariants are well-behaved on manifolds with nice geometric properties, like positive scalar curvature or symplectic, but their use for closed manifolds has mostly plateaued. In this talk, we will discuss a slightly different perspective on invariants of four-manifolds and, if time, more topology-intrinsic constructions of four-manifolds. This is joint work with Adam Levine and Lisa Piccirillo.

# The geometry of k-Ricci curvature and a Monge-Ampere equation

Lei Ni

UC San Diego

## Time:

Tuesday, May 31, 2022 - 4:00pm

## Location:

ISEB 1200

A joint Geometry and Analysis seminar.

Abstract: The k-Ricci curvature interpolates between the Ricci curvature and holomorphic sectional curvature. For the positive case, a recent result asserts that the compact Kaehler manifolds with positive k-Ricci are  projective and rationally connected. This generalizes the previous results of Campana, Kollar-Miyaoka-Mori for the Fano case and the Heirer-Wong and Yang for holomorphic sectional curvature case. For the negative case, all compact Kaehler manifolds with negative k-Ricci admit a Kaehler-Einstein metric with negative scalar curvature. I shall explain how to get this result by solving a complex Monge-Ampere equation.