On the moduli spaces of ALH*-gravitational instantons

Speaker: 

Yu-Shen Lin

Institution: 

Boston University

Time: 

Monday, June 5, 2023 - 3:30pm

Location: 

RH 340N

Gravitational instantons are defined as non-compact hyperKahler
4-manifolds with L^2 curvature decay. They are all bubbling limits of K3
surfaces and thus serve as stepping stones for understanding the K3 metrics.
In this talk, we will focus on a special kind of them called
ALH*-gravitational instantons. We will explain the Torelli theorem, describe
their moduli spaces and some partial compactifications of the moduli spaces.
This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi,
X. Zhu and S. Soundararajan.

Earlier time and joint seminar with Differential Geometry Seminar.

Homological Mirror Symmetry for Theta Divisors

Speaker: 

Catherine Cannizzo

Institution: 

UC Riverside

Time: 

Tuesday, April 11, 2023 - 4:00pm

Location: 

ISEB 1200

Symplectic geometry is a relatively new branch of geometry.
However, a string theory-inspired duality known as “mirror symmetry” reveals
more about symplectic geometry from its mirror counterparts in complex
geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry
called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS
results were then proved for symplectic mirrors to Calabi-Yau and Fano
manifolds. Those mirror to general type manifolds have been studied in more
recent years, including my research. In this talk, we will introduce HMS
through the example of the 2-torus T^2. We will then outline how it relates
to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam,
Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to
hypersurfaces of higher dimensional tori, otherwise known as “theta
divisors.” This is also joint with Azam, Lee, and Liu.

 

Joint with Differential Geometry Seminar.

Constructing Derived Motivic Measures from Six Functors

Speaker: 

Josh Lieber

Institution: 

UCI

Time: 

Monday, April 24, 2023 - 4:00pm to 5:00pm

Location: 

RH 340N

In this talk, we will show how six functors formalisms (which are central to algebraic geometry) may be used to define derived motivic measures (maps from the K-theory of varieties to other spectra).  In particular, we will use this to construct a derived motivic measurement which lifts the Gillet-Soulé motivic measure.  This addresses a conjecture of Campbell-Wolfson-Zakharevich (in fact, there are potentially several derived lifts).

Towards knot homology for 3-manifolds

Speaker: 

Aaron Mazel-Gee

Institution: 

Caltech

Time: 

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

Host: 

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

Speaker: 

Federico Scavia

Institution: 

UCLA

Time: 

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

Host: 

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

Normalization in the integral models of Shimura varieties of abelian type

Speaker: 

Yujie Xu

Institution: 

MIT

Time: 

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

Host: 

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

Speaker: 

Jesse Wolfson

Institution: 

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

Speaker: 

Tye Lidman

Institution: 

North Carolina State University

Time: 

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

Host: 

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

Speaker: 

Lei Ni

Institution: 

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.

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