• Title: Uncertainty Quantification of Physics-constrained Problems – Data Assimilation and Parameter Estimation
• Date: 01/09/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: 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 Physics-constrained 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 hydrogen-oxygen combustion are provided.

• Title: From Hilbert's Nullstellensatz to quotient categories
• Date: 01/12/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: 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.

• Title: Invariant and characteristic random subgroups and their applications
• Date: 01/30/2018
• Time: 10:20 AM - 11:10 AM
• Place: C304 Wells Hall
• Speaker: 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.

• Title: Reaction-diffusion equations in biology
• Date: 01/31/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Olga Turanova, UCLA

Reaction-diffusion equations describe a variety of physical and biological phenomena. In this talk, I begin by presenting the classical Fisher-KPP equation and its significance to ecology. I then describe recent results on other PDEs of reaction-diffusion type, including non-local 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.

• Title: Quantum Integrable Systems and Enumerative Geometry
• Date: 02/05/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: 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 K-theory. Around 10 years ago, physicists Nekrasov and Shatashvili proposed an unexpected relation between quantum K-theory and quantum integrable systems based on quantum groups within their studies of 3-dimensional 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 Nekrasov-Shatashvili 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 K-theory of the cotangent bundles to Grassmanians and the Heisenberg XXZ spin chain. We will also discuss relation of equivariant quantum K-theory of flag varieties and many-body integrable systems of Ruijsenaars-Schneider and Toda.

• Title: Spectral Optimization and Free Boundary Problems
• Date: 02/12/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: 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 vector-valued free boundary problems of Bernoulli type.

• Title: Algorithms for mean curvature motion of networks
• Date: 03/01/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: 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.

• Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes (Second Phillips Lecture)
• Date: 03/13/2018
• Time: 4:00 PM - 5:00 PM
• Place: 115 International Center
• Speaker: Alex Lubotzky, Hebrew University

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago, a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others. In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d greater than 1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders. By developing new isoperimetric methods for “locally minimal small” F_2-co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov’s original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders. We will describe these developments and the general area of high dimensional expanders and some of its open problems.

• Title: Quadratic forms and Clifford algebras
• Date: 03/29/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Parimala Raman, Emory University

Clifford algebras play an important role in the classification of quadratic forms over number fields. Surprisingly, the also play a critical role in studying the isotropy (existence of nontrivial zeros) of quadratic forms over function fields of curves over totally imaginary number fields. We shall explain some open questions concerning isotropy of quadratic forms over function fields of curves over number fields and their connection to Clifford algebras.

• Title: Number theory in geometry
• Date: 04/05/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Gopal Prasad, University of Michigan

Historically, it is geometry which led to important developments in several areas of mathematics including number theory. But recently there have been several instances of number theory being applied to settle important questions in geometry. I will talk about two problems in whose solution number theory has been used in a crucial way. The first one, settled in collaboration with Sai-Kee Yeung, is classification of fake projective planes and their higher dimensional analogs. (I recall that fake projective planes are smooth projective complex surfaces with same Betti numbers as the complex projective plane, but which are not isomorphic to the complex projective plane. The first such surface was constructed by David Mumford.) The second problem concerns compact Riemannian manifolds and it has the following very interesting formulation due to Mark Kac: “Can one hear the shape of a drum?”. In precise mathematical terms, the question asks whether two compact Riemannian manifolds with same spectrum (i.e., the set of eigenvalues counted with multiplicities) are isometric. The answer is in general “no”. However, Andrei Rapinchuk and I investigated Kac’s question, using number theoretic results and tools, for a particularly nice class of manifolds, namely locally symmetric spaces. The answer turned out to be very interesting and has led to several other developments which, if time permits, I will mention.

• Title: TBA
• Date: 04/11/2018
• Time: 10:20 AM - 11:10 AM
• Place: C304 Wells Hall
• Speaker: David Tannor

TBA

• Title: An Auction Dynamics Approach to Semi-Supervised Data Classification
• Date: 04/17/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Ekaterina Rapinchuk, Michigan State University

We reinterpret the semi-supervised data classification problem using an auction dynamics framework inspired by real life auctions. This novel forward and reverse auction procedure for data classification requires remarkably little training/labeled data and readily incorporates volume/class size constraints. We prove that the algorithm always terminates with the right properties for any choice of parameters and derive its computational complexity. Experimental results on benchmark machine learning datasets show that our approach exceeds the performance of current state-of-the-art methods, while requiring a fraction of the computational time. This is joint work with Matt Jacobs and Selim Esedoglu.

• Title: TBA
• Date: 04/26/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Speaker: Ben McReynolds, Purdue Univeresity

No abstract available.