Colloquia

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

January 10, 2022, Monday at 4pm in B239 + Live stream + Chat over Zoom, Reza Gheissari (UC Berkeley)

(reserved by the hiring committee)

Surface phenomena in the 2D and 3D Ising model

Since its introduction in 1920, the Ising model has been one of the most studied models of phase transitions in statistical physics. In its low-temperature regime, the model has two thermodynamically stable phases, which, when in contact with each other, form an interface: a random curve in 2D and a random surface in 3D. In this talk, I will survey the rich phenomenology of this interface in 2D and 3D, and describe recent progress in understanding its geometry in various parameter regimes where different surface phenomena and universality classes emerge.

January 17, 2022, Monday at 4pm in B239 + Live stream + Chat over Zoom, Marissa Loving (Georgia Tech)

(reserved by the hiring committee)

Symmetries of surfaces: big and small

We will introduce both finite and infinite-type surfaces and study their collections of symmetries, known as mapping class groups. The study of the mapping class group of finite-type surfaces has played a central role in low-dimensional topology stretching back a hundred years to work of Max Dehn and Jakob Nielsen, and gaining momentum and significance through the celebrated work of Bill Thurston on the geometry of 3-manifolds. In comparison, the study of the mapping class group of infinite-type surfaces has exploded only within the past few years. Nevertheless, infinite-type surfaces appear quite regularly in the wilds of mathematics with connections to dynamics, the topology of 3-manifolds, and even descriptive set theory -- there is a great deal of rich mathematics to be gained in their study! In this talk, we will discuss the way that the study of surfaces intersects and interacts with geometry, algebra, and number theory, as well as some of my own contributions to this vibrant area of study.

January 21, 2022, Friday at 4pm in B239 + Live stream + Chat over Zoom, Nicholas Marshall (Princeton)

(reserved by the hiring committee)

Laplacian quadratic forms, function regularity, graphs, and optimal transport

In this talk, I will discuss two different applications of harmonic analysis to problems motivated by data science. Both problems involve using Laplacian quadratic forms to measure the regularity of functions. In both cases the key idea is to understand how to modify these quadratic forms to achieve a specific goal. First, in the graph setting, we suppose that a collection of m graphs G_1 = (V,E_1),...,G_m=(V,E_m) on a common set of vertices V is given, and consider the problem of finding the 'smoothest' function f : V -> R with respect to all graphs simultaneously, where the notion of smoothness is defined using graph Laplacian quadratic forms. Second, on the unit square [0,1]^2, we consider the problem of efficiently computing linearizations of 2-Wasserstein distance; here, the solution involves quadratic forms of a Witten Laplacian.

January 24, 2022, Monday at 4pm in B239 + Live stream + Chat over Zoom, Rachel Skipper (Ohio State)

(reserved by the hiring committee)

From simple groups to symmetries of surfaces

We will take a tour through some families of groups of historic importance in geometric group theory, including self-similar groups and Thompson’s groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces.

February 11, 2022, at 4pm in B239 + Live stream + Chat over Zoom, Mariya Soskova (UW-Madison)

The e-verse

Computability theory studies the relative algorithmic complexity of sets of natural numbers and other mathematical objects. Turing reducibility and the induced partial order of the Turing degrees serve as the well-established model of relative computability. Enumeration reducibility captures another natural relationship between sets of natural numbers in which positive information about the first set is used to produce positive information about the second set. The induced structure of the enumeration degrees can be viewed as an extension of the Turing degrees, as there is a natural way to embed the second partial order in the first. In certain cases, the enumeration degrees can be used to capture the algorithmic content of mathematical objects, while the Turing degrees fail. Certain open problems in degree theory present as more approachable in the extended context of the enumeration degrees, e.g. first order definability. We have been working to develop a richer “e-verse”: a system of classes of enumeration degrees with interesting properties and relationships, in order to better understand the enumeration degrees. I will outline several research directions in this context.

February 18, 2022, at 4pm in B239 + Video over Zoom, Andreas Seeger (UW-Madison)

Spherical maximal functions and fractal dimensions of dilation sets

We survey old and new problems and results on spherical means, regarding pointwise convergence, $L^p$ improving and consequences for sparse domination.

February 25, 2022, at 4pm in B239 + Live Stream, Rohini Ramadas (Warwick)

(hosted by WIMAW)

Dynamics on the moduli space of point-configurations on the Riemann sphere

A degree-$d$ rational function $f(z)$ in one variable with complex coefficients defines a holomorphic self-map of the Riemann sphere. A rational function is called post-critically finite (PCF) if every critical point is (pre)-periodic. PCF rational functions have been central in complex dynamics, due to their special dynamical behavior, and their special distribution within the parameter space of all rational maps.

By work of Koch building on a result of Thurston, every PCF map arises as an isolated fixed point of an algebraic dynamical system on the moduli space $M_{0,n}$ of point-configurations on the Riemann sphere. I will introduce PCF maps and $M_{0,n}$. I will then present results characterizing the ensuing dynamics on $M_{0,n}$.

This talk includes joint work with Nguyen-Bac Dang, Sarah Koch, David Speyer, and Rob Silversmith.

March 1, 2 and 4, 2022 (Tuesday, Wednesday and Friday), Robert Lazarsfeld (Stony Brook)

(Departmental Distinguished Lecture series)

Public Lecture: Pythagorean triples and parametrized curves

Tuesday, March 1, 4:00pm (Humanities 3650 + Live Stream). Note unusual time and location!

In this lecture, aimed at advanced undergraduate and beginning graduate students, I will discuss the question of when a curve in the plane admits a parameterization by polynomials or rational functions.

Colloquium: How irrational is an irrational variety?

Wednesday, March 2, 4:00pm (VV B239 + Live Stream).

Recall that an algebraic variety is said to be rational if it has a Zariski open subset that is isomorphic to an open subset of projective space. There has been a great deal of recent activity and progress on questions of rationality, but most varieties aren't rational. I will survey a body of work concerned with measuring and controlling “how irrational” a given variety might be.

Seminar: Measures of association for algebraic varieties

Friday, March 4, 4:00pm (VV B239 + Live Stream)

I will discuss some recent work with Olivier Martin that attempts to quantify how far two varieties are from being birationally isomorphic. Besides presenting a few results, I will discuss many open problems and avenues for further investigation.