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Francisco Villacis, Department of Pure Mathematics, University of Waterloo

"An Overview of the Gelfand-Cetlin System"

The Gelfand-Cetlin system is an integrable system first introduced by V. Guillemin and S. Sternberg in 1983 in order to study the quantization of complex flag varieties. This integrable system shares many properties with moment maps coming from torus actions, such as having a polytope as image and the fibres above the interior points of the polytope are Lagrangian tori. There is one big difference between toric moment maps and the Gelfand-Cetlin system: the latter allows for Lagrangian fibres which are not tori. This phenomenon makes this system an important object to study in the context of mirror symmetry.  In this talk, I will give a brief overview of the Gelfand-Cetlin system and discuss the classification by Y. Cho, Y. Kim and Y-G. Oh of the Lagrangian fibres.

MC 5417

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Effective Descriptive Set Theory 4"

We will continue to introduce effective descriptive set theory following Andrew Marks's notes.

MC 5479

Frederik Benirschke, University of Chicago

"Isometric embeddings and totally geodesic submanifolds of Teichmüller spaces"

Classical results by Royden, Earle, and Kra imply that the biholomorphism group of Teichmüller space, the isometry group of the Teichmüller metric, and the mapping class group of the underlying surface are all isomorphic. In other words, every isometry of Teichmüller space is induced by a homeomorphism of the underlying surface.

In this talk, we present a generalization, obtained in joint work with Carlos Serván, where we relax isometries to isometric embeddings. The main result is that isometric embeddings of Teichmüller spaces are coverings constructions, except for some low-dimensional special cases. In other words: Isometric embeddings are induced by branched coverings of the underlying surfaces.

Time permitting, we explain how our techniques can be used to rule out the existence of certain totally geodesic submanifolds.

QNC 2501

Sun Woo Park, University of Wisconsin-Madison

"On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields"

Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer, Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erd\"os-Kac theorem, and the geometric ergodicity of Markov chains.  

MC 5501

Jason Bell, Department of Pure Mathematics, University of Waterloo

"Hsia kernels"

We’ll look at the Hsia kernel, a basic tool for potential theory on the Berkovich affine line.

MC 5479

Frederik Benirschke, University of Chicago

"An introduction to bi-algebraic geometry, with a view towards strata of differentials"

I will introduce some of the main ideas in bi-algebraic geometry, focusing on the case of algebraic tori and Abelian varieties. Afterward, we explore these ideas for a natural bi-algebraic structure on the moduli space of marked points on the sphere that arises from integrating differential forms.

MC 5403

Steven Rayan, University of Saskatchewan

“Moduli Spaces and Quantum Matter: From Materials to Pure Mathematics and Back"

The advent of topological materials, a form of physical matter with unusual but useful properties, has brought with it unexpected new connections between physics and pure mathematics. As the name suggests, topology has played a significant role in understanding and classifying these materials. In this talk, I will offer a brief look at a vast extension to this story, arising from my work with various collaborators over the last three years, that sees complex algebraic geometry — in particular, Riemann surfaces and moduli spaces associated to them — being used to anticipate new models of quantum matter. There will be lots of pictures.

MC 5501

Natasha Dobrinen, University of Notre Dame

"Cofinal types of ultrafilters on measurable cardinals"

Nicole Kitt, Department of Pure Mathematics, University of Waterloo

"Characterization of Cofree Representations of SL_n\times SL_m"

Michael Lipnowski, Ohio State University

"Rigid Meromorphic Cocycles for Orthogonal Groups"