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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)
Gradient variational problems
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
Colloquium
Wieslawa Niziol, Sorbonne Universite'
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
Colloquium
Sijue Wu, University of Michigan, Ann Arbor
Thu Feb 18

Colloquium

5:00pm - Via Zoom
Colloquium
Takeshi Saito, University of Tokyo
Thu Feb 25

Colloquium

3:30pm - Via Zoom
Colloquium
Thomas Lam, University of Michigan, Ann Arbor
Thu Mar 04

Colloquium

3:30pm - Via Zoom
Colloquium
Xinwen Zhu, California Institute of Technology
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

Layers of two-dimensional materials stacked with a small twist-angle give rise to periodic beating patterns on a "moiré superlattice" scale much larger than the original lattice. This effective large-scale fundamental domain allows phenomena such as the fractal Hofstadter butterfly to be observed in crystalline materials at experimental magnetic fields. More recently, this new length scale has allowed experimentalists to observe new correlated electronic phases such as superconductivity at a lower electron density than previously accessible and has motivated an intense focus by theorists to develop models for this correlated behavior.
We will present an introduction to some mathematical and computational models for moiré physics. Almost every area of mathematics including partial differential equations, harmonic analysis, functional analysis, dynamical systems, numerical analysis, geometry, topology and number theory has been utilized to gain insight into new phenomena at the moiré scale.

Thu Mar 25

Colloquium

10:00am - Via Zoom (note special time)
Colloquium
Carola-Bibiane Schönlieb, Cambridge University
Thu Apr 01

Colloquium

3:30pm - Via Zoom
Colloquium
David Ben-Zvi, University of Texas, Austin
Thu Apr 15

Colloquium

3:30pm - Via Zoom
Colloquium
Kavita Ramanan, Brown University
Thu Apr 29

Colloquium

3:30pm - Via Zoom
Colloquium
Tatiana Toro, University of Washington