Colloquia: Past Seminars

May 4, 2023 – How curved is a combinatorial graph?

Stefan Steinerberger, University of Washington

Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and completely explicit linear system of equations. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related open problems. No prior knowledge of differential geometry (or graphs) is required.

April 27, 2023 – The algebraic structure of homeomorphism groups of surfaces

Dan Cristofaro-Gardiner, University of Maryland

Abstract: In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Kirby, Mather, Fathi, Thurston,  and many others.  A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces.  For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of  Fathi from the 1970s, but the answer in the two-dimensional case was not known.   My talk will be about some recent joint work, solving many of the mysteries of the two-dimensional case by using ideas from symplectic geometry.  In particular, we resolved the "Simplicity conjecture", which stated that the group of area-preserving homeomorphisms of the two-disc  that are the identity near the boundary is not simple, in contrast to the situation in higher dimensions.  I will also explain some mysteries that remain unresolved.  A key role in our arguments is played by a kind of Weyl law, relating the asymptotics of some new "spectral invariants" to more classical invariants.  No prior knowledge of symplectic geometry will be assumed.

April 20, 2023 – Fourier analysis beyond vector spaces

Jayce Getz, Duke University

Abstract: The Fourier transform and Poisson summation formula on a vector space have a venerable place in mathematics.  It has recently become clear that they are but the first case of general phenomena.  Namely, conjectures of Braverman, Kazhdan, L. Lafforgue, Ngo and Sakellaridis suggest that one can define Fourier transforms and prove Poisson summation formulae whenever the vector space is replaced by a so-called spherical variety satisfying certain desiderata.  In this talk I will focus on what has been proven in this direction for a particular family of spherical varieties related to flag varieties.  A simple (but nontrivial) example is the zero locus of a nondegenerate quadratic form.  Kazhdan believes that these generalized Fourier transforms and Poisson summation formulae will eventually have many applications throughout mathematics.  I agree, and to expedite these applications I will present them in a format that is as accessible as possible.

April 13, 2023 – Certain recent progress on Atiyah-Floer conjecture

Kenji Fukaya, Simons Center for Geometry and Physics, SUNY, Stone Brook

Abstract: Atiyah-Floer conjecture concerns a relation between two different version of Floer homologies, one in gauge theory and the other in symplectic geometry.  Based on joint works with A. Daemi and M. Lipyanskiy, I explain a certain partial results and generalizations on this conjecture.

April 6, 2023 – Some new results in Higher Teichmüller Theory

Sara Maloni, University of Virginia

Abstract: The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, the Teichmüller space can also be seen as a connected component of (conjugacy classes of) representations from the fundamental group of S into PSL(2,R), consisting entirely of discrete and faithful representations. Generalizing this point of view, Higher Teichmüller Theory studies connected components of (conjugacy classes of) representations from the fundamental group of S into more general semisimple Lie groups which consist entirely of discrete and faithful representations.

We will give a survey of some aspects of Higher Teichmüller Theory, and will make links with the recent powerful notion of Anosov representation. We will conclude by focusing on two separate questions:

1. Do these representations correspond to deformation of geometric structures?
2. Can we generalize the important notion of pleated surfaces to higher rank Lie groups like PSL(d, C)?

The answer to question 1 is joint work with Alessandrini, Tholozan and Wienhard, while the answer to question 2 is joint work with Martone, Mazzoli and Zhang.

March 30, 2023 – Neighborhood growth models

David Sivakoff, Ohio State University

Abstract: Neighborhood growth cellular automata were introduced over 40 years ago as easy to describe models that exhibit the complex phenomena of nucleation and metastability.  The most well-known of these is the threshold-2 growth model on the 2d integer lattice, wherein an initially occupied set of vertices is iteratively enlarged by occupying all vertices with at least two occupied neighbors.  If the initially occupied set of vertices is chosen by randomly including each vertex independently with small probability p>0, then surprisingly all vertices eventually become occupied, but it typically takes exponentially long in (1/p) for the origin to become occupied.  I will discuss the history and intuition behind results like these.  I will also mention more recent progress on neighborhood growth models where polynomial scaling laws appear and the growth mechanism is quite different.

March 23, 2023 –  Geometric variational problems: regularity vs singularity formation

Yannick Sire, Johns Hopkins University

Abstract: I will describe in a very informal way some techniques to deal with the existence ( and more qualitatively regularity vs singularity formation) in different geometric problems and their heat flows motivated by (variations of) the harmonic map problem, the construction of Yang-Mills connections or nematic liquid crystals. I will emphasize in particular on recent results on the construction of very fine asymptotics of blow-up solutions via a new gluing method designed for parabolic flows. I’ll describe several open problems and many possible generalizations, since the techniques are rather flexible.

March 16, 2023 –  Geometric Graph-Based Methods for High Dimensional Data

Andrea Bertozzi, University of California, Los Angeles

Abstract: High dimensional data can be organized on a similarity graph - an undirected graph with edge weights that measure the similarity between data assigned to nodes. We consider problems in semi-supervised and unsupervised machine learning that are formulated as penalized graph cut problems. There are a wide range of problems including Cheeger cuts, modularity optimization on networks, and semi-supervised learning. We show a parallel between these modern problems and classical minimal surface problems in Euclidean space.  
This analogy allows us to develop a suite of new algorithms for machine learning that are both very fast and highly accurate.  These are analogues of well-known pseudo-spectral methods for partial differential equations.

March 2, 2023 – Categorical diagonalization

Ben Elias, University of Oregon

Abstract: We know what it means to diagonalize an operator in linear algebra.  What might it mean to diagonalize a functor?

Given a linear operator f whose characteristic polynomial is multiplicity-free, we can construct projection to each eigenspace as a polynomial in f, using a technique known as Lagrange interpolation.  We think of the process of finding a complete family of orthogonal idempotents as the diagonalization of f.  After reviewing this we provide a categorical analogue: given a functor F with some additional data (akin to the set of eigenvalues), we construct idempotent functors projecting to "eigencategories."  Along the way we'll explain some of the basic concepts in categorification.

Diagonalization is incredibly important in every field of mathematics.  I am a representation theorist, so I will briefly indicate some of the important applications of categorical diagonalization to representation theory.  I'll also indicate applications to algebraic geometry.

In this talk we will follow a running example involving modules over the ring Z[x]/(x^2 - 1), in other words, the group algebra of the group of size 2.  If you know what a complex of modules is (and what chain maps and homotopies are) then you have all the prerequisites needed for this talk.

This is all joint work with Matt Hogancamp.

February 9, 2023 – Quantitative closing lemmas

Michael Hutchings, University of California, Berkeley

We consider the dynamics of an area-preserving diffeomorphism of a surface, or (what turns out to be closely related) a Reeb vector field on a three-manifold. One of the basic questions in dynamics is to understand periodic orbits of a diffeomorphism or vector field. A "closing lemma" is a statement asserting that one can make a small perturbation of a diffeomorphism or vector field to arrange that there is a periodic orbit passing through a given nonempty open set. The
goal of this talk is to describe a new approach to proving "quantitative" closing lemmas, which gives upper bounds on how much one needs to perturb in order to obtain a periodic orbit with a given upper bound on the period. This is based in part on joint work with Oliver Edtmair.

February 2, 2023 – The Dirichlet Problem with Lᵖ Data: When Can It Be Solved?

Steven Hoffmann, University of Missouri

Abstract: For a domain Ω ⊂ Rᵈ, a classical criterion of Wiener characterizes the domains for which one can solve the Dirichlet problem (originally, for Laplace’s equation) with continuous bound- ary data. What happens if we allow singular data, say in Lᵖ (with respect to surface measure on the boundary) for some finite expo- nent p? It turns out that solvability in the latter setting is equivalent to a quantitative, scale invariant version of absolute continuity of harmonic measure with respect to surface measure on ∂Ω. In turn, to determine what sort of boundaries are permitted in the presence of such absolute continuity, involves a version of a classical 1-phase free boundary problem. In this talk, we shall discuss the question of characterizing Lᵖ solvability, and we shall give an answer that is rather definitive (i.e., we find a characterization in the presence of some natural “best possible” background hypotheses), in the case of Laplace’s equation. Time permitting, we shall also discuss recent partial progress in the caloric case.

January 19, 2023 –  Nonlinear Model Reduction for Slow-Fast Stochastic Systems near Unknown Invariant Manifolds

Mauro Maggioni, Johns Hopkins University

Abstract:  We are interested in model reduction for certain classes of high-dimensional stochastic dynamical systems. Model reduction consists in estimating a low-dimensional stochastic dynamical system that approximates, in a suitable sense (e.g. at certain spatial or temporal time-scale), the original system, or at least some observables of the original system. Typically such reduced models may be faster to simulate (e.g. because of their lower dimensionality) and may offer important insights on the dynamical behavior of the original system.
Motivated by examples and applications, including molecular dynamics, we consider a special, well-studied class of stochastic dynamical systems that exhibit two important properties. The first one is that the dynamics can be split into two timescales, a slow and a fast timescale, and the second one is that the slow dynamics takes place on, or near to, a low-dimensional manifold M, while the fast dynamics can be thought of as consisting of fast oscillations off that manifold.
Given only access to a black-box simulator from which short bursts of simulations can be obtained, and a (possibly small) set of reasonable initial conditions, we design an algorithm that outputs an estimate of the manifold M, a process representing the effective stochastic dynamics on M, which has averaged out the fast modes, and a simulator of such process. The fast modes are not assumed to be small, nor orthogonal to M.
This simulator is efficient in that it exploits the low dimension of the manifold M, and takes time steps of size dependent only the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes in high dimensions. Furthermore, the algorithm and the estimation can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them. This is joint work with X.-F. Ye and S. Yang.

November 17, 2022 – Local Multiplicity for spherical varieties

Chen Wan, Rutgers University, Newark

Abstract: Let G be a reductive group and H be a closed subgroup of G.  We say H is a spherical subgroup of G if there exists a Borel subgroup B of G such that BH is Zariski open in G.  (I will explain what the above terminologies mean in my talk.)  One of the fundamental problems in the relative Langlands program is to study the multiplicity problem for the pair (G,H), i.e. to study the restriction of a representation of G to H.  In this talk, I will first recall the multiplicity problem in the finite group case and in the Lie group case.  Then I will go over the general conjecture and all the known results for the multiplicity problem of spherical varieties.  Lastly, I will explain how to use the trace formula method to study this problem.

November 10, 2022 –  Representations of reductive groups over local fields

Tasho Kaletha, University of Michigan, Ann Arbor

Abstract: The pioneering work of Langlands has established the theory of reductive algebraic groups and their representations as a key part of modern number theory.  I will survey classical and modern results in the representation theory of reductive groups over local fields (the fields of real, complex, or p-adic numbers, or of Laurent series over finite fields) and discuss how they relate to Langlands' ideas, as well as to the various reflections of the basic mathematical idea of symmetry in arithmetic and geometry.

October 27, 2022 –  Skein valued counts of holomorphic curves

Tobias Ekholm, Uppsala University

Abstract: We show that counting holomorphic curves by the values of their boundaries in the HOMFLY skein module gives rise to invariant counts of holomorphic curves with boundary in a Maslov zero Lagrangian in a 3-dimensional Calabi-Yau manifold.  This leads to simple and powerful recursion relations for curves that we use to prove the Ooguri-Vafa relation between HOMFLY polynomials of knots in the 3-sphere and Gromov-Witten theory in the resolved conifold.  The talk reports on joint work with Vivek Shende.

October 20, 2022 – Minimal surfaces and the isoperimetric inequality

Simon Brendle, Columbia University

Abstract: The isoperimetric inequality in Euclidean space has a long history in geometry, going back to the legend of Queen Dido.  Over the past century, mathematicians have sought to generalize the isoperimetric inequality to various curved spaces.  In this lecture, I will discuss how the isoperimetric inequality can be generalized to minimal surfaces.  The proof is inspired by, but does not actually use, optimal transport.

October 13, 2022 – Equidistribution and reciprocity in number theory

Jack Thorne, University of Cambridge

Abstract: A famous result in number theory is Dirichlet's theorem that there exist infinitely many prime numbers in any given arithmetic progression a, a + N, a + 2 N, ... where a, N are coprime.  In fact, a stronger statement holds: the primes are equidistributed in the different residue classes modulo N.  In order to prove his theorem, Dirichlet introduced Dirichlet L-functions, which are analogues of the Riemann zeta function which depend on a choice of character of the group of units modulo N.

More general L-functions appear throughout number theory and are closely connected with equidistribution questions, such as the Sato--Tate conjecture (concerning the number of solutions to y^2 = x^3 + a x + b in the finite field with p elements, as the prime p varies).  L-functions also play a central role in both the motivation for and the formulation of the Langlands conjectures in the theory of the automorphic forms.

I will give a gentle introduction to some of these ideas and discuss some recent theorems in the area.

September 15, 2022 – Inverse Problems for Nonlinear Equations

Ru-Yu Lai, University of Minnesota 

Abstract: Inverse problems arise from the need to extract knowledge from incomplete measurements.  Lying at the heart of modern scientific inquiry, inverse problems find extensive applications in various fields, including geophysics, medical imaging, biology, and many others.  In this talk, we will first introduce the mathematical
background and several interesting applications.  We will also demonstrate the high order linearization approach to solve several inverse problems for nonlinear PDEs, including nonlinear magnetic Schrodinger equations and transport equations.  Unique determination of unknown coefficients from measurements will be discussed.

May 5, 2022 — Eigenfunctions, Unique Continuation, and Homogenization

Charles Smart, Yale University

Abstract: I will survey some recent results on unique continuation and homogenization. In particular, I will discuss the Landis conjecture, the embedded eigenvalue problem for periodic elliptic equations, and applications of homogenization to quantitative unique continuation.

April 28, 2022 — Isoperimetric Multi-Bubble Problems - Old and New

Emanuel Milman, Technion — Israel Institute of Technology and Oden Institute, The University of Texas at Austin

Abstract: The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$, known to the ancient Greeks in dimensions two and three, states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the sphere $\mathbb{S}^n$ and on Gauss space $\mathbb{G}^n$. Furthermore, one may consider the ``multi-bubble" partitioning problem, where one partitions the space into $q \geq 2$ (possibly disconnected) bubbles, so that their total common surface-area is minimal. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $q=2$; the case $q=3$ is called the double-bubble problem, and so on.

In 2000, Hutchings, Morgan, Ritoré and Ros resolved the Double-Bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently resolved in $\mathbb{R}^n$ as well) -- the optimal partition into two bubbles of prescribed finite volumes (and an exterior unbounded third bubble) which minimizes the total surface-area is given by three spherical caps, meeting at 120-degree angles. A more general conjecture of J.~Sullivan from the 1990's asserts that when $q \leq n+2$, the optimal Multi-Bubble partition of $\mathbb{R}^n$ (as well as $\mathbb{S}^n$) is obtained by taking the Voronoi cells of $q$ equidistant points in $\mathbb{S}^{n}$ and applying appropriate stereographic projections to $\mathbb{R}^n$ (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous Multi-Bubble conjecture on the optimal partition of Gauss space $\mathbb{G}^n$ into $q \leq n+1$ bubbles -- the unique optimal partition is given by the Voronoi cells of (appropriately translated) $q$ equidistant points. In this talk, we will describe our approach in that work, as well as recent progress on the Multi-Bubble problem on $\mathbb{R}^n$ and $\mathbb{S}^n$. In particular, we show that minimizing partitions are always spherical when $q \leq n+1$, and we resolve the latter conjectures when in addition $q \leq 6$ (e.g. the triple-bubble conjecture in $\mathbb{R}^3$ and $\mathbb{S}^3$, and the quadruple-bubble conjecture in $\mathbb{R}^4$ and $\mathbb{S}^4$).

Based on joint work (in progress) with Joe Neeman.

April 14, 2022 — Spatial Ecology & Singularly Perturbed Reaction-Diffusion Equations

Arjen Doelman, Leiden University

Abstract: Pattern formation in ecological systems is driven by counteracting feedback mechanisms on widely different spatial scales. Moreover, ecosystem models typically have the nature of reaction-diffusion systems: the dynamics of ecological patterns can be studied by the methods (geometric) singular perturbation theory. In this talk we give an overview of the surprisingly rich cross-fertilization between ecology, the physics of pattern formation and the mathematics of singular perturbations. We show how a mathematical approach uncovers mechanisms by which real-life ecosystems may evade (catastrophic) tipping under slowly varying climatological circumstances. This insight is based on two crucial ingredients: the careful study of Busse balloons in (parameter, wavenumber)-space associated to spatially periodic patterns and the validation of the model predictions by field observations. Vice versa, ecosystem models motivate the study of classes of singularly perturbed reaction-diffusion equations that exhibit much more complex behavior than the models so far studied by mathematicians: we present several novel research directions initiated by ecology.

April 7, 2022 — Complex multiplication for real quadratic fields

Henri Darmon, McGill University

Abstract: The theory of complex multiplication developed in the 19th century for imaginary quadatic fields via the moduli of elliptic curves with complex multiplication, and extended by Shimura, Taniyama and Weil to CM fields, leads to (a partial) explicit class field theory for these base fields. The case where the base field is not a CM field appears more mysterious. I will describe a largely conjectural approach, for the simplest case of real quadratic fields, which rests on replacing modular functions by “rigid meromorphic cocycles”. This is an account of ongoing joint work with Alice Pozzi and Jan Vonk.

April 5, 2022 — Rational points on intersections of two quadrics

Wei Zhang, Massachusetts Institute of Technology

Abstract: In the first talk, we will discuss the Hasse principle for rational points on intersection of two quadrics in P^N over function fields (of algebraic curves over a finite field). Among other things, the new tools here include an on-going work with Zhiwei Yun to establish a function field analog of (a strengthened) Kolyvagin's theorem for elliptic curves and our previous work on a Higher Gross-Zagier formula relating intersection numbers of certain cycles on moduli space of Drinfeld Shtukas to L-functions of elliptic curves.

March 31, 2022 — Unlikely intersections and the Andre-Oort conjecture

Jacob Tsimerman, University of Toronto

Abstract: The Andre-Oort conjecture concerns special points of a Shimura variety S --- points that are in a certain sense "maximally symmetric".  It states that if a variety V in S contains a zariski-dense set of such points, then V must itself be a Shimura variety.  It is an example of the field now known as "unlikely intersections theory" which seeks to explain "arithmetic coincidences" using geometry.  In fact, there is a very natural sense in which the Andre-Oort conjecture can be seen as an analogue of Faltings theorem concerning rational points on curves.  The proof of this conjecture involves a wide range of disparate mathematical ideas --- functional transcendence, mondromy, point counting in transcendental sets, upper bounds for arithmetic complexity (heights of special points), and p-adic hodge theory.  We will survey these concepts and how they relate to each other in the proof, aiming to give an overview of the relevant ideas.  We will also discuss the current status of the field, now spearheaded by the (still extremely open!) Zilber-Pink conjecture, and what is required to make further progress.

March 24, 2022 — Dirac operators and scalar curvature rigidity

Guoliang Yu, Texas A&M

Abstract: I will give an introduction to some recent progress on scalar curvature rigidity using the Dirac operator method. In particular, I will discuss how the Dirac operator method can be used to resolve several of Gromov's conjectures on scalar curvature. I will make this talk accessible to graduate students and non-experts.

March 17, 2022 — Grain Structure, Grain Growth and Evolution of the Grain Boundary Network

Yekaterina Epshteyn, University of Utah

Abstract: Cellular networks are ubiquitous in nature. Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small monocrystalline cells or grains, separated by interfaces, or grain boundaries of crystallites with different lattice orientations. A central problem in materials science is to develop technologies capable of producing an arrangement of grains that provides for a desired set of material properties. One method by which the grain structure can be engineered is through grain growth (also termed coarsening) of a starting structure.

The evolution of grain boundaries and associated grain growth is a very complex multiscale process. It involves, for example, dynamics of grain boundaries, triple junctions, and the dynamics of lattice misorientations/grains rotations. In this talk, we will discuss recent progress in mathematical modeling, simulation and analysis of the evolution of the grain boundary network in polycrystalline materials.

March 3, 2022 — The unbounded denominators conjecture

Yunqing Tang, Princeton 

Abstract: The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer in 1968, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. Our proof of this conjecture is based on a new arithmetic algebraization theorem, which has its root in the classical Borel--Dwork rationality criterion. In this talk, we will discuss some ingredients in the proof and a variant of our arithmetic algebraization theorem, which we will use to prove the irrationality of certain 2-adic zeta value. This is joint work with Frank Calegari and Vesselin Dimitrov.

February 24, 2022 — The leapfrogging and the Vortex filament conjecture for Euler equations

Jun-cheng Wei, University of British Columbia

Abstract: The existence and stability of  concentrated vorticity solutions (vortices in 2D and vortex filaments in 3D) to the Euler equation is a long-standing problem.  In this talk I will discuss our new approach--the inner-outer-gluing-scheme-- to this problem.  I will first report the result on desingularization of vortices in 2D: Let $ \xi (t)$ be any solution of Kirtchhoff-Ruth dynamical system. Then one can construct solutions to 2D Euler equations with concentrating vorticity along the trajectory of $\xi (t)$. Then I will discuss  recent work on the Leapfrogging phenomenon in 3D Euler: Let $ (r_j, z_j)$ be any solution of Leapfrogging Dynamics. Then one can construct an axially symmetric 3D Euler solution with vortex filaments following the Leapfrogging dynamics.  Vortex Filament Conjecture and relation of 3D Euler with generalized Adler-Moser polynomials will be mentioned.

February 17, 2022 — Higher-dimensional Heegaard Floer homology and Hecke algebras

Ko Honda, University of California, Los Angeles

Abstract: Hecke algebras are ubiquitous in number theory and geometric representation theory.  In this talk we describe the appearance of various Hecke algebras such as the affine Hecke algebra and the double affine Hecke algebra (DAHA) in Floer theory, through the higher-dimensional analog of Heegaard Floer homology.  This is joint work with Yin Tian and Tianyu Yuan.

February 10, 2022 — Teaching proofs to a computer

Kevin Buzzard, Imperial College London

Abstract: After around 25 years as an algebraic number theorist, I switched fields in 2017 and started to teach number theory to a computer proof system called Lean. I now believe that these systems will be playing an inevitable role in the future of mathematical research. I'll tell the story of what I've been doing and why I think it's important. I will assume no prior knowledge of computer proof systems.

February 3, 2022 — The mapping class group of a surface

Andrew Putman, University of Notre Dame

Abstract: The mapping class group of a surface is a remarkable group with connections to low-dimensional topology, algebraic geometry, dynamics, and many other areas. I will give an elementary survey of some aspects of this topic with a focus on finiteness properties.

January 27, 2022 — Probing Electron Transport with Light: A Flavor of Localized Edge Modes in 2D Materials

Dio Margetis, University of Maryland

Abstract: Recent experimental studies in the properties of atomically thin materials such as graphene and black phosphorus have offered insights into useful aspects of the collective motion of electrons in 2D. A wealth of intriguing optical phenomena can arise in these systems because of the coupling of the electron motion with incident electromagnetic fields.

In many applications of photonics at the nanoscale, 2D materials such as graphene may behave as conductors, and allow for the excitation and propagation of electromagnetic waves with surprisingly small length scales. These surface waves are tightly confined to the material. They can possibly beat the optical diffraction limit, in the sense that the wavelength of the excited surface waves can be much smaller than that of the incident wave in a frequency range of practical interest. A broad goal in mathematical modeling is to understand how distinct kinetic regimes of 2D electron transport can be probed, and even controlled, by electromagnetic signals.

I will discuss recent work in describing the dispersion of electromagnetic modes that may propagate along edges of flat, anisotropic conducting sheets. Some emphasis will be placed on an emergent concept for the existence of such modes. The starting point is a boundary value problem for Maxwell’s equations coupled with the physics of the moving electrons in monolayer and bilayer structures.

January 20, 2022 — Learning Interaction laws in particle- and agent-based systems

Mauro Maggioni, Johns Hopkins University

Abstract: Interacting agent-based systems are ubiquitous in science, from modeling of particles in Physics to prey-predator and colony models in Biology, to opinion dynamics in economics and social sciences. Oftentimes the laws of interactions between the agents are quite simple, for example they depend only on pairwise interactions, and only on pairwise distance in each interaction. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be “any” function of pairwise distances. We consider  this problem both the mean-field limit (i.e. the number of particles going to infinity) and in the case of a finite number of agents, with an increasing number of observations, albeit in this talk we will mostly focus on the latter case. We cast this as an inverse problem, and present a solution in the simplest yet interesting case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, we construct estimators for the interaction kernels with provably good statistically and computational properties, and discuss extensions to second-order systems, more general interaction kernels, and stochastic systems. We measure empirically the performance of our techniques on various examples, that include extensions to agent systems with different types of agents, second-order systems, and families of systems with parametric interaction kernels. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior. This is joint work with F. Lu, J.Miller, S. Tang and M. Zhong.

December 14, 2021 — Cycles, Cobordisms and Vector Bundles from the Motivic Viewpoint (Special Colloquium)

Elden Elmanto, Harvard University

Abstract: In 1957, Grothendieck defined the K-groups of vector bundles on an algebraic variety to formulate his Riemann-Roch theorem, bringing an algebraic perspective to a theorem in complex analysis. The higher K-groups of varieties were discovered in 1973 by Quillen and have remained a source of both mystery and inspiration in mathematics, and interacts with many vastly different subjects. In this talk, I will provide a panoramic tour on K-theory and its cousins - cycles and cobordisms. In particular I will explain how the birational geometry of Hilbert schemes (joint with Bachmann) and p-adic Hodge theory (joint with Morrow) can be used to shed new light on these creatures. 

December 10, 2021 — Arithmetic Hyperbolic Manifolds and their Finite Covers

Michelle Chu, University of Illinois Chicago

Abstract: Arithmetic hyperbolic manifolds are constructed as quotients of hyperbolic space by subgroups of isometries commensurable with integer points in algebraic groups. The rich connection between the geometry and arithmetic give these manifolds a special beauty. In this talk, I will introduce arithmetic methods to construct hyperbolic manifolds and describe how arithmeticity helps us understand the geometry and topology of these manifolds and their finite covers.

December 9, 2021 — Uncovering the Rules of Crumpling with a Data-driven Approach (Special Colloquium)

Christopher Rycroft, Harvard University

Abstract: When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].

This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3].
[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).
[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).
[3] J. Hoffmann et al., Sci. Advances 5, 6792 (2019).

December 8, 2021 — Inferring Latent Structure in Random Graphs (Special Colloquium)

Miklos Racz, Princeton University

Abstract: From networks to genomics, large amounts of data are abundant and play critical roles in helping us understand complex systems. In many such settings, these data take the form of large discrete structures with important combinatorial properties. The interplay between structure and randomness in these systems presents unique mathematical challenges. In this talk I will highlight these through two vignettes on inferring latent structure in random graphs: (1) inference of latent high-dimensional geometry, and (2) improved recovery of communities using multiple correlated graphs.

First, I will talk about a canonical random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels. We are particularly interested in the high-dimensional case when d is large and, in the dense regime, we determine the phase transition for when geometry is detectable/lost. The proofs highlight novel graph statistics, as well as connections to random matrices. Next, I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph.

About the Speaker: Miklos Z. Racz is an assistant professor at Princeton University in the ORFE department, as well as an associated faculty member at the Center for Statistics and Machine Learning (CSML). Before coming to Princeton, he received his PhD in Statistics from UC Berkeley and was then a postdoc in the Theory Group at Microsoft Research, Redmond. Miki’s research interests lie broadly at the interface of probability, statistics, information theory, and computer science. Miki's research and teaching has been recognized by Princeton's Howard B. Wentz, Jr. Junior Faculty Award, a Princeton SEAS Innovation Award, and an Excellence in Teaching Award.

December 7, 2021 — Efficient Learning Algorithms through Geometry, and Applications in Cancer Research (Special Colloquium)

Caroline Moosmueller, University of California, San Diego

Abstract: In this talk, I will discuss how incorporating geometric information into classical learning algorithms can improve their performance. The main focus will be on optimal mass transport (OMT), which has evolved as a major method to analyze distributional data. In particular, I will show how embeddings can be used to build OMT-based classifiers, both in supervised and unsupervised learning settings. The proposed framework significantly reduces the computational effort and the required training data. Using OMT and other geometric data analysis tools, I will demonstrate applications in cancer research, focusing on the analysis of gene expression data and on protein dynamics.

December 2, 2021 — Braid Group Actions, Crystals, and Cacti (Special Colloquium)

Iva Halacheva, Northeastern University

Abstract: The representations of Lie algebras and their associated quantum groups are known to have symmetries captured by the braid group. When considering their combinatorial shadows known as crystals, an action of a closely related group—the cactus group—emerges. I will describe how this action has surprising appearances throughout representation theory: it can be realized both geometrically as a monodromy action coming from a family of “shift of argument” algebras, as well as categorically by studying the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.

November 18, 2021 — Mathematical methods in evolutionary dynamics

Natalia Komarova, University of California, Irvine

Abstract: Evolutionary dynamics are at the core of carcinogenesis. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to shed light into cancer origins, progression, and mechanisms of treatment. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics  -- including stochastic tunneling and calculating the rate of evolution. Applications include origins of cancer, and development of resistance against treatment. I will also talk about evolution in random environments.  The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Of particular interest are the dynamics of non-selected mutants, which exhibit counterintuitive properties.

November 16, 2021 — Simplifying and Comparing Derived Categories (Special Colloquium)

David Favero, University of Alberta

Abstract: Derived categories are a construction used across many mathematical disciplines, including algebraic geometry, topology, and non-commutative algebra.  I will introduce derived categories and illustrate how to use them to recover a well-known topological invariant.  By discussing some open conjectures revolving around derived categories in algebraic geometry, I will then explain how to decompose derived categories into smaller, more manageable pieces and compare them to one another through my research in geometric invariant theory.  To conclude, I will outline my program to solve these conjectures and discuss some works in progress.

November 11, 2021 — Newell-Littlewood numbers

Alexander Yong, University of Illinois Urbana Champaign

Abstract: The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question: Which multiplicities are nonzero?

In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers.

This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152.

November 4, 2021 — Diffeomorphisms of discs

Alexander Kupers, University of Toronto Scarborough

Abstract: The simplest manifolds are arguably the disc, but their diffeomorphism groups remain mysterious. I will discuss why they are fundamental objects of geometric topology, what is known about them, as well as joint work with Oscar Randal-Williams, which has the goal of understanding the rational homotopy type of the group of diffeomorphisms of even-dimensional discs.

October 28, 2021 —Wave maps into the sphere

Carlos Kenig, University of Chicago

Abstract: We will introduce wave maps and discuss some of their basic properties, leading to recent works on the soliton resolution conjecture for critical wave maps and related equations.

October 21, 2021 — Abelian and Non-Abelian X-ray transforms. Sharp mapping properties and Bayesian inversion

Francois Monard, University of California, Santa Cruz

Abstract: Abelian and Non-Abelian X-ray transforms are examples of integral-geometric transforms with applications to X-ray Computerized Tomography and the imaging of magnetic fields inside of materials (Polarimetric Neutron Tomography). Their study uses tools from classical inverse problems (assessments of injectivity, stability and inversions), and mathematical statistics to deal with cases with noisy data. 
After giving a brief introduction to the topic, I plan on covering the following recent results: 

1. We will first discuss a sharp description of the mapping properties of the X-ray transform (and its associated normal operator I*I) on the Euclidean disk, associated with a special L2 topology on its co-domain.
2. We will then focus on how to use this framework to show that attenuated X-ray transforms (with skew-hermitian attenuation matrix), more specifically their normal operators, satisfy similar mapping properties. 
3. Finally, we will discuss an important application of these results to the Bayesian inversion of the problem of reconstructing an attenuation matrix (or Higgs field) from its scattering data corrupted with additive Gaussian noise. Specifically, I will discuss a Bernstein-VonMises theorem on the 'local asymptotic normality' of the posterior distribution as the number of measurement points tends to infinity, useful for uncertainty quantification purposes. Numerical illustrations will be given throughout.

(2) and (3) are joint work with R. Nickl and G.P.Paternain (Cambridge).

October 14, 2021 — Equivariant log concavity in the cohomology of configuration spaces

Nicholas Proudfoot, University of Oregon

Abstract:  Consider the space whose points consist of n-tuples of distinct complex numbers.  The Betti numbers of this space are called Stirling numbers, and they form a log concave sequence by a theorem of Isaac Newton.  I will state a conjectural generalization of this result that takes into account the action of the symmetric group by permuting the points.  The full conjecture is open, but I will explain how to leverage the theory of representation stability to prove infinitely many cases.

October 7, 2021 — On stability problems in fluid dynamics

Hao Jia, University of Minnesota

Abstract: Real world fluid flows are often turbulent. However, remarkably, certain flow patterns, such as two dimensional shear flows and vortices, can be quite stable even at high Reynolds numbers. These flow patterns can be observed in large scale flows, e.g. the great red spot on Jupiter, and the analysis of their stability is a fascinating mathematical problem. 

The stability of fluid flows is an old problem, considered already in the late 19th and early 20th century by Kelvin, Rayleigh, Orr, among many others. The classical works concern discrete eigenvalues of the linearized problem whose existence often indicate violent (exponential) instability. The continuous spectrum was only studied recently,  and leads to a new phenomenon called ``inviscid damping": the perturbation converges, weakly but not strongly as time approaches infinity. This weak convergence is consistent with 2d turbulence theory, which predicts a partial transfer of energy to high frequencies and a reverse cascade of energy to low frequencies. 

In this talk we will consider the two dimensional Euler equation focusing on shear flows and vortices, review recent progresses and discuss some important open problems on both linear and nonlinear inviscid damping. This is based on joint work with A. Ionescu.

September 30, 2021 — Community detection in random graphs and hypergraphs

Ioana Dumitriu, University of California, San Diego

Abstract: The problem of community detection is one of the fundamental problems in machine learning, with applications from recommending systems to matrix completion and biogenomics. While random graph and hypergraph models are being widely and successfully used to build and benchmark algorithms for clustering and community detection, analyzing these algorithms relies crucially on tools from random matrix and spectral graph theory. In this talk, I will give an overview of existing threshold bounds for various detection regimes in graph and hypergraph models, and mention some of the important tools used. This talk will touch upon joint work with Gerandy Brito, Christopher Hoffman, Shirshendu Ganguly, Kameron Harris, Linh Tran, Haixiao Wang, and Yizhe Zhu.

September 23, 2021 — Angular momentum in general relativity

Mu-Tao Wang, Columbia University

Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960’s, due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same.

In this talk, I shall show how the mathematical theory of optimal isometric embedding and quasilocal angular momentum identifies a correction term, and leads to a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau. 

September 16, 2021 — Variational methods for gradient flow

Li Wang, University of Minnesota

Abstract: In this talk, I will introduce a general variational framework for nonlinear evolution equations with a gradient flow structure, which arise in material science, animal swarms, chemotaxis, and deep learning, among many others. Building upon this framework, we develop numerical methods that have built-in properties such as positivity preserving and entropy decreasing, and resolve stability issues due to the strong nonlinearity. Two specific applications will be discussed. One is the Wasserstein gradient flow, where the major challenge is to compute the Wasserstein distance and resulting optimization problem. I will show techniques to overcome these difficulties. The other is to simulate crystal surface evolution, which suffers from significant stiffness and therefore  prevents simulation with traditional methods on fine spatial grids. On the contrary, our method resolves this issue and is proved to converge at a rate independent of the grid size. 

April 22, 2021 — Scalable spaces

Fedor Manin, University of California, Santa Barbara

Abstract: 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.

April 15, 2021 — Large deviations for lacunary trigonometric sums

Kavita Ramanan, Brown University

Abstract: 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.

April 1, 2021 — Electric-magnetic duality between periods and L-functions

David Ben-Zvi, University of Texas, Austin

Abstract: 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.

March 25, 2021 — From differential equations to deep learning for image analysis

Carola-Bibiane Schönlieb, University of Cambridge

Abstract: 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.

March 18, 2021 — Mathematics and physics of moiré patterns

Mitchell Luskin, University of Minnesota, Twin Cities

Abstract: 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.

March 4, 2021 — Bessel F-crystals for reductive groups

Xinwen Zhu, California Institute of Technology

Abstract: 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.

February 25, 2021 — From Grassmannians to Catalan numbers

Thomas Lam, University of Michigan, Ann Arbor

Abstract: 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.

February 18, 2021 — Wild ramification and cotangent bundle in mixed characteristic

Takeshi Saito, University of Tokyo

Abstract: 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.

February 11, 2021 — The quartic integrability and long time existence of steep water waves in 2d

Sijue Wu, University of Michigan, Ann Arbor

Abstract: 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.

February 4, 2021 — Pastures, polynomials, and matroids

Matthew Baker, Georgia Institute of Technology

Abstract: 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.

January 28, 2021 — Hodge theory of p-adic varieties

Wieslawa Niziol, Sorbonne Université

Abstract: 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.

December 1, 2020 — Galois symmetries of the stable homology of integer symplectic groups

Akshay Venkatesh, Institute for Advanced Study

Abstract: 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.

November 19, 2020 — On the Ramanujan conjecture and its generalisations

Ana Caraiani, Imperial College London

Abstract: 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.

November 12, 2020 — Small scale creation and singularity formation in fluid mechanics

Alexander Kiselev, Duke University

Abstract: 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.

November 5, 2020 — Low-degree hardness of random optimization problems

David Gamarnik, Massachusetts Institute of Technology

Abstract: 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.

October 29, 2020 — The Weyl group and the nilpotent orbit

Zhiwei Yun, Massachusetts Institute of Technology

Abstract: 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.

October 22, 2020 — An analytic version of the Langlands correspondence for complex curves

Edward Frenkel, University of California, Berkeley

Abstract: 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.

October 15, 2020 — The diffeomorphism group of a 4-manifold

Daniel Ruberman, Brandeis University

Abstract: 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.

October 8, 2020 — Mean-field disordered systems and Hamilton-Jacobi equations

Jean-Christophe Mourrat, Courant Institute of Mathematical Sciences, New York University

Abstract: 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.

October 1, 2020 — Gradient variational problems

Richard Kenyon, Yale University

Abstract: 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.

September 24, 2020 — Random walks in graph-based learning

Jeff Calder, University of Minnesota, Twin Cities

Abstract: 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).

September 17, 2020 — 25 years since Fermat's Last Theorem

Frank Calegari, University of Chicago

Abstract: 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.

March 5, 2020 — Cluster formation and self-assembly in stratified fluids: a novel mechanism for particulate aggregation

Richard McLaughlin, University of North Carolina at Chapel Hill

Abstract: The experimental and mathematical study of the motion of bodies immersed in fluids with variable concentration fields (e.g. temperature or salinity) is a problem of great interest in many applications, including delivery of chemicals in laminar micro-channels, or in the distribution of matter in the ocean. In this lecture we present some recent experimental and mathematical advances we have made for several such problems. First, we review results on how the shape of a tube can be used to sculpt the profile of chemical delivery in pressure driven laminar shear flows. Then, we explore recent results for the behavior of matter trapped vertically in a variable density water column. For this second problem, we experimentally observe and mathematically model a new attractive mechanism we have found in our laboratory by which particles suspended within stratification may self-assemble and form large aggregates without need for short range binding effects (adhesion). This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and the geometry of the aggregate, which produces toroidal flows. We show that these flows yield attractive horizontal forces between particles. We experimentally observe that many particles demonstrate a collective motion revealing a system which self-assembles, appearing to solve jigsaw-like puzzles on its way to organizing into a large scale disc-like shape, with the effective force increasing as the collective disc radius grows. We overview our modeling and simulation campaign towards understanding this intriguing dynamics, which may play an important role in the formation of particle clusters in lakes and oceans.

February 11, 2020 — Understanding maps between Riemann surfaces

Felix Janda, Institute for Advanced Study in Princeton

Abstract: Moduli spaces of Riemann surfaces are a fundamental object in algebraic geometry. Their geometry is rich and holds many outstanding mysteries. One way to probe genus g Riemann surfaces is to understand the maps they admit to the simplest Riemann surface, the Riemann sphere. In my talk, I will describe one facet of this approach, a formula for the double ramification cycle (joint work with R. Pandharipande, A. Pixton and D. Zvonkine). Along the way, we will see connections to combinatorics, number theory and symplectic geometry.

January 23, 2020 — Modularity and the Hodge/Tate conjectures for some self-products

Laure Flapan, Massachusetts Institute of Technology

Abstract: If X is a smooth projective variety over a number field, the Hodge and Tate conjectures describe how information about the subvarieties of X is encoded in the cohomology of X. We explore the role that certain automorphic representations, called algebraic Hecke characters, can play in understanding which cohomology classes of X arise from subvarieties. We use this to deduce the Hodge and Tate conjectures for certain self-products of varieties, including some self-products of K3 surfaces. This is joint work with J. Lang.

January 21, 2020 — Optimal Transport as a Tool in Analytic Number Theory and PDEs

Stefan Steinerberger, Yale University

Abstract: Optimal Transport is concerned with the question of how to best move one measure to another (this could be sand on a beach or products from a warehouse to consumers). I will explain the basic definition of Wasserstein distance and then describe how it can be used as a tool to say interesting things in other fields. (1) How to get new regularity statements for classical objects in number theory almost for free (irrational rotations on the torus, quadratic residues in finite fields). (2) How to best distribute coffee shops over downtown Minneapolis. (3) Finally, how to obtain higher dimensional analogues of classical Sturm-Liouville theory: simply put, Sturm-Liouville theory says that eigenfunctions of the operator Ly = -y''(x) +p(x)y(x) (think of sin(kx) and cos(kx)) cannot have an arbitrary number of roots; we present a generalization to higher dimensions that is based on a simple (geometric) inequality.

December 5, 2019 — Modular forms on exceptional groups

Aaron Pollack, Duke University

Abstract: When G is a reductive (non-compact) Lie group, one can consider automorphic forms for G. These are functions on the locally symmetric space X_G associated to G that satisfy some sort of nice differential equation. When X_G has the structure of a complex manifold, the _modular forms_ for the group G are those automorphic forms that correspond to holomorphic functions on X_G. They possess close ties to arithmetic and algebraic geometry. For certain exceptional Lie groups G, the locally symmetric space X_G is not a complex manifold, yet nevertheless possesses a very special class of automorphic functions that behave similarly to the holomorphic modular forms above. Building upon work of Gan, Gross, Savin, and Wallach, I will define these modular forms and explain what is known about them.

December 3, 2019 — Mirror symmetry and canonical bases for quantum cluster algebras

Travis Mandel, University of Edinburgh

Abstract: Mirror symmetry is a phenomenon which relates the symplectic geometry of one space X to the algebraic geometry of another space Y. One consequence is that a canonical basis of regular functions on Y can be defined in terms of certain counts of holomorphic curves in X. I'll discuss the application of this to (quantum) cluster algebras --- certain combinatorially defined algebras whose definition was motivated by the appearance of canonical bases in representation theory and Teichmüller theory.

December 2, 2019 — Analysis and geometry of free boundaries: recent developments

Mariana Smit Vega Garcia, Western Washington University

Abstract: In the applied sciences one is often confronted with free boundaries, which arise when the solution to a problem consists of a pair: a function u (often satisfying a partial differential equation (PDE)), and a set where this function has a specific behavior. Two central issues in the study of free boundary problems and related problems in calculus of variations and geometric measure theory are:

1. What is the optimal regularity of the solution u?
2. How smooth is the free boundary (or how smooth is a certain set related to u)?

In this talk, I will overview recent developments in obstacle type problems and almost minimizers of Bernoulli-type functionals, illustrating techniques that can be used to tackle questions (1) and (2) in various settings.

The study of the classical obstacle problem - one of the most renowned free boundary problems - began in the ’60s with the pioneering works of G. Stampacchia, H. Lewy and J. L. Lions. During the past five decades, it has led to beautiful and deep developments in the calculus of variations and geometric partial differential equations. Nowadays obstacle type problems continue to offer many challenges and their study is as active as ever.

While the classical obstacle problem arises from a minimization problem (as many other PDEs do), minimizing problems with noise lead to the notion of almost minimizers. Interestingly, though deeply connected to "standard" free boundary problems, almost minimizers do not satisfy a PDE as minimizers do, requiring additional tools from geometric measure theory to address (1) and (2).

November 26, 2019 — Schrodinger solutions on sparse and spread-out sets

Xiumin Du, University of Maryland

Abstract: If we want the solution to the Schrodinger equation to converge to its initial data pointwise, what's the minimal regularity condition for the initial data should be? I will present recent progress on this classic question of Carleson. This pointwise convergence problem is closely related to other problems in PDE and geometric measure theory, including spherical average Fourier decay rates of fractal measures, Falconer's distance set conjecture, etc. All these problems essentially ask how to control Schrodinger solutions on sparse and spread-out sets, which can be partially answered by several recent results derived from induction on scales and Bourgain-Demeter's decoupling theorem.

November 26, 2019 — K-stability and moduli spaces of Fano varieties

Yuchen Liu, Yale University

Abstract: Fano varieties are positively curved algebraic varieties which form one of the three building blocks in the classification. Unlike the case of negatively curved varieties, moduli spaces of Fano varieties (even smooth ones) can fail to be Hausdorff. K-stability was originally invented as an algebro-geometric notion characterizing the existence of K\"ahler-Einstein metrics on Fano varieties. Recently, people have found strong evidence toward constructing compact Hausdorff moduli spaces of Fano varieties using K-stability. In this talk, I will discuss recent progress in this approach, including an algebraic proof of the existence of Fano K-moduli spaces, and describing these moduli spaces explicitly. This talk is partly based on joint works with H. Blum and C. Xu.

November 19, 2019 — Random matrix theory and supersymmetry techniques

Tatyana Shcherbyna, Princeton University

Abstract: Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a spacial structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY) . SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width.

November 14, 2019 — $p$-adic estimates for exponential sums on curves

Joe Kramer-Miller, University of California, Irvine

Abstract: A central problem in number theory is that of finding rational or integer solutions to systems of polynomials in several variables. This leads one naturally to the slightly easier problems of finding solutions modulo a prime $p$. Using a discrete analogue of the Fourier transformation, this modulo $p$ problem can be reformulated in terms of exponential sums. We will discuss $p$-adic properties of such exponential sums in the case of higher genus curves as well as connections to complex differential equations.

November 14, 2019 — Applications of Frobenius beyond prime characteristic

Daniel Hernández, University of Kansas

Abstract: Recall that the Frobenius morphism is simply the map sending an element in a ring of prime characteristic $p>0$ -- say, a polynomial with coefficients in a finite field -- to its $p$-th power. Though simple to define, Frobenius has proven to be a useful and effective tool in algebraic geometry, representation theory, number theory, and commutative algebra. Furthermore, and remarkably, some of the most interesting applications of Frobenius are to the study of objects defined over the complex numbers, and more generally, over a field of characteristic zero! In this talk, we will discuss some of these applications, with an eye towards classical singularity theory and birational algebraic geometry, both over the complex numbers.

November 12, 2019 — Unraveling Local Cohomology

Emily Witt, University of Kansas

Abstract: Local cohomology modules are fundamental tools in commutative algebra, due to the algebraic and geometric information they carry. For instance, they can help determine the number of equations necessary to define an affine variety. Unfortunately, however, the application of local cohomology is limited by the fact that these modules are typically very large (e.g., not finitely generated), and can be difficult to determine explicitly. In this talk, we discuss new techniques developed to understand the structure of local cohomology (e.g., coming from invariant theory). We also describe recently-discovered "connectedness properties" of spectra that local cohomology encodes.

November 7, 2019 — Eisenstein Series on Loop Groups and their Metaplectic Covers

Manish Patnaik, University of Alberta

Abstract: Both the Langlands-Shahidi method of studying automorphic L-functions and approach via the theory of Weyl group multiple Dirichlet series to studying moments of L-functions now require new classes of groups with which to work. In this talk, I will explain our progress on extending these techniques to certain infinite-dimensional Kac-Moody groups, namely loop groups (and their metaplectic covers). Of note in our work is the presence of two quite different types of Eisenstein series that exist on the same group and which need to be considered in conjunction with one other. This is a report on joint work in progress with H. Garland, S.D. Miller, and A. Puskas.

November 7, 2019 — On various questions (and answers) in High-dimensional probability

Galyna Livshyts, Georgia Tech

Abstract: In this talk, several topics from High-dimensional probability shall be discussed. This fascinating area is rich in beautiful problems, and several easy-to-state questions will be outlined. Further, some connections between them will be explained throughout the talk.

I shall discuss several directions of my research. One direction is invertibility properties of inhomogeneous random matrices: I will present sharp estimates on the small ball behavior of the smallest singular value of a very general ensemble of random matrices, and will briefly explain the new tools I developed in order to obtain these estimates.

Another direction is isoperimetric-type inequalities in high-dimensional probability. Such inequalities are intimately tied with concentration properties of probability measures. Among other results, I will present a refinement of the concavity properties of the standard gaussian measure in an n-dimensional euclidean space, under certain structural assumptions, such as symmetry. This result constitutes the best known to date estimate in the direction of the conjecture of Gardner and Zvavitch from 2007.

The above topics will occupy most of the time of the presentation. In addition, I shall briefly mention other directions of my research, including noise-sensitivity estimates for convex sets, or, in other words, upper bounds on perimeters of convex sets with respect to various classes of probability distributions. If time permits, I will discuss my other results, such as small ball estimates for random vectors with independent coordinates, and partial progress towards Levi-Hadwiger illumination conjecture for convex sets in high dimensions.

October 31, 2019 — Differential operators on invariant rings

Anurag Singh, University of Utah

Abstract: Work of Levasseur and Stafford describes the rings of differential operators on various classical invariant rings of characteristic zero; in each of these cases, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of reductive groups over the complex numbers, Smith and Van den Bergh asked if reduction modulo p works for differential operators in this context. In joint work with Jack Jeffries, we establish that this is not the case for various classical groups.

October 17, 2019 — Hopf monoids relative to a hyperplane arrangement

Marcelo Aguiar, Cornell University

Abstract: The talk is based on recent and ongoing work with Swapneel Mahajan. We will introduce a notion of Hopf monoid relative to a real hyperplane arrangement. When the latter is the braid arrangement, the notion is closely related to that of a Hopf monoid in Joyal's category of species, and to the classical notion of connected graded Hopf algebra. We are able to extend many concepts and results from the classical theory of connected Hopf algebras to this level. The extended theory connects to the representation theory of a certain finite dimensional algebra, the Tits algebra of the arrangement. This perspective on Hopf theory is novel even when applied to the classical case. We will outline our approach to generalizations of the classical Leray-Samelson, Borel-Hopf, and Cartier-Milnor-Moore theorems to this setting. Background on hyperplane arrangements will be reviewed.

September 19, 2019 — The Langlands Program: An Introduction and Recent Progress

Solomon Friedberg, Boston College

Abstract: The Langlands Program, connecting algebra, analysis and geometry in diverse ways, is foundational to modern number theory. I will introduce this program and indicate some recent progress. As we shall see, a great deal still remains to be done.

May 2, 2019 — Singularity Formation in 3D Euler Equations and Related Models

Thomas Hou, Ordway Vsitor, California Institute of Technology

Abstract: Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is a long-standing open question in mathematical fluid dynamics. Recent computations have provided strong numerical evidence that the 3D Euler equations develop a finite time singularity from smooth initial data. I will report some recent progress in providing a rigorous justification of the singularity formation in the 3D Euler equations and related models.

April 30, 2019 — The NTRU Class of cryptosystems, lattices, and post quantum cryptograph

Jeffrey Hoffstein, Brown University

Abstract: I'll discuss the nature and origins of public key cryptography, and the dangers that the development of quantum computers pose to the security of the internet and virtually all cryptosystems presently in widespread use. All public key cryptographic systems are based on a hard mathematical problem, and I'll explain how, while originally despised, hard problems based on lattices have evolved to become the most promising direction for the development of quantum resistant cryptography. I'll focus on NTRU, the earliest effective and efficient public key cryptosystem, which was developed in 1996 by myself, Jill Pipher and Joe Silverman. I will also discuss the recent history of public key cryptography. The talk will require no previous knowledge of lattices or cryptography, and will be aimed at a wide audience.

April 25, 2019 — Recent progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures for unitary groups

Raphael Beuzart-Plessis, CNRS and Columbia University

Abstract: A celebrated result of Waldspurger from the eighties express the central value of certain base-change L-functions for $GL_2$ as toric periods of modular forms or generalizations thereof. In the mid 2000s Gan, Gross and Prasad have formulated conjectural generalizations to higher rank classical groups relating the non-vanishing of central values of certain automorphic L-functions to the non-vanishing of certain explicit integrals of automorphic forms that are called 'automorphic periods'. These predictions have been subsequently refined by Ichino-Ikeda and N.Harris into precise identities relating the two invariants. These conjectures also have local counterparts which concern certain branching laws in the representation theory of real or p-adic groups. Most of these conjectures have now been established for unitary groups. This talk aims to give an introduction to this circle of ideas and to review recent results on the subject.

March 14, 2019 — Short-Time Asymptotic Methods In Financial Mathematics

Jose Figueroa-Lopez, Washington University

In this talk, we will be concerned with the average values of certain random functionals of the path of a stochastic process during a given time period. High-order asymptotic characterizations of such values when the time period shrinks to 0 have a wide range of applications. In statistics, they are instrumental in establishing infill asymptotic properties of high-frequency based statistical methods of stochastic processes. In finance, they have been used as model selection and calibration tools based on near expiration option prices. In some Engineering problems, they also show up as a method to solve a problem in continuous time by looking at the analogous problem in discrete time and shrinking the time step to 0. These short-time asymptotic methods are especially useful in the study of complex models with jumps and stochastic volatility due to the lack of tractable formulas and efficient statistical and numerical procedures. In this talk, I will discuss some recent advances in the area and illustrate their broad relevance in several contexts.

Bio: Dr. Figueroa-López is a Professor in the Department of Mathematics and Statistics at Washington University in St. Louis. He currently serves as part of the executive committee and the chair of the statistics committee of the Department. Formerly he was Associate Professor of Statistics at Purdue University, where he served as Associate Director of the Computational Finance Program and as a member of the University Senate. Professor Figueroa’s ongoing research includes short-time asymptotics of jump-diffusion models, diffusion limits of Limit Order Book models, optimal limit order placement problems, market making via reinforced learning, and optimal tuning of high-frequency based econometric methods. He was awarded the NSF career award in 2012 and currently has two active NSF grants on the interplay of finance, statistics, and probability. He is an Associate Editor of the SIAM Journal on Financial Mathematics (SIFIN) and a former Associate Editor of Electronic Journal of Statistics.

March 5, 2019 — Cohomology of Shimura varieties

Sug Woo Shin, University of California, Berkeley

Abstract: Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.

February 5, 2019 — A Multiscale/Multiphysics Coupling Framework for Heart Valve Damage

Yue Yu, Lehigh University

Abstract: Bioprosthetic heart valves (BHVs) are the most popular artificial replacements for diseased valves that mimic the structure of native valves. However, the life span of BHVs remains limited to 10-15 years, and the mechanisms that underlie BHVs failure remain poorly understood. Therefore, developing a unifying mathematical framework which captures material damage phenomena in the fluid-structure interaction environment would be extremely valuable for studying BHVs failure. Specifically, in this framework the computational domain is composed of three subregions: the fluid (blood) , the fracture structure (damaged BHVs) modeled by the recently developed nonlocal (peridynamics) theory, and the undamaged thin structure (undamaged BHVs). These three subregions are numerically coupled to each other with proper interface boundary conditions.

In this talk, I will introduce two sub-problems and the corresponding numerical methods we have developed for this multiscale/multiphysics framework. In the first problem the coupling strategy for fluid and thin structure is investigated. This problem presents unique challenge due to the large deformation of BHV leaflets, which causes dramatic changes in the fluid subdomain geometry and difficulties on the traditional conforming coupling methods. To overcome the challenge, the immersogeometric method was developed where the fluid and thin structure are discretized separately and coupled through penalty forces. To ensure the capability of the developed method in modeling BHVs, we have verified and validated this method. The second problem focuses on the nonlocal Neumann-type interface boundary condition which plays a critical role in the fluid—peridynamics coupling framework. In the nonlocal models the Neumann-type boundary conditions should be defined in a nonlocal way, namely, on a region with non-zero volume outside the surface, while in fluid—structure interfaces the hydrodynamic loadings from the fluid side are typically provided on a sharp co-dimensional one surface. To overcome this challenge, we have proposed a new nonlocal Neumann-type boundary condition which provides an approximation of physical boundary conditions on a sharp surface, with an optimal asymptotic convergence rate to the local counter part. Based on this new boundary condition, we have developed a fluid—peridynamics coupling framework without overlapping regions.

January 31, 2019 — From infinite random matrices over finite fields to square ice

Leonid Petrov, University of Virginia

Abstract: Asymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.

January 22, 2019 — Multiscale Problems in Cell Biology

Chuan Xue, The Ohio State University

Abstract: Complex biological systems involve multiple space and time scales. To get an integrated understanding of these systems involves multiscale modeling, computation and analysis. In this talk, I will discuss two such examples in cell biology and illustrate how to use multiscale methods to explain experimental data. The first example is on chemotaxis of bacterial populations. I will present recent progress on embedding information of single cell dynamics into models of cell population dynamics. I will clarify the scope of validity of the well-known Patlak-Keller-Segel chemotaxis equation and discuss alternative models when it breaks down. The second example is on the axonal cytoskeleton dynamics in health and disease. I will present a stochastic multiscale model that gave the first mechanistic explanation for the cytoskeleton segregation phenomena observed in many neurodegenerative diseases.

December 6, 2018 — Number theory over function fields and geometry

Will Sawin, Columbia University

Abstract: The function field model involves taking problems in classical analytic number theory and replacing the integers with polynomials over a finite field. This preserves most of the complexity of these problems while giving them a stronger relationship to geometry, allowing new techniques to be applied. We will explain how this works using a recent example involving sums of the divisor function where, thanks to good luck, particularly simple methods can prove a particularly powerful result.

November 29, 2018 — Interpolative decomposition and its applications

Lexing Ying, Stanford (Ordway Visitor)

Abstract: Interpolative decomposition is a simple and yet powerful tool for approximating low-rank matrices. After discussing the theory and algorithm, I will present a few new applications of interpolative decomposition in numerical partial differential equations, quantum chemistry, and machine learning.

November 29, 2018 — The role of Energy in Regularity

Max Engelstein, Massachusetts Institute of Technology

Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.

However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.

We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.

November 27, 2018 — Surface bundles, monodromy, and arithmetic groups

Bena Tshishiku, Harvard University

Abstract: Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to "compute" (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber over a surface. This is joint work with Nick Salter.

November 15, 2018 — (Log)-Epiperimetric Inequality and the Regularity of Variational Problems

Luca Spolaor, Massachusetts Institute of Technology

Abstract: In this talk I will present a new method for studying the regularity of minimizers to variational problems. I will start by introducing the notion of blow-up, using as a model case the so-called Obstacle problem. Then I will state the (Log)-epiperimetric inequality and explain how it is used to prove uniqueness of the blow-up and regularity results for the solution near its singular set. I will then show the flexibility of this method by describing how it can be applied to other free-boundary problems and to (almost)-area minimizing currents. Finally I will describe some future applications of this method both in regularity theory and in other settings.

October 11, 2018 — Propagation of bistable fronts through a perforated wall

Hiroshi Matano, Meiji University, Tokyo (Ordway Visitor)

Abstract: We consider a bistable reaction-diffusion equation on ${\bf R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with periodically arrayed holes. More precisely, $K$ is a closed subset of ${\bf R}^N$ with smooth boundary such that its projection onto the $x_1$-axis is bounded, while it is periodic in the rest of variables $(x_2,\ldots, x_N)$. We assume that ${\bf R}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from $x_1 = +\infty$ meets the wall $K$.

We first show that there is clear dichotomy between `propagation' and `blocking'. In other words, the traveling front either completely penetrates through the wall or is totally blocked, and that there is no intermediate behavior. This dichotomy result will be proved by what we call a De Giorgi type lemma for an elliptic equation on ${\bf R}^N$. Then we will discuss sufficient conditions for blocking, and those for propagation. This is joint work with Henri Berestycki and Francois Hamel.

September 25, 2018 — Microlocal codimension-three conjecture

Kari Vilonen, University of Melbourne (Ordway Visitor)

Abstract: Special functions (or distributions) can be understood and analyzed in terms of the systems of differential equations they satisfy. To this end, a general theory of systems of linear (micro) differential equations was developed by the Sato school in Kyoto. This point of view, in its various incarnations, is now ubiquitous in many parts of mathematics. For example, in the geometric Langlands program and representation theory it allows us to replace functions and group representations by geometric objects, perverse sheaves or D-modules. It has been well-known for a long time that the description of these objects gains more symmetry when one passes to the cotangent bundle. We explain the shape of the general microlocal structure of these objects and discuss, in particular, the role played by the codimension-three conjecture which was proved by Masaki Kashiwara and the speaker a few years ago.

September 13, 2018 — Singularities of the Ricci flow and Ricci solitons

Huai-Dong Cao, Lehigh University (Ordway Visitor)

Abstract: Understanding formation of singularities has been an important subject in the study of the Ricci flow and other geometric flows. It turns out generic singularities in the Ricci flow are modeled on shrinking Ricci solitons. In this talk, I will discuss some of the recent progress on classifications of shrinking Ricci solitons and their stability/instability with respect to Perelman's $\nu$-entropy.

May 3, 2018 — Fractal solutions of dispersive PDE on the torus

Burak Erdogan, University of Illinois

Abstract: In this talk we discuss qualitative behavior of certain solutions to linear and nonlinear dispersive partial differential equations such as Schrodinger and Korteweg-de Vries equations. In particular, we will present results on the fractal dimension of the solution graph and the dependence of solution profile on the algebraic properties of time.

April 26, 2018 — Phase transitions and conic geometry

Joel Tropp (Ordway Visitor)

Abstract: A phase transition is a sharp change in the behavior of a mathematical model as one of its parameters changes. This talk describes a striking phase transition that takes place in conic geometry. First, we will explain how to assign a notion of "dimension" to a convex cone. Then we will use this notion of "dimension" to see that two randomly oriented convex cones share a ray with probability close to zero or close to one. This fact has implications for many questions in signal processing. In particular, it yields a complete solution of the "compressed sensing" problem about when we can recover a sparse signal from random measurements. This talk is designed for a general mathematical audience.

Based on joint works with Dennis Amelunxen, Martin Lotz, Mike McCoy, and Samet Oymak.

April 26, 2018 — Bounding torsion in cohomology

Bhargav Bhatt, University of Michigan (Ordway Visitor)

Abstract: The integral cohomology groups of a complex algebraic variety are one of the most fundamental invariants associated to the variety. The ranks of these groups are well understood in terms of the equations defining the variety, thanks to Hodge theory. However, the torsion tends to be more "transcendental" in nature and is not easily accessible via algebraic techniques. Torsion cohomology classes have played a pivotal role in many recent advances in number theory, algebraic geometry, and representation theory, so it is important to better understand torsion from an algebraic perspective. In this talk, I'll discuss my recent work with Morrow and Scholze that explains how to bound the torsion explicitly in terms of the equations defining the variety.

April 12, 2018 — Perfectoid spaces, homological conjectures, and singularities in mixed characteristic

Linquan Ma, University of Utah

Abstract: The homological conjectures have been a focus of research in commutative algebra since 1960s. They concern a number of interrelated conjectures concerning various homological properties of commutative rings to their internal ring structures. These conjectures had largely been resolved for rings that contain a field, but several remained open in mixed characteristic---until recently Yves Andre proved Hochster's direct summand conjecture and the existence of big Cohen-Macaulay algebras, which lie in the heart of the homological conjectures. The main new ingredient in the solution is to systematically use Scholze's theory of perfectoid spaces, which leads to many further developments in the study of mixed characteristic singularities. For example, using perfectoid algebras and big Cohen-Macaulay algebras, we can define the mixed characteristic analog of rational/F-rational and log terminal/F-regular singularities, and they turn out to have many applications to singularities over arithmetic families (this is based on recent joint work with Karl Schwede). In this talk, we will survey all these results.

April 12, 2018 — Matryoshka Dolls and Tinkertoys: Calabi-Yau Manifolds and Supersymmetry

Charles Doran, University of Alberta

Abstract: The geometry of Calabi-Yau manifolds and supersymmetry are key ingredients in string theory. We will take a non-traditional point of view on both of these. The hereditary structure of nested Calabi-Yau manifolds underlies string dualities and motivates an algebraic reinterpretation. Dimensional reduction of supermultiplets produces a discrete scaffolding (Adinkra graphs) that nevertheless possesses an emergent form of geometry. Surprisingly, we find that supermultiplets themselves bear the stamp of Calabi-Yau geometry. Our proof of this uses work of H.S.M. Coxeter inspired by a woodcut by M.C. Escher. The talk is designed to be broadly accessible to graduate students.

April 10, 2018 — Hankel Transforms, Langlands Functoriality and Functional Equation of Automorphic L-functions (I)

Ngo Bao Chau, University of Chicago (Ordway Visitor)

Abstract: Since the beginning of the century, several approaches to Langlands functoriality conjecture have been proposed by Langlands himself, by Braverman-Kazhdan and Lafforgue, ....
In this lecture I will explain how these ideas may be combined and connected to recents works on singularities of certain arc spaces.

April 5, 2018 — Boundary Conditions for Crystalline Defects

Christoph Ortner, University of Warwick (Ordway Visitor)

Abstract: A key problem of atomistic materials modelling is to determine properties of crystalline defects, such as geometries, formation energies, or mobility, from which meso-scopic material properties or coarse-grained models (e.g., diffusion, dislocation dynamics, fracture) can be derived.In this lecture I will focus on the most basic task: determining the equilibrium configuration of a crystalline defect (time permitting I can comment on other properties). Even the very first question, "What is the <exact> model?", warrants discussion. To answer it I will present a thermo-dynamic limit argument which is interesting in its own right but more importantly provides a machinery for quantifying approximation errors in typical computational models, both classical as well as modern multi-scale schemes. It quickly transpires that the error is always due to an approximate descriptions of the crystalline far-field, resulting in an error in the boundary condition. This perspective naturally leads to an exceptionally promising new class of computational schemes where much of the hard work is done analytically in deriving higher-order continuum descriptions of the crystalline far-field.

March 22, 2018 — Statistics of the Riemann Zeta Function and L-functions

Michael Rubinstein, University of Waterloo

Abstract: The Riemann Zeta function and related L-functions stand at the epicenter of number theory. They have played a crucial role in our understanding of fundamental arithmetic and algebraic objects. But many basic questions about their properties elude us. In my talk l will give an overview of various fascinating statistics, some proven and others conjectural, concerning the Riemann Zeta function and L-functions.

March 8, 2018 — SRB measures for infinite-dimensional dynamical systems with potential applications to PDE

Alex Blumenthal, University of Maryland

Abstract: I will talk about the extension to the setting of Banach space mappings a concept which has proven highly useful in the study of finite-dimensional dynamical systems exhibiting chaotic behavior, that of SRB measures. This extended notion of SRB measure and our results potentially apply to a large class of dissipative PDE, including dissipative parabolic and dispersive wave equations.

We generalize two results known in the finite-dimensional setting. The first is a geometric result, absolute continuity of the stable foliation, which in particular implies that an SRB measure with no zero exponents is visible, in the sense of time averages converging to spatial averages, with respect to a large subset of phase space. The second is the characterization of the SRB property in terms of the relationship between a priori different quantifications of chaotic behavior, Lyapunov exponents and metric entropy.

Complications of our infinite-dimensional environment include: (1) the absence of Lebesgue measure as a reference measure, not even k-dimensional volume elements (whereas the finite dimensional theory heavily involves the notion of volume growth along unstable leaves); and (2) mappings in our setting are not locally onto or differentiably invertible, possibly exhibit arbitrarily strong rates of contraction (even near attractors).

This work is joint with Lai-Sang Young.

March 1, 2018 — Associativity and Integrability

Rui Loja Fernandes, University of Illinois at Urbana-Champaign (Ordway Visitor)

Abstract: A fundamental result of Lie theory is Lie’s Third Theorem which states that every finite dimensional Lie algebra integrates to a Lie group. This result fails for infinite dimensional Lie algebras (e.g., Banach Lie algebras) and it also fails for (finite dimensional) Lie algebroids. But every reasonable Lie algebra (finite or infinite dimensional) integrates to a local Lie group and every Lie algebroid integrates to a local Lie groupoid. On the other hand, a classical theorem of Mal’cev states that a local group is enlargeable to a group if and only if it is global associative. This talk will be an introduction to Lie algebroids and groupoids, focusing on the failure of Lie’s Third Theorem and its relationship to the failure of associativity.

February 22, 2018 — Front capturing schemes for nonlinear PDEs with a free boundary limit

Li Wang, State University of New York at Buffalo

Abstract: Evolution in physical or biological systems often involves interplay between nonlinear interaction among the constituent “particles,” and convective or diffusive transport, which is driven by density dependent pressure. When the pressure-density relationship becomes highly nonlinear, the evolution equation converges to a free boundary problem as a stiff limit. In terms of numerics, the nonlinearity and degeneracy bring great challenges, and there is lack of standard mechanism to capture the propagation of the front in the limit. In this talk, I will introduce a numerical scheme for tumor growth models based on a prediction-correction reformulation, which naturally connects to the free boundary problem in the discrete sense. As an alternative, I will present a variational method for a class of continuity equations (such as the Keller-Segel model) using the gradient flow structure, which has built-in stability, positivity preservation and energy decreasing property, and serves as a good candidate for capturing the stiff pressure limit.

February 13, 2018 — Fundamental groups in arithmetic and geometry

Daniel Litt, Columbia University

Abstract: Let X be an algebraic variety -- that is, the solution set to a system of polynomial equations. Then the *fundamental group* of X has several incarnations, reflecting the geometry, topology, and arithmetic of X. This talk will discuss some of these incarnations and the subtle relationships between them, and will describe an ongoing program which aims to apply the study of the fundamental group to classical problems in
algebraic geometry and number theory.

February 6, 2018 — Topological Vistas in Neuroscience

Kathryn Hess, École polytechnique fédérale de Lausanne

Abstract: I will describe results obtained in collaboration with the Blue Brain Project on the topological analysis of the structure and function of digitally reconstructed microcircuits of neurons in the rat cortex and outline our on-going work on topology and synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use. If time allows, I will also briefly sketch other collaborations with neuroscientists in which my group is involved.

February 1, 2018 — A rational blowdown surgery on 4-manifolds

Jongil Park, Seoul National University (Ordway Visitor)

Abstract: Since gauge theory was introduced in 1982, people working on 4- manifolds have developed various techniques and surgeries and they have obtained many fruitful and remarkable results on 4-manifolds in last 35 years. Among them, a rational blowdown surgery technique initially introduced by R. Fintushel and R. Stern and later generalized by J. Park turned out to be one of the simple but powerful techniques to construct a new family of 4-manifolds.

In this talk, first I’d like to briefly review what we have obtained in 4-manifold topology by using a rational blowdown surgery. And then I’ll explain in some details that any minimal symplectic filling of the link of a quotient surface singularity can be obtained by a sequence of rational blowdowns and blowing-ups from the minimal resolution the corresponding quotient surface singularity.

January 18, 2018 — Heat kernels on Riemannian manifolds

Alexander Grigor'yan, Universität Bielefeld

Abstract: Plan of the talk:

1. Heat kernels for elliptic operators in $R^n$.
2. Laplace-Beltrami operator and its heat kernel.
3. Gaussian estimate of the heat kernel in integrated form (Davies-Gaffney).
4. Li-Yau estimate of the heat kernel. Necessary and sufficient condition in terms of volume doubling and Poincare inequality.
5. Examples of manifolds satisfying Li-Yau estimate.
6. Manifolds with ends.
7. Parabolic and non-parabolic manifolds.
8. Heat kernels on manifolds with ends.