Seeing Space(time)

Speaker: 

Steve Trettel

Institution: 

University of San Francisco

Time: 

Thursday, May 15, 2025 - 2:15pm to 3:05pm

Host: 

Location: 

ISEB 1200

For two thousand years geometry was synonymous with the perfectly flat expanse imagined by Euclid. But nineteenth‑century investigations into the parallel postulate lifted a veil from our eyes, revealing the richer realms charted by Gauss and Riemann. In this talk we’ll take an “insider’s tour” of those curved landscapes.

We begin by thinking carefully about what it means to see, and use this to step inside new geometries, by tracing light rays along their geodesics. Modern computing affords us the ability to make this thought experiment a reality, with interactive ray-traced demos and experiments.  Using these, we will explore curved spaces important to modern mathematics, and physics, including the curved spacetime we live in.

 

 

 

The second rational homology of the Torelli group

Speaker: 

Dan Minahan

Institution: 

UChicago

Time: 

Monday, May 12, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

The Torelli subgroup of the mapping class group of a surface is the subgroup acting trivially on the first homology of the surface.  We will discuss recent joint work with Andrew Putman where we compute the second rational homology of the Torelli group for closed, orientable surfaces of genus at least 6.  Along the way we compute the first twisted homology of the Torelli group with coefficients in the abelianization of the maximal abelian cover of the closed surface.  We also prove some new purely representation theoretic results about certain infinitely presented representations of the symplectic group

Swimming with the current: the impact of research atmosphere on mathematical progress

Speaker: 

Bianca Viray

Institution: 

University of Washington

Time: 

Tuesday, March 11, 2025 - 2:00pm to 3:00pm

Host: 

Location: 

RH 440R

Just as a current impacts the effort a swimmer must make, so too does the research atmosphere in a community or conference affect research output.  In this talk, I will discuss various examples of this, both long-standing programs of others, and many examples that I have experienced or witnessed.  In particular, I will discuss different branches of my research program and how their development was impacted by the atmosphere in conferences, seminars, and research communities. I also discuss what I have learned from times when my actions have created counter currents for others.

Content note: This talk will include some descriptions of harassment.

New Special Lagrangians in Calabi-Yau 3-Folds with Fibrations

Speaker: 

Yu-Shen Lin

Institution: 

Boston University

Time: 

Monday, May 19, 2025 - 4:00pm

Location: 

RH 340P

Special Lagrangian submanifolds, introduced by Harvey and Lawson, are an important class of minimal submanifolds in Calabi-Yau manifolds. In this talk, I will explain common constructions of special Lagrangians and then a gluing construction of a special Lagrangian in Calabi-Yau manifolds with K3-fibrations when the K3-fibres are collapsing. Furthermore, these special Lagrangians converge to an interval or loop of the base of the fibration at the collapsing limit. This phenomenon is similar to holomorphic curves collapsing to tropical curves in special Lagrangian fibrations and is only a tip of the iceberg of the Donaldson-Scaduto conjecture. This is a joint work with Shih-Kai Chiu.

Hochschild Cohomology and Higher Centers

Speaker: 

Sonja Farr

Institution: 

University of Nevada, Reno

Time: 

Monday, April 21, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Kontsevich’s formality theorem in deformation theory asserts that the Hochschild-Kostant-Rosenberg map lifts to a quasi-isomorphism of homotopy Gerstenhaber algebras between polyvector fields and polydifferential operators. By exhibiting the sheaf of polydifferential operators as a universal enveloping algebra, this can be interpreted as a type of Duflo isomorphism. In this talk, I will present my work towards understanding this geometric Duflo isomorphism in the context of ∞-operads and higher centers. In particular, I will explain how higher centers can be used to equip the Hochschild cochain complex with a universal property.

Computations in Representation Stability

Speaker: 

Emil Geisler

Institution: 

UCLA

Time: 

Monday, April 7, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

We recall the notion of representation stability in the context of the cohomology of ordered configuration space of the plane. We give examples of the computation of stable multiplicities by arithmetic methods using the Grothendieck-Lefschetz fixed point formula and describe how these methods lead to a general algorithm and proofs of specific asymptotics. 

Norms and Hermitian K-theory

Speaker: 

Brian Shin

Institution: 

UCLA

Time: 

Monday, February 3, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340N

Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative (i.e. $\mathbb{E}_\infty$) ring spectra. In this talk, I will discuss an algebro-geometric analogue of this framework, called the theory of normed motivic ring spectra. As a particular example of interest, I'll show that (very effective) Hermitian K-theory can be equipped with a normed ring structure.

Renormalized energy, harmonic maps and random matrices

Speaker: 

Antoine Song

Institution: 

Caltech

Time: 

Monday, March 3, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340N

I will introduce a geometric way to study unitary representations of surface groups. I will discuss a notion of renormalized energy, its corresponding harmonic maps into sphere, and their asymptotic or random behavior. The results connect harmonic maps to random matrix theory and representation theory

Cohomology of definable coherent sheaves and definable Picard groups

Speaker: 

Patrick Brosnan

Institution: 

Maryland

Time: 

Monday, January 13, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

Rh 340N

Definable coherent sheaves (with respect to an o-minimal structure) were introduced by Bakker, Brunebarbe and Tsimerman  (BBT) and used as an essential tool in their proof of Griffiths' conjecture that the image of the period map is algebraic.   The category of these definable sheaves on a complex algebraic variety X sits in between the category of algebraic and analytic sheaves.  More precisely, there is a definablization functor taking coherent algebraic sheaves to definable coherent sheaves and an analytification functor going from the category of definable coherent sheaves to the category of coherent analytic sheaves.  This makes them useful for answering questions about analytic maps involving algebraic varieties.  I'll explain these two functors and the concept of o-minimality necessary to define the BBT category of definable coherent sheaves.  Then I'll state a couple of results I obtained recently with Adam Melrod on the cohomology groups of definable coherent sheaves both in the case where X is projective (when, for reasonable o-minimal structures,  the groups are the same as the usual cohomology groups) and the general case (when they very much aren't).

Algebraic points on curves

Speaker: 

Bianca Viray

Institution: 

University of Washington

Time: 

Monday, March 10, 2025 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340N

The Mordell Conjecture (proved by Faltings in 1983) is a landmark result exemplifying the philosophy "Geometry controls arithmetic". It states that the genus of an algebraic curve, a purely topological invariant that can be computed over the complex numbers, determines whether the curve may have infinitely many rational points. However, it also implies that we can never hope to understand the arithmetic of a higher genus curve solely by studying its rational points over a fixed number field. In this talk, we will introduce the concepts of parametrized points and density degree sets and show how they, together with the Mordell-Lang conjecture (proved by Faltings in 1994), allow us to organize all algebraic points on a curve.

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