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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"

Rahim Moosa, Department of Pure Mathematics, University of Waterloo

"NIP"

This learning seminar will be reading through parts of Pierre Simon’s book “A Guide to NIP Theories”. In this talk I will give a non-technical introduction to the subject, explaining some of the history and motivation behind considering these theories.

MC 5403

Aleksandar Milivojevic, Department of Pure Mathematics, University of Waterloo

"Formality and non-zero degree maps"

Natasha Dobrinen, University of Notre Dame

"Infinite-dimensional Ramsey theory on binary relational homogeneous structures"