Talk_id  Date  Speaker  Title 
8242

Monday 1/8 4:10 PM

David Hansen, Columbia University

Elliptic curves and padic Lfunctions
 Elliptic curves and padic Lfunctions
 01/08/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 David Hansen, Columbia University
I'll explain the notion of a padic Lfunction, try to
motivate why one might care about such a gadget, and give some history of their construction and applications. At the end of the talk I'll discuss a recent joint work with John Bergdall in which (among other things) we construct canonical padic Lfunctions associated with modular elliptic curves over totally real number fields.

8231

Tuesday 1/9 4:10 PM

Yoonsang Lee, Lawrence Berkeley National Laboratory

Uncertainty Quantification of Physicsconstrained Problems – Data Assimilation and Parameter Estimation
 Uncertainty Quantification of Physicsconstrained Problems – Data Assimilation and Parameter Estimation
 01/09/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Yoonsang Lee, Lawrence Berkeley National Laboratory
Observation data along with mathematical models play a crucial role in improving prediction skills in science and engineering. In this talk we focus on the recent development of uncertainty quantification methods, data assimilation and parameter estimation, for Physicsconstrained problems that are often described by partial differential equations. We discuss the similarities shared by the two methods and their differences in mathematical and computational points of view and future research topics. As applications, numerical weather prediction for geophysical flows and parameter estimation of kinetic reaction rates in the hydrogenoxygen combustion are provided.

8230

Wednesday 1/10 4:10 PM

Preston Wake, UCLA

Quantifying congruences between Eisenstein series and cusp forms
 Quantifying congruences between Eisenstein series and cusp forms
 01/10/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Preston Wake, UCLA
Consider the following two problems in algebraic number theory:
1. For which prime numbers p can we easily show that the Fermat equation x^p + y^p =z^p has no nontrivial integer solutions?
2. Given an elliptic curve E over the rational numbers, what can be said about the group of rational points of finite order on E?
These seem like very different problems, but, surprisingly, they share a common theme: they are both related to the existence of congruences between two types of modular forms, Eisenstein series and cusp forms. We will explain these examples, and discuss a new technique for giving quantitative information about these congruences (for example, counting the number of cusp forms congruent to an Eisenstein series). We will explain how this can give finer arithmetic information than simply knowing the existence of a congruence. This is joint work with Carl WangErickson.

8232

Friday 1/12 4:10 PM

John Calabrese, Rice University

From Hilbert's Nullstellensatz to quotient categories
 From Hilbert's Nullstellensatz to quotient categories
 01/12/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 John Calabrese, Rice University
A common theme in algebraic geometry is the interplay between algebra and geometry. In this talk I will discuss a few "reconstruction theorems", in which the algebra determines the geometry.

9248

Wednesday 1/17 4:10 PM

Tristan Collins, Harvard University

SasakiEinstein metrics and Kstability
 SasakiEinstein metrics and Kstability
 01/17/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Tristan Collins, Harvard University
I will discuss the connection between SasakiEinstein metrics and algebraic geometry in the guise of Kstability. In particular, I will give a differential geometric perspective on Kstability which arises from the Sasakian view point, and use Kstability to find infinitely many nonisometric SasakiEinstein metrics on the 5sphere. This is joint work with G. Szekelyhidi.

9259

Tuesday 1/30 10:20 AM

Rostyslav Kravchenko, Northwestern University

Invariant and characteristic random subgroups and their applications
 Invariant and characteristic random subgroups and their applications
 01/30/2018
 10:20 AM  11:10 AM
 C304 Wells Hall
 Rostyslav Kravchenko, Northwestern University
The invariant random subgroups (IRS) were implicitly used by Stuck and Zimmer in 1994 and defined explicitly by Abert, Glasner and Virag in 2012. We recall the definition of IRS and discuss their properties. We also define the notion of characteristic random subgroups (CRS) which are a natural analog of IRSs for the case of the group of all automorphisms. We determine CRS for free abelian groups and for free groups of finite rank. Using our results on CRS of free groups we show that for some groups of geometrical nature there are infinitely many continuous ergodic IRS.

9261

Tuesday 1/30 4:10 PM

Kate Juschenko, Northwestern University

Amenability of discrete groups and their actions
 Amenability of discrete groups and their actions
 01/30/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Kate Juschenko, Northwestern University
The subject of amenability essentially begins in 1900's with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. This led to the study of positive, finitely additive and translation invariant measure on reals as well as on other spaces. In particular the study of isometryinvariant measure led to the BanachTarski decomposition theorem in 1924. The class of amenable groups was introduced by von Neumann in 1929, who explained why the paradox appeared only in dimensions greater or equal to three, and does not happen when we would like to decompose the twodimensional ball. In 1940's, M. Day formally defined a class of elementary amenable groups as the largest class of groups amenability of which was known to von Naumann. He asked whether there are other groups then that. Currently there are many groups that answer von NeumannDay's question. However, in each particular case it is algebraically difficult to show that the group is not elementary amenable, and analytically difficult to show that it is amenable. The talk is aimed to discuss recent developments and approaches in the field. In particular, it will be shown how to prove amenability of all known nonelementary amenable groups using only one single approach. We will also discuss techniques coming from random walks of groups.

9257

Wednesday 1/31 4:10 PM

Olga Turanova, UCLA

Reactiondiffusion equations in biology
 Reactiondiffusion equations in biology
 01/31/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Olga Turanova, UCLA
Reactiondiffusion equations describe a variety of physical and biological phenomena. In this talk, I begin by presenting the classical FisherKPP equation and its significance to ecology. I then describe recent results on other PDEs of reactiondiffusion type, including nonlocal equations arising in evolutionary ecology, as well as ones that model tumor growth (joint with Inwon Kim). I will highlight the mathematical challenges and techniques that arise in the analysis of these PDEs.

9263

Thursday 2/1 4:10 PM

Daniel Thompson, Ohio State University

Geodesic flow in nonpositive curvature: An inspiration for new techniques in ergodic theory
 Geodesic flow in nonpositive curvature: An inspiration for new techniques in ergodic theory
 02/01/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Daniel Thompson, Ohio State University
We discuss some recent progress in the smooth ergodic theory of geodesic flows. This talk will be suitable for a general mathematical audience, and will start with an intuitive overview of the classic results developed by luminaries such as Anosov, Bowen and Ruelle in the well understood setting of surfaces with variable negative curvature. Efforts to understand the much more difficult case of nonpositive curvature were initiated by Pesin in the 1970’s. However, despite substantial successes, the picture has remained far from complete. There has been a great deal of recent progress in this area, which has required, and motivated, the development of new machinery in the abstract theory. I will give an overview of some recent developments, including:
1) General machinery developed by Vaughn Climenhaga and myself, which gives “nonuniform" dynamical criteria for uniqueness of equilibrium measures;
2) Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher, where we apply this machinery to geodesic flow on nonpositive curvature manifolds;
3) If time permits, I will also mention related joint work with JeanFrancois Lafont and Dave Constantine, where we develop the theory of equilibrium measures for geodesic flow on locally CAT(1) spaces; these are geodesic metric spaces which generalize negative curvature Riemannian manifolds by having the “thin triangle” property.

9265

Monday 2/5 4:10 PM

Anton M. Zeitlin, Louisiana State University

Quantum Integrable Systems and Enumerative Geometry
 Quantum Integrable Systems and Enumerative Geometry
 02/05/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Anton M. Zeitlin, Louisiana State University
The correspondence between integrable systems and enumerative geometry
started roughly 25 years ago in the works of Givental and his collaborators,
studying quantum cohomology and quantum Ktheory. Around 10 years ago,
physicists Nekrasov and Shatashvili proposed an unexpected relation between
quantum Ktheory and quantum integrable systems based on quantum groups
within their studies of 3dimensional gauge theories. Their bold proposal
led to a lot of interesting developments in mathematics, bringing a new life
to older ideas of Givental, and enriching it with flavors of geometric
representation theory via the results of Braverman, Maulik, Nakajima, Okounkov
and many others. In this talk I will focus on recent breakthroughs in the
subject, leading to the proper mathematical understanding of NekrasovShatashvili
original papers as well as some other subsequent conjectures made by physicists.
Our main illustration of such a relation is an interplay between equivariant quantum Ktheory of the cotangent bundles to Grassmanians and the Heisenberg XXZ spin chain. We will also
discuss relation of equivariant quantum Ktheory of flag varieties and
manybody integrable systems of RuijsenaarsSchneider and Toda.

9266

Friday 2/9 4:10 PM

Anna Mazzucato, Pennsylvania State University

Optimal mixing and irregular transport by incompressible flows
 Optimal mixing and irregular transport by incompressible flows
 02/09/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Anna Mazzucato, Pennsylvania State University
I will discuss transport of passive scalars by incompressible flows (such as a die in a fluid) and measures of optimal mixing and stirring under physical constraint on the flow. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with a prescribed bound on certain Sobolev norms of the associated velocity, such as under an energy or an enstrophy budget. These examples are related to examples of (instantaneous) loss of Sobolev regularity for solutions to linear transport equation with nonLipschitz velocity.

9270

Monday 2/12 4:10 PM

Dennis Kriventsov, NYU Courant

Spectral Optimization and Free Boundary Problems
 Spectral Optimization and Free Boundary Problems
 02/12/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Dennis Kriventsov, NYU Courant
A classic subject in analysis is the relationship between the spectrum of the Laplacian on a domain and that domain's geometry. One approach to understanding this relationship is to study domains which extremize some function of their spectrum under geometric constraints. I will explain how to attack these problems using tools from the calculus of variations to find solutions. A key difficulty with this method is showing that the optimizers (which are a priori very weak) are actually smooth domains, and I address this issue in some recent work with Fanghua Lin. Our results are based on relating spectral optimization problems to certain vectorvalued free boundary problems of Bernoulli type.

9271

Friday 2/16 4:10 PM

Brent Nelson, UC Berkeley

Nontracial free transport
 Nontracial free transport
 02/16/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Brent Nelson, UC Berkeley
Von Neumann algebras are certain *subalgebras of bounded operators acting on a Hilbert space. They are generally thought of as noncommutative measure spaces and offer connections to many fields of mathematics (e.g. group theory, lowdimensional topology, logic, ergodic theory, and random matrix theory to name a few). In some instances an analogy with probability spaces is more appropriate, and indeed this is precisely what informs the field of free probability, wherein one uses noncommutative analogs of probabilistic notions to study the structure of von Neumann algebras. One particular example of this is free transport. In probability theory, transport refers to a measurable map between probability spaces that pushes one measure onto the other. Following work of Brenier in 1991, transportation theory has known great success. Free transport, the noncommutative analog that was introduced by Guionnet and Shlyakhtenko in 2014, offers methods for proving isomorphisms between von Neumann algebras. In this talk, I will discuss these ideas as well my work, which used free transport to prove isomorphisms between certain socalled "nontracial" von Neumann algebras.

11277

Thursday 3/1 4:10 PM

Selim Esedoglu, University of Michigan

Algorithms for mean curvature motion of networks
 Algorithms for mean curvature motion of networks
 03/01/2018
 4:10 PM  5:00 PM
 C304 Wells Hall
 Selim Esedoglu, University of Michigan
Motion by mean curvature for networks of surfaces arises in a variety of
applications, such as the dynamics of foam and the evolution of
microstructure in polycrystalline materials. It is steepest descent
(gradient flow) for an energy: the sum of the areas of the surfaces
constituting the network.
During the evolution, surfaces may collide and junctions (where three or
more surfaces meet) may merge and split off in myriad ways as the
network coarsens in the process of decreasing its energy. The first idea
that comes to mind for simulating this evolution  parametrizing the
surfaces and explicitly specifying rules for cutting and pasting when
collisions occur  gets hopelessly complicated. Instead, one looks for
algorithms that generate the correct motion, including all the necessary
topological changes, indirectly but automatically via just a couple of
simple operations.
An almost miraculously elegant such algorithm, known as threshold
dynamics, was proposed by Merriman, Bence, and Osher in 1992. Extending
this algorithm, while preserving its simplicity, to more general
energies where each surface in the network is measured by a different,
possibly anisotropic, notion of area requires new mathematical
understanding of the original version, which then elucidates a
systematic path to new algorithms.
