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.

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.

Contact topology and geometry in high dimensions

Speaker: 

Bahar Acu

Institution: 

Pitzer College and Claremont Graduate University

Time: 

Monday, November 4, 2024 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340N

A very useful strategy in studying topological manifolds is to factor them into “smaller" pieces. An open book decomposition of an n-manifold (the open book) is a fibration that helps us study our manifold in terms of its (n-1)-dimensional fibers (the pages) and (n-2)-dimensional boundary of these submanifolds (the binding). Open books provide a natural framework for studying topological properties of certain geometric structures on smooth manifolds such as "contact structures". Thanks to open books, contact manifolds, odd dimensional smooth manifolds carrying these geometric structures, can be studied from an entirely topological viewpoint. For example, every contact 3-manifold can be presented as an open book whose pages are surfaces and binding is a knot/link. In this talk, we will talk about higher-dimensional contact manifolds and provide a setting where we study these manifolds in terms of 3D open books. We also present various results along with examples concerning geometric and topological aspects of contact and symplectic manifolds along with upcoming work concerning these special fibrations.

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