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Department of Mathematics

Upcoming Colloquia

Memorial Union on sunny day

How do you lift a modular representation of a Galois group?

KEC 1001

Speaker: Ashwin Iyengar

This talk is about the role of Galois representations in the context of the Langlands program. First I will give a historical overview of the types of questions that come up in this area, and how they relate to number theory. Then I'll introduce the deformation theory of Galois representations and present a result, obtained jointly with G. Böckle and V. Paškūnas, which describes the space of lifts of modular representations of the Galois group of a p-adic local field. I will aim to be as non-technical and possible, and attempt to illustrate the broader ideas which motivate my research area. Read more.

An introduction to the Langlands program: Understanding symmetry through number theory

OWEN 101

Speaker: Peter Dillery

The Langlands program predicts a deep connection between representationtheory and number theory. At its heart is the idea that spaces offunctions that are highly symmetric can be broken into "atomic" pieces,and that number theory governs how this decomposition works. This idea isa vast generalization of the decomposition of periodic functions intofrequencies using Fourier analysis.In this talk, I will start by discussing how to organize representationsusing linear algebra and harmonic analysis. I will then explain howintroducing number theory into the picture leads to the proof of Fermat'sLast Theorem, sheds light on the Ramanujan Conjecture, and culminates inthe Arthur multiplicity formula. This last conjecture completely describesa spectral decomposition for representations of Lie groups defined overthe rational numbers. Along the way, I will illustrate how modern numbertheory provides a unifying framework linking these seemingly differentproblems and conclude by discussing my… Read more.

TBA

Speaker: Sergei Pilyugin

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TBA

Speaker: Andrew Hanlon

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Multiscale Modeling, Simulation, and Analysis of Microstructure Evolution in Polycrystalline Materials

Speaker: Yekaterina Epshteyn

In this talk, we will present recent perspectives on mathematical modeling, numerical simulation, and mathematical analysis of the evolution of the grain boundary network in polycrystalline microstructures. This evolution is a very complex, multiscale, and multiphysics process. Our efforts support the solutions of the central problems in materials science, the design of technologies delivering an arrangement of grains that produces a desired set of material properties.  Relevant recent experiments, along with current and future research, will be discussed as well.  Most technologically useful materials–spanning the length scale from meters to nanometers, from aircraft to microprocessors–are polycrystalline. Polycrystals are composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. The changes in the grain and grain boundary structure of polycrystalline materials highly influence their properties, including, but… Read more.

TBA

Speaker: Daniel Nakano

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TBA

Speaker: Ian Zemke

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A Stroll Through Geometric Ideas

Speaker: Laura Schaposnik

During the first half of the talk, we will introduce Higgs bundles, their integrable system, and motivate why they become useful tools to further our understanding in different geometric settings. After describing some dualities they satisfy (not only from mirror symmetry but also via other correspondences such as low-rank isogenies), we will then focus on different methods to understand the Hitchin fibration and branes it contains, and especially its singular fibers (monodromy, transitional geometries, Cayley correspondences). Then, we shall move on to more applied realms and look at how geometric insights can be used to classify viruses, understand the spread of fake news, examine the relationship between COVID and dengue, and address other questions about the world. Read more.

TBA

Speaker: William Leeb

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