Spring 2022 Colloquiums

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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 )

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.

March 11, 2022, David Anderson (UW-Madison)

Stochastic models of reaction networks and the Chemical Recurrence Conjecture

Cellular, chemical, and population processes are all often represented via networks that describe the interactions between the different population types (typically called the species).

If the counts of the species are low, then these systems are most often modeled as continuous-time Markov chains on $Z^d$ (with d being the number of species), with rates determined by stochastic mass-action kinetics. A natural (broad) question is: how do the qualitative properties of the dynamical system relate to the properties of the network? One specific conjecture, called the Chemical Recurrence Conjecture, and that has been open for decades, is the following: if each connected component of the network is strongly connected, then the associated stochastic model is positive recurrent (meaning the model is quite stable).

I will give a general introduction to this class of models and will present the latest work towards a proof of the Chemical Recurrence Conjecture. I will make this talk accessible to graduate students, regardless of their field of study. Some of the new results presented are joint with Daniele Cappelletti, Andrea Agazzi, and Jonathan Mattingly.

March 25, 2022, Friday at 4pm on Zoom. Richard Canary (Michigan)

(hosted by Zimmer)

Hitchin representations of Fuchsian groups

Abstract: The Teichmüller space $\mathcal T(S)$ of all hyperbolic structures on a fixed closed surface $S$ is a central object in geometry, topology and dynamics. It may be viewed as the orbifold universal cover of the moduli space of algebraic curves of fixed genus and also as a component of the space of (conjugacy classes of) representations of $\pi_1(S)$ into $\mathsf{PSL}(2,\mathbb R)$ which is topologically a cell. Hitchin discovered a component $\mathcal H_d(S)$ of the space of (conjugacy classes of) representations of $\pi_1(S)$ into $\mathsf{PSL}(d,\mathbb R)$ which is topologically a cell. Subsequently, many striking analogies between the Hitchin component $\mathcal H_d(S)$ and Teichmüller space $\mathcal T(S)$ were found. For example, Labourie showed that all representations in $\mathcal H_d(S)$ are discrete, faithful quasi-isometric embeddings.

In this talk, we will begin by gently reviewing the parallel theories of Teichmüller space and the Hitchin component. We will finish by reviewing a long term project to develop a geometric theory of the augmented Hitchin component which parallels the classical theory of the augmented Hitchin component (which one may view as the "orbifold universal cover" of the Deligne-Mumford compactification of Teichmüller space). This program includes joint work with Harry Bray, Nyima Kao, Giuseppe Martone, Tengren Zhang and Andy Zimmer.

April 1, 2022, Friday at 4pm in B239 + Zoom broadcast, Priyam Patel (Utah)

(hosted by WIMAW)

Infinite-type surfaces

Surfaces fall into two categories: finite-type and infinite-type. The theory of infinite-type surfaces has been historically less developed than that of finite-type surfaces, but in the last few years, there has been a surge of interest in surfaces of infinite type and their mapping class groups (informally thought of as the groups of topological symmetries of these surfaces). In this talk, I will survey some of the biggest open problems in this quickly growing subfield of geometric group theory and topology, and discuss some of my own recent joint work towards resolving them.

April 8, 2022, Friday at 4pm in B239 + Live stream, Matthew Stover (Temple University)

(hosted by Zimmer)

A geometric characterization of arithmeticity

An old, fundamental problem is classifying closed n-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature -1, that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not at all evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will give an overview, assuming only basic differential topology, of how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics originating in groundbreaking work of Margulis to prove our characterization. Given time, I will mention some more recent developments and open questions.

April 15, 2022, Friday at 4pm in B239 + Live stream, Bernhard Lamel, (Texas A&M University at Qatar)

(hosted by Gong)

Convergence and Divergence of Formal Power Series Maps

Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.

April 25-26-27 (Monday [VV B239], Tuesday [Chamberlin 2241], Wednesday [VV B239]) 4 pm Larry Guth (MIT)

(Departmental Distinguished Lecture series)

Reflections on decoupling and Vinogradov's mean value problem.

Decoupling is a recent development in Fourier analysis that has solved several longstanding problems. The goal of the lectures is to describe this development to a general mathematical audience.

We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory. This problem is about the number of solutions of a certain system of diophantine equations. It was raised in the 1930s and resolved in the last decade.

We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof. The method of decoupling came as a big surprise to me, and I think to other people working in the field. The main idea in the proof of decoupling is to combine estimates from many different scales. We will describe this process and reflect on why it is helpful.

Lecture 1:Introduction to decoupling and Vinogradov's mean value problem. Live stream

In this lecture, we introduce Vinogradov's problem and give an overview of the proof.

Lecture 2: Features of the proof of decoupling.

In this lecture, we look more closely at some features of the proof of decoupling.  The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scales.  The second feature we examine is called the wave packet decomposition.  This structure has roots in quantum physics and in information theory.

Lecture 3: Open problems.

In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*.  In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems.  Hopefully this will give a sense of some of the issues and difficulties involved in these problems.

May 10+12, 2022, Tuesday+Thursday, 12pm on Zoom. Gil Kalai (Hebrew University)

(Hilldale Lectures / Special colloquium)

The argument against quantum computers

In the first lecture I will introduce quantum computers and present an argument for why quantum computers are impossible. From my analysis I will derive general principles for the behavior of noisy quantum systems and will also briefly discuss the recent announcements concerning "quantum computational supremacy,” which conflict with my theory. In the second lecture I will discuss the connection between the possibility of quantum computers, the predictability of complex quantum systems in nature, and the issue of free will. Both lectures are self-contained, intended for a wide audience and assume no background on quantum computers or philosophy.

Lecture I: The Argument Against Quantum Computers

A quantum computer is a new type of computer based on quantum physics. When it comes to certain computational objectives, the computational ability of quantum computers is tens, and even hundreds of orders of magnitude faster than that of the familiar digital computers, and their construction will enable us to factor large numbers and to break most of the current cryptosystems.

We will describe a computational complexity argument against the feasibility of quantum computers. We identify a very low complexity class of probability distributions described by noisy intermediate-scale quantum computers (NISQ computers), and explain why it will allow neither good-quality quantum error-correction nor a demonstration of "quantum supremacy."

The analysis also shows that for a wide range of noise rates NISQ computers are inherently chaotic in the strong sense that their output cannot be predicted even probabilistically. Some general principles governing the behavior of noisy quantum systems in a "world devoid of quantum computers" will be derived.

I will briefly discuss the recent announcements regarding "quantum computational supremacy" by scientists from Google ("Sycamore") and from USTC, which conflict with my theory.

The lecture is going to be self-contained, it is intended for a wide audience, and we assume no prior knowledge of quantum computers.

Relevant papers are: https://arxiv.org/abs/1908.02499 https://arxiv.org/abs/2008.05188 https://arxiv.org/abs/1409.3093 https://arxiv.org/abs/2008.05177

Lecture II: Quantum Computers, Predictability and Free Will

We will discuss the connection between the possibility of quantum computers, the predictability of complex quantum systems in nature, and the issue of free will. The argument regarding the impossibility of quantum computers implies that the future of complex quantum systems in nature cannot be predicted. A more involved argument shows that the impossibility of quantum computation supports the view whereby the laws of nature do not in fact contradict free will. For this philosophical journey, we discuss in parallel the Google “Sycamore” quantum computer of 12 computational units (qubits), and the human- being Alice, whose free will we attempt to analyze.

At the center of the argument is the ambiguity inherent in the way the future is determined by the past; ambiguity that is not expressed in terms of the mathematical laws of physics (which are fully deterministic) but rather in terms of the physical description of the objects we refer to.

The lecture will be self-contained and we will not assume prior background regarding quantum computers or philosophy. (It will also not rely on the first lecture.)

A relevant paper is: https://arxiv.org/abs/2204.02768