## Seminar Categories

This page lists seminar series that have events scheduled between two months ago and twelve months from now and have speaker information available.

## Current Series

[View Past Series]

Thu Sep 10

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Kinetic theory of structured populations: demographics, cell size control, and stochastic hierarchies
Tom Chou, University of California, Los Angeles

We will briefly review, through two examples, classic deterministic PDE models of population dynamics structured according to attributes such as age and/or size. First, we describe how the original McKendrick model was used to motivate China's one-child policy, and generalize it to include an imposed, finite interbirth refractory period. We quantify the effectiveness of this softer, staggered birth policy and discuss its predicted effectiveness. We then review sizer-timer-adder-type models used to quantify proliferating cell populations. Here, blow-up in mean cell sizes can arise, which represents a challenging numerical problem. Finally, we extend these classic deterministic models to allow for both demographic and growth-rate stochasticity by developing a fully kinetic theory. Marginalization of the full density functions results in a set of coupled kinetic models similar to the BBGKY hierarchy. We map out the different combinations of stochastic descriptions and show how the classic age-dependent population models are connected to this hierarchy, the lowest order of which is a master equation for the total stochastic population. Differences in the stochastic description of birth through budding or splitting are explored.

Thu Sep 17

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
25 years since Fermat's Last Theorem
Frank Calegari, University of Chicago

Wiles's proof of Fermat's Last Theorem was published 25 years ago. Wiles's paper introduced many new ideas and methods which have since shaped the field of algebraic number theory. This colloquium talk intends to give a (biased) tour of these developments, especially with regard to questions that might be of interest to non-specialists.

Thu Sep 24

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Random walks in graph-based learning
Jeff Calder, University of Minnesota, Twin Cities

I will discuss several applications of random walks to graph-based learning, both for theoretical analysis and algorithm development. Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of high-dimensional data for tasks like semi-supervised classification, clustering and dimension reduction. Our first application will be to use the random walk interpretation of the graph Laplacian to characterize the lowest label rate at which Laplacian regularized semi-supervised learning is well-posed. Second, we will show how analysis via random walks leads to a new algorithm that we call Poisson learning for semi-supervised learning at very low label rates. Finally, we will show how stochastic coupling of random walks can be used to prove Lipschitz estimates for graph Laplacian eigenfunctions on random geometric graphs, leading to new spectral convergence results.

This talk will cover joint work with many people, including Brendan Cook (UMN), Nicolas-Garcia Trillos (Wisconsin-Madison), Marta Lewicka (Pittsburgh), Dejan Slepcev (CMU), Matthew Thorpe (University Manchester).

Thu Oct 01

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Richard Kenyon, Yale University

This is joint work with Istvan Prause. Many well-known random tiling models such as domino tilings and square ice lead to variational problems for functions h:R^2->R which minimize a functional depending only on the gradient of h. Other examples of such variational problems include minimal surfaces and surfaces satisfying the "p-laplacian". We give a representation of solutions of such a problem in terms of kappa-harmonic functions: functions which are harmonic for a laplacian with a varying conductance kappa.

Thu Oct 08

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Mean-field disordered systems and Hamilton-Jacobi equations
Jean-Christophe Mourrat, Courant Institute of Mathematical Sciences, New York University

The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that interact through locally simple rules and are influenced by some amount of noise. A celebrated model in this class is the Ising model, where spins can take the values +1 and -1, and the local interaction favors the alignement of the spins.

In this talk, I will mostly focus on the situation where the interactions are themselves disordered, with some pairs having a preference for alignement, and some for anti-alignement. These models, often called "spin glasses", are already surprisingly difficult to analyze when all spins directly interact with each other. I will describe a fundamental result of the theory called the Parisi formula. I will then explain how this result can be recast using suitable Hamilton-Jacobi equations, and what benefits this new point of view may bring to the topic.

Thu Oct 15

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
The diffeomorphism group of a 4-manifold
Daniel Ruberman, Brandeis University

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor's discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group. The lowest dimension for such classical phenomena is 5.

I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)-parameter gauge theory. The construction uses a topological technique.

Thu Oct 22

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
An analytic version of the Langlands correspondence for complex curves
Edward Frenkel, University of California, Berkeley

The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Robert Langlands asked whether it is possible to construct a function-theoretic version. Together with Pavel Etingof and David Kazhdan, we have formulated a function-theoretic version as a spectral problem for (a self-adjoint extension of) an algebra of commuting differential operators on the moduli space of G-bundles of a complex algebraic curve.

I will start the talk with a brief introduction to the Langlands correspondence. I will discuss both the geometric and the function-theoretic versions for complex curves, and the relations between them. I will then present some of the results and conjectures from my joint work with Etingof and Kazhdan.

Thu Oct 29

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
The Weyl group and the nilpotent orbits
Zhiwei Yun, Massachusetts Institute of Technology

The Weyl group and the nilpotent orbits are two basic objects attached to a semisimple Lie group. The interplay between the two is a key ingredient in the classification of irreducible representations in various contexts. In this talk, I will describe two different constructions to relate these two objects due to Kazhdan-Lusztig, Lusztig and myself. I will concentrate on the construction using the loop geometry of the group. The main result is that the two seemingly different give the same maps between conjugacy classes in the Weyl group and the set of nipotent orbits.

Thu Nov 05

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
Low-degree hardness of random optimization problems
David Gamarnik, Massachusetts Institute of Technology

We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model, and (b) finding a large independent set in a sparse Erdos-Renyi graph, both to be introduced in the talk. We consider the family of algorithms based on low-degree polynomials of the input. This is a general framework that captures methods such as approximate message passing and local algorithms on sparse graphs, among others. We show this class of algorithms cannot produce nearly optimal solutions with high probability. Our proof uses two ingredients. On the one hand both models exhibit the Overlap Gap Property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions close to optimality are either close or far from each other. The second proof ingredient is the stability of the algorithms based on low-degree polynomials: a small perturbation of the input induces a small perturbation of the output. By an interpolation argument, such a stable algorithm cannot overcome the OGP barrier thus leading to the inapproximability. The stability property is established using concepts from Gaussian and Boolean Fourier analysis, including noise sensitivity, hypercontractivity, and total influence.

Joint work with Aukosh Jagannath and Alex Wein.

Thu Nov 12

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
Small scale creation and singularity formation in fluid mechanics
Alexander Kiselev, Duke University

The Euler equation describing motion of ideal fluid goes back to 1755. The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its potential connection with the arguably greatest unsolved problem of classical physics, turbulence. Results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

Thu Nov 19

## Colloquium

10:00am - Via Zoom ID 91514486597 (contact faculty for pw)
On the Ramanujan conjecture and its generalisations
Ana Caraiani, Imperial College London

In 1916, Ramanujan made a conjecture that can be stated in completely elementary terms: he predicted an upper bound on the coefficients of a power series obtained by expanding a certain infinite product. In this talk, I will discuss a more sophisticated interpretation of this conjecture, via the Fourier coefficients of a highly symmetric function known as a modular form. I will give a hint of the idea in Deligne's proof of the conjecture in the 1970's, who related it to the arithmetic geometry of smooth projective varieties over finite fields. Finally, I will discuss generalisations of this conjecture and some recent progress on these using the machinery of the Langlands program. The last part is based on joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne.

Tue Dec 01

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
Galois symmetries of the stable homology of integer symplectic groups
Akshay Venkatesh, Institute for Advanced Study

There are many natural sequences of moduli spaces in algebraic geometry whose homology approaches a "limit", despite the fact that the spaces themselves have growing dimension. If these moduli spaces are defined over a field K, this limiting homology carries an extra structure -- an action of the Galois group of K -- which is arithmetically interesting.

In joint work with Feng and Galatius, we compute this action (or rather a slight variant) in the case of the moduli space of abelian varieties. I will explain the answer and why I find it interesting. No familiarity with abelian varieties will be assumed -- I will emphasize topology over algebraic geometry.

Thu Dec 03

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
Shock formation and vorticity creation for compressible Euler
Vlad Vicol, Courant Institute of Mathematical Sciences, New York University

We discuss the formation of singularities (shocks) for the compressible Euler equations with the ideal gas law. We provide a constructive proof of stable shock formation from smooth initial datum, of finite energy, and with no vacuum regions. Via modulated self-similar variables, the blow-up time and location can be explicitly computed, the geometry of the shock set can be understood, and at the blow-up time the solutions can be shown to have precisely Holder 1/3 regularity. Additionally, for the non-isentropic problem we show that sound waves interact with entropy waves to produce vorticity at the shock. This talk is based on joint work with Tristan Buckmaster and Steve Shkoller.

Thu Dec 10

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
The Dehn complex: scissors congruence, K-theory, and regulators
Inna Zakharevich, Cornell University

Hilbert's third problem asks: do there exist two polyhedra with the same volume which are not scissors congruent? In other words, if $P$ and $Q$ are polyhedra with the same volume, is it always possible to write $P = \bigcup_{i=1}^n P_i$ and $Q = \bigcup_{i=1}^nQ_i$ such that the $P$'s and $Q$'s intersect only on the boundaries and such that $P_i \cong Q_i$? In 1901 Dehn answered this question in the negative by constructing a second scissors congruence invariant now called the "Dehn invariant," and showing that a cube and a regular tetrahedron never have equal Dehn invariants, regardless of their volumes. We can then restate Hilbert's third problem: do the volume and Dehn invariant separate the scissors congruence classes? In 1965 Sydler showed that the answer is yes; in 1968 Jessen showed that this result extends to dimension 4, and in 1982 Dupont and Sah constructed analogs of such results in spherical and hyperbolic geometries. However, the problem remains open past dimension 4. By iterating Dehn invariants Goncharov constructed a chain complex, and conjectured that the homology of this chain complex is related to certain graded portions of the algebraic K-theory of the complex numbers, with the volume appearing as a regulator. In joint work with Jonathan Campbell, we have constructed a new analysis of this chain complex which illuminates the connection between the Dehn complex and algebraic K-theory, and which opens new routes for extending Dehn's results to higher dimensions. In this talk we will discuss this construction and its connections to both algebraic and Hermitian K-theory, and discuss the new avenues of attack that this presents for the generalized Hilbert's third problem.

Thu Jan 28

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
Wieslawa Niziol, Sorbonne Universite'

p-adic Hodge theory is one of the most powerful tools in modern arithmetic geometry. In this talk, I will review p-adic Hodge theory of algebraic varieties, present current developments in p-adic Hodge theory of analytic varieties, and discuss some of its applications to problems in number theory.

Thu Feb 04

## Colloquium

3:30pm - via Zoom ID 91514486597 (contact faculty for pw)
Pastures, polynomials, and matroids
Matthew Baker, Georgia Institute of Technology

A pasture is, roughly speaking, a field in which addition is allowed to be both multivalued and partially undefined. I will describe a theorem about univariate polynomials over pastures which simultaneously generalizes Descartes' Rule of Signs and the theory of Newton Polygons. I will also describe a novel approach to the theory of matroid representations which revolves around a universal pasture, called the "foundation", which one can attach to any matroid. This is joint work with Oliver Lorscheid.

Thu Feb 11

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
The quartic integrability and long time existence of steep water waves in 2d
Sijue Wu, University of Michigan, Ann Arbor

It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals $\mathfrak E_j(t)$, directly in the physical space, that involves material derivatives of order $j$ of the solutions for the 2d water wave equation, so that $\frac{d}{dt} \mathfrak E_j(t)$ is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than $\varepsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\varepsilon^{-3})$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\varepsilon$, then the lifespan of the solution is at least of order $O(\varepsilon^{-5/2})$. Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.

Thu Feb 18

## Colloquium

5:00pm - Via Zoom ID 91514486597 (contact faculty for pw)
Wild ramification and cotangent bundle in mixed characteristic
Takeshi Saito, University of Tokyo

As an algebraic analogue of micro local analysis, the singular support and characteristic of an etale sheaf on a smooth algebraic variety over a perfect field is defined on the cotangent bundle. We discuss this geometric theory and some recent progress in the arithmetic context.

Thu Feb 25

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
From Grassmannians to Catalan numbers
Thomas Lam, University of Michigan, Ann Arbor

The binomial coefficients have a well-studied q-analogue known as Gaussian polynomials. These polynomials appear as Poincare polynomials (or point counts) of the Grassmannian of k-planes in C^n (or F_q^n).

Another family of important combinatorial numbers is the Catalan numbers, and they have two well-studied q-analogues from the 1960s, due to Carlitz and Riordan and to MacMahon respectively. I will explain how these q-analogues appear as the Poincare polynomial and point count, respectively, of an open (non-compact) subvariety of the Grassmannian known as the top positroid variety. The story involves connections to knot theory and to the geometry of flag varieties.

The talk is based on joint work with Pavel Galashin.

Thu Mar 04

## Colloquium

3:30pm - via Zoom ID 91514486597 (contact faculty for pw)
Bessel F-crystals for reductive groups
Xinwen Zhu, California Institute of Technology

I will first review the relationship between the classical Bessel differential equation

z^2f''(z)+zf'(z)+zf(z)=0

and the classical Kloosterman sum

\sum_{x=1}^{p-1} e((x+x*)/p), where e(-)=exp(2\pi i -) and x* is the inverse of x mod p

following the work of Deligne, Dwork and Katz. Then I will discuss a generalization of this story from the point of view of Langlands duality, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, myself, and the recent joint work with Daxin Xu. In particular, the joint work with Xu gives (probably) the first example of a p-adic version of the geometric Langlands correspondence. It allows us to prove a conjecture of Heinloth-Ngo-Yun on the functoriality of some specific automorphism forms.

Thu Mar 18

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Mathematics and physics of moiré patterns
Mitchell Luskin, University of Minnesota, Twin Cities

Placing a two-dimensional lattice on another with a small rotation gives rise to periodic "moire" patterns on a superlattice scale much larger than the original lattice. This effective large-scale fundamental domain allows phenomena such as the fractal Hofstadter butterfly spectrum in Harper's equation to be observed in real crystals. Experimentalists have more recently observed new correlated phases at "magic" twist angles predicted by theorists.

We will give mathematical and computational models to predict and gain insight into new physical phenomena at the moiré scale including our recent mathematical and experimental results for twisted trilayer graphene.

Thu Mar 25

## Colloquium

10:00am - Zoom ID 91514486597 (contact faculty for pw)
From differential equations to deep learning for image analysis
Carola-Bibiane Schönlieb, University of Cambridge

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about some of these mathematical problems, about variational models for image analysis and their connection to partial differential equations and deep learning. The talk is furnished with applications to art restoration, forest conservation and cancer research.

Thu Apr 01

## Colloquium

3:30pm - Via Zoom ID 91514486597 (contact faculty for pw)
Electric-magnetic duality between periods and L-functions
David Ben-Zvi, University of Texas, Austin

I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory.

A fundamental tool in number theory, the relative Langlands program, is centered on the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index these period integrals (spherical varieties for a reductive group G) and the L-functions (representations of the Langlands dual group G^) don't seem to line up, making the search for integral representations somewhat of an art.

We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric gauge theory. Namely, we rewrite the *relative* Langlands program as duality in the presence of boundary conditions. As a result the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups.

Thu Apr 15

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Large deviations for lacunary trigonometric sums
Kavita Ramanan, Brown University

Lacunary trigonometric sums are known to exhibit several properties that are typical of sums of iid random variables such as the central limit theorem, established by Salem and Zygmund, and the law of the iterated logarithm, due to Erdos and Gal. We study large deviation principles for such sums, and show that they display several interesting features, including sensitivity to the arithmetic properties of the corresponding lacunary sequence. This is joint work with C. Aistleitner, N. Gantert, Z. Kabluchko and J. Prochno.

Thu Apr 22

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Scalable spaces
Fedor Manin, University of California, Santa Barbara

I will discuss topological invariants arising in metric geometry. In the 1970s, Gromov remarked that the n-sphere has self-maps of degree L^n whose Lipschitz constant is O(L), for every integer L. These should be thought of as maps of maximal geometric efficiency; we say a closed n-manifold is scalable if it admits efficient self-maps in infinitely many degrees. How can we decide which manifolds are scalable? Recently, Sasha Berdnikov and I showed that for simply connected manifolds, scalability is an invariant of rational homotopy type, and gave some equivalent conditions. For example, we found that the connected sum of three CP^2's is scalable but the connected sum of four is not.

Thu Apr 29

## Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Colloquium
Tatiana Toro, University of Washington