# Colloquium

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday February 6, 2023 at 16:00, Wachman 617
An aspect of the geometry of p-Schatten spaces and its relation to the hyperplane conjecture

Kavita Ramanan, Brown University

The hyperplane conjecture in convex geometry is a statement about the volume of a convex body and that of its hyperplane sections. Taking a measure-theoretic perspective to this problem, Bourgain highlighted the importance of the notion of a $\psi_2$-convex body, which captures integrability properties of linear images of the volume measure on the body. Despite this notion being introduced more than a quarter century ago, there are not many examples of such bodies. We describe several results on the $\psi_2$ (or more generally, $\psi_\alpha$) behavior of Schatten balls and their marginals, and their relation to the hyperplane conjecture. Along the way, we also establish some properties of the Haar measure on the orthogonal group that may be of independent interest. This is joint work with Grigoris Paouris.

• Monday February 13, 2023 at 16:00, Wachman 617
From graphs to algebra and geometry

Ralph Kaufmann, Purdue University

Graphs are a universal tool in mathematics. Their study is interesting in its own right. They also naturally appear in the geometry of surfaces. When coupled with additional structures such as local orders or other local information in the form of so-called ribbon graphs. We will present a unified approach to these structures and explore their relationship with algebra, topology and geometry.

• Monday March 13, 2023 at 16:00, Wachman 617
Stark's Conjectures and Hilbert's 12th Problem

Samit Dasgupta, Duke University

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field; this question lies at the core of Hilbert's 12th Problem. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to the question of explicit class field theory for these fields.

• Monday March 27, 2023 at 16:00, Wachman 617
Lifting Galois representations to the Grothendieck-Teichmueller group and beyond

Frauke Bleher, University of Iowa

This is joint work with A. Lubotzky and T. Chinburg. It is a longstanding problem in Galois theory to give an explicit description of the absolute Galois group G_Q of the rationals. A famous theorem by Belyi from 1979 says that G_Q can be embedded into the automorphism group A of a free profinite group on two generators. This theorem led Grothendieck, Drinfel'd and others to try to identify G_Q inside A as a subgroup satisfying various extra relations. Each set of such relations defines a so-called Grothendieck-Teichmueller group GT inside A that is a candidate for being G_Q. It has been an open problem since the 1990's to identify natural non-abelian representations of G_Q that can be shown to lift to such GT groups. We will show that after passing to subgroups of finite index, this can be done for various natural families of representations, e.g. for the actions of subgroups of finite index in G_Q on all of the finite subgroups of the points of an elliptic curve defined over a number field. The main tool is a construction of linear representations of the automorphism group of a free profinite group on finitely many generators that generalizes work of Grunewald and Lubotzky on linear representations of the automorphism group of a finitely generated free discrete group.

• Monday April 3, 2023 at 16:00, Wachman 617
Algebraic methods in extremal graph theory

Michael Tait, Villanova University

How many edges may a graph with no triangle have? Given a graph F, the Turan problem asks to maximize the number of edges in a graph on n vertices subject to the constraint that it does not contain F as a subgraph. In this talk, we will discuss constructions for this problem coming from finite geometry (eg using projective planes), additive combinatorics, and "random polynomials".

• Monday April 10, 2023 at 16:00, Wachman 617
Rational points on elliptic curves (and their p-adic construction)

Negative answer to Hilbert's 10th problem tells us that determining whether or not an algebraic variety should carry any rational points is impossibly hard (literally!). The same problem even for curves is very difficult: For elliptic curves, this is the subject of the celebrated Birch and Swinnerton-Dyer conjecture. I will survey recent results on this problem, and explain briefly an explicit p-adic analytic construction of rational points of infinite order on elliptic curves of rank one (settling a conjecture of Perrin-Riou). These final bits of the talk will be a report on joint works with Rob Pollack & Shu Sasaki, and with Denis Benois.

• Monday April 24, 2023 at 16:00, Wachman 617
TBA

Natesh Pillai, Harvard University

TBA

• Monday September 11, 2023 at 16:00, Wachman 617
TBA

Ricardo Nochetto, University of Maryland

TBA

• Monday October 16, 2023 at 16:00, Wachman 617
TBA

Ian Tobasco, University of Illinois Chicago

TBA