# Difference between revisions of "Colloquia"

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(reserved by the hiring committee) | (reserved by the hiring committee) | ||

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+ | '''Degenerations of curves, rational points, and arithmetic topology''' | ||

+ | |||

+ | Number theory has a rich history of long standing open problems that are fairly easy to state, but are notoriously difficult to answer. The most famous among these is Fermat's Last Theorem, whose solution spurred the development of many technical tools in use today. The quest to find explicit methods to solve other Diophantine equations continues. | ||

+ | |||

+ | A recent method that has had spectacular success in finding rational points on curves that were previously out of reach is the "Quadratic Chabauty" method. The explicit implementation of the Quadratic Chabauty method is a formidable computational challenge. This talk will feature a simplification of the Quadratic Chabauty method using geometric ideas, developed jointly with Besser and Mueller. Using ideas inspired by topology, we will outline new results (joint with Li, Litt and Salter) that show that most curves have no rational solutions at all, guided by Grothendieck's Section Conjecture. The key is to study degenerations in families of curves. The talk will close with various natural ways of measuring degenerations in families of curves (such as the conductor and the discriminant) and their interrelationship. | ||

== December 10, 2021, Friday at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [http://www-personal.umich.edu/~apisa/ Paul Apisa] (University of Michigan) == | == December 10, 2021, Friday at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [http://www-personal.umich.edu/~apisa/ Paul Apisa] (University of Michigan) == |

## Revision as of 13:00, 3 December 2021

**UW Madison mathematics Colloquium is on Fridays at 4:00 pm. **

# Fall 2021

## September 17, 2021, Social Sciences 5208 + Live Stream, Mark Shusterman (Harvard)

(hosted by Gurevich)

**Finitely Presented Groups in Arithmetic Geometry**

I will report on recent works, in part joint with Esnault—Srinivas, and with Jarden, on the finite presentability of several (profinite) groups arising in algebraic geometry and in number theory. These results build on a cohomological criterion of Lubotzky involving Euler characteristics. I will try to explain the analogy, rooted in arithmetic topology, between these results and classical facts about fundamental groups of three-dimensional manifolds.

## September 24, 2021, B239 + Zoom stream, Sean Paul (UW-Madison)

**The Tian-Yau-Donaldson conjecture for general polarized manifolds**

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle L on a compact complex manifold X should be equivalent to an algebro-geometric "stability condition" satisfied by the pair (X,L). The cscK metrics are the critical points of Mabuchi's K-energy functional M, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff M satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical general account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometryin the spirit of Arakelov, Faltings, and Bismut-Gillet- Soule.

## October 1, 2021, B239 + Live stream, Andrei Caldararu (UW-Madison)

**Yet another Moonshine**

The j-function, introduced by Felix Klein in 1879, is an essential ingredient in the study of elliptic curves. It is Z-periodic on the complex upper half-plane, so it admits a Fourier expansion. The original Monstrous Moonshine conjecture, due to McKay and Conway/Norton in the 1980s, relates the Fourier coefficients of the j-function around the cusp to dimensions of irreducible representations of the Monster simple group. It was proved by Borcherds in 1992.

In my talk I will try to give a rudimentary introduction to modular forms, explain Monstrous Moonshine, and discuss a new version of it obtained in joint work with Yunfan He and Shengyuan Huang. Our version involves studying the j-function around CM points (so-called Landau-Ginzburg points in the physics literature) and expanding with respect to a coordinate which arises naturally in string theory.

## October 8, 2021, Zoom + live video on the 9th floor, Jon Chapman (University of Oxford)

(**Wasow lecture**; hosted by Thiffeault)

**Asymptotics beyond all orders: the devil's invention?**

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.

The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, and Hele-Shaw flow.

## October 11, 13, 15, 2021, Zoom, **[Mon, Wed, Fri 4-5pm]**, Geordie Williamson (University of Sydney)

(**Distinguished Lecture Series**; hosted by Gurevich)

**Geometric representation theory and modular representations**

Representation theory is the study of linear symmetry. We are interested in all ways in which a group can arise as the symmetries of a vector space. Representation theory is a remarkably rich subject, with deep connections to number theory, combinatorics, algebraic geometry, differential geometry, theoretical physics and beyond. This lecture series will focus on modular representations, i.e. those representations where our vector spaces are over a field of characteristic p. I will try to highlight some of the main questions in the field and why we are interested in answering them. It is remarkable how much is still unknown and how hard some of these questions are. I will explain the role played by geometric representation theory in our attempts to understand these questions. A fascinating blend of algebra, algebraic geometry, category theory and algebraic topology is informing our understanding of basic questions. Much remains to be understood!

## October 22, 2021, Zoom, Vera Serganova (UC Berkeley)

(hosted by Gurevich/Gorin)

**Supersymmetry and tensor categories**

I will explain how representation theory of supergroups and supergeometry are related to general theory of tensor categories, present old and new results and open questions in the field. We will see how universal tensor categories can be constructed using supergroups and discuss analogy between super representation theory and representation theory over the fields of positive characteristic.

## October 29, 2021, Zoom, Alexandru Ionescu (Princeton University)

(hosted by Wainger)

**Polynomial averages and pointwise ergodic theorems on nilpotent groups**

I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.

## November 5, 2021, B239 + Live stream, Jayadev S. Athreya (University of Washington)

(hosted by Uyanik)

**Surfaces and Point Processes**

We'll give several concrete examples of how to go from the geometry of surfaces to the study of point processes, following work of Siegel, Veech, Masur, Eskin, Mirzakhani, Wright, and others. We'll discuss how this "probabilistic" perspective helps inform both the direction of questions one asks, as well as providing ideas of how to prove things. We'll discuss some pieces of joint work with Cheung-Masur, Margulis, and Arana-Herrera.

## November 12, 2021, Zoom, Kasso Okoudjou (Tufts University)

(hosted by Stovall)

**An exploration in analysis on fractals **

Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.

## November 19, 2021 , B239 + Live stream, Albert Ai(UW-Madison)

(reserved by the hiring committee)

** Low regularity solution for quasilinear PDEs**

In this talk, we will consider the low regularity well-posedness problem for a pair of quasilinear dispersive PDEs: the nonlinear wave equation, and the water waves equations. Two classical methods, energy estimates and Strichartz estimates, have historically yielded substantial but partial results toward advancing the low regularity theory. We will see how, using a special structure of the equations known as a normal form structure, combined with tools from harmonic and microlocal analysis, we can refine these classical methods to drastically improve the known results for low regularity well-posedness.

## December 1, 2021, Wednesday at 4pm in B239 + Zoom stream, Brian Lawrence (UCLA)

(reserved by the hiring committee)

**Integral points on moduli spaces**

Mordell's conjecture, now a theorem of Faltings, states that an algebraic curve of genus at least two has only finitely many rational points. Recent work with Venkatesh gives a new proof of Mordell's conjecture; the method gives some hope of proving finiteness results for any variety (even of higher dimension) that can be realized as a moduli space. I'll discuss some recent results in this direction.

## December 3, 2021, Friday at 4pm on ZOOM + live video in B239, Martino Lupini (Victoria University of Wellington)

(reserved by the hiring committee)

**Borel-definable Algebraic Topology**

In this talk, I will explain how ideas and methods from logic can be used to obtain refinements of classical invariants from homological algebra and algebraic topology. I will then present some applications to classification problems in topology. This is joint work with Jeffrey Bergfalk and Aristotelis Panagiotopoulos.

## December 6, 2021, Monday at 4pm on ZOOM, Michael Lipnowski (McGill)

(reserved by the hiring committee)

**Story about a dodecahedron**

The Seifert-Weber dodecahedral space is a famous closed hyperbolic 3-manifold, one of the first to be discovered. I'll describe some computations that I've done, together with Francesco Lin, on the dodecahedral space and some questions about small eigenvalues on hyperbolic manifolds which motivated them in the first place. I'll also raise a question about (unlikely) intersections of geodesics on hyperbolic manifolds inspired by these computations.

## December 8, 2021, Wednesday at 4pm in B239, Padmavathi Srinivasan (University of Georgia)

(reserved by the hiring committee)

**Degenerations of curves, rational points, and arithmetic topology**

Number theory has a rich history of long standing open problems that are fairly easy to state, but are notoriously difficult to answer. The most famous among these is Fermat's Last Theorem, whose solution spurred the development of many technical tools in use today. The quest to find explicit methods to solve other Diophantine equations continues.

A recent method that has had spectacular success in finding rational points on curves that were previously out of reach is the "Quadratic Chabauty" method. The explicit implementation of the Quadratic Chabauty method is a formidable computational challenge. This talk will feature a simplification of the Quadratic Chabauty method using geometric ideas, developed jointly with Besser and Mueller. Using ideas inspired by topology, we will outline new results (joint with Li, Litt and Salter) that show that most curves have no rational solutions at all, guided by Grothendieck's Section Conjecture. The key is to study degenerations in families of curves. The talk will close with various natural ways of measuring degenerations in families of curves (such as the conductor and the discriminant) and their interrelationship.

## December 10, 2021, Friday at 4pm in B239 + Zoom stream, Paul Apisa (University of Michigan)

(reserved by the hiring committee)

**Billiards, dynamics, and the moduli space of Riemann surfaces**

The Hodge bundle is the space whose points correspond to a Riemann surface equipped with a holomorphic 1-form. This space admits a GL(2, R) action whose dynamics govern the geometry of the moduli space of Riemann surfaces, an object of central importance in geometry, algebra, and physics. I will describe work, joint with Alex Wright, that classifies roughly half of all GL(2, R) orbit closures. I will also describe applications to deceptively simple sounding problems about billiards in polygons. Along the way I will highlight connections to algebraic geometry, homogeneous dynamics, and more.

## December 13, 2021, Monday at 4pm in B239, Nicole Looper (Brown)

(reserved by the hiring committee)

## December 15, 2021, Wednesday at 4pm in B239, Chris Rycroft (Harvard)

(reserved by the hiring committee)

**Uncovering the rules of crumpling with a data-driven approach**

When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].

This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3].

[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018). [2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021). [3] J. Hoffmann et al., Sci. Advances 5, eaau6792 (2019).

## December 17, 2021, Friday at 4pm on ZOOM, Dmitry Chelkak (ENS Paris)

(reserved by the hiring committee)

## December 20, 2021, Monday at 4pm in B239 + Live stream, Matthew Novack (IAS)

(reserved by the hiring committee)

**Turbulent Weak Solutions of the 3D Euler Equations**

The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.