The concept of entropy was introduced into ergodic theory by Kolmogorov in the late 1950s and has become a pervasive tool in dynamics with applications to areas such as Riemannian geometry, analytic number theory, and Diophantine approximation. Kolmogorov's approach is based on Shannon's theory of information from the 1940s and is most generally applicable to actions of groups satisfying a kind of internal finite approximation property called amenability.
A new approach to entropy in dynamics was initiated by Bowen some ten years ago and then further developed by Li and myself. In this case one externalizes the finite approximation of the dynamics so that it occurs outside the acting group, and then counts these models in the spirit of Boltzmann's work in statistical mechanics. This notion of entropy applies to the much larger class of acting groups satisfying the property of soficity, which includes free groups. In fact it is not known whether nonsofic groups exist.
I will discuss some of the basic ideas at play in both amenable and sofic entropy, and describe how the passage from single transformations to actions of general amenable and sofic groups marks a shift in applications away from geometry and smooth dynamics and more towards noncommutative harmonic analysis and operator algebras.
September 14 3:30 PM SDRICH 207
Marcelo Disconzi (Vanderbilt University) Viscous Fluids in General Relativity
We consider the problem of describing relativistic viscous fluids. More specifically, we study Einstein's equations coupled to a relativistic version of the Navier-Stokes equations. After motivating the problem and reviewing its history, we present recent results about well-posedness and causality of the equations of motion. If time allows, applications to physics will be briefly discussed.
October 11 3:30 PM SDRICH 207
Frank Morgan (Williams College) The Isoperimetric Problem with Density
The fact that the round sphere provides the least-perimeter way to enclose given volume in Euclidean space has inspired millennia of theory and applications. What happens if you give space a density, weighting both perimeter and volume? Interest in such densities has exploded since their appearance in Perelman's proof of the Poincaré Conjecture. We'll discuss some conjectures and recent results, some by undergraduates.
October 31 3:30 PM SDRICH 207
Alain Bensoussan (UT Dallas) Systems of Quasilinear Parabolic Equations in R^{n} and Systems of Quadratic BSDE
The objective of this work is twofold. On the one hand, we complete the work of A. Bensoussan - J.Frehse , devoted to systems of PDE on a bounded domain. One can then expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R^{n} , because of the lack of bounds. We give conditions so that our previous results can be extended to R^{n} . The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of H. Xing-G. Žitković. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R^{n} and not on a bounded domain. Xing and Žitković develop a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after translation into the analytic framework. This is in particular true for the uniqueness result.
November 9 3:30 PM SDRICH 207
Lotfi Hermi (Florida International University) Isoperimetric Inequalities for Wedge-Like Domains
By introducing geometric factors and physical parameters that lend themselves to the Payne interpretation in Weinstein fractional space, we prove new isoperimetric inequalities for the fundamental eigenvalue of the Dirichlet problem for wedge-like membranes in two dimensions.
These inequalities generalize and improve sharp inequalities that date back to the works of Payne-Weinberger (1960), Payne-Rayner (1973), Crooke-Sperb (1978), and Chiti (1982). We also motivate, conjecture, and prove an isoperimetric inequality relating the fundamental eigenvalue of a wedge-like membrane to its ``relative torsional rigidity'', which we introduce. This new sharp universal inequality beats both Faber-Krahn and Payne-Weinberger at the isoperimetric game, which we will show numerically for certain triangles. The central tool of this recent development is the use of a new weighted form of the Kohler-Jobin symmetrization method, which we develop. Similar considerations can be undertaken for convex cones in higher dimensions.
December 5 2:00 PM SDRICH 207
Bertram Duering (University of Sussex) Title: A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations
Nonlinear diffusion equations whose dynamics are driven by internal energies and given external potentials, e.g., the porous medium equation and the fast diffusion equation, have received a lot of interest in mathematical research and practical applications alike in recent years. Many of them have been interpreted as gradient flows with respect to some metric structure. When it comes to solving partial differential equations of gradient flow type numerically, it is natural to ask for appropriate schemes that preserve the equations' special structure at the discrete
level.
In this talk we present a Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in multiple space dimensions. The
scheme applies to a large class of nonlinear diffusion equations. The key ingredient in our approach is the gradient flow structure of the
dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler
method in time. Thanks to that time discretization, the fully discrete solution inherits energy estimates from the original gradient flow, and these
lead to weak compactness of the trajectories in the continuous limit. We discuss the consistency of the scheme in two space dimensions and present
numerical experiments for the porous medium equation.
January 18 3:30 PM SDRICH 207
Ken Ono (Emory University) Title: Polya's Program for the Riemann Hypothesis and Related Problems
In 1927, Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has only been proved for degrees d=1,2,3. We prove the hyperbolicity of 100% of the Jensen polynomials of every degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia and Wang on the partition function. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
February 22 3:30 PM SDRICH 207
Rudi Weikard (University of Alabama Birmingham) Title: Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients
We discuss the spectral theory of the first-order system Ju'+qu=wf of differential equations on the real interval (a,b) when J is a constant, invertible skew-Hermitian matrix and q and w are matrices whose entries are distributions of order zero with q Hermitian and w non-negative. We do not require the definiteness condition customarily made on the coefficients of the equation. Specifically, we construct associated minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we construct Green's function and prove the existence of a spectral (or generalized Fourier) transformation.
February 26 3:30 PM SDRICH 344
Ridgway Scott (University of Chicago & Rice University) Title: Automated Modeling with FEniCS
The FEniCS Project develops both fundamental software components and end-user codes to automate numerical solution of partial differential equations (PDEs). FEniCS enables users to translate scientific models quickly into efficient finite element code and also offers powerful capabilities for more experienced programmers.
FEniCS uses the variational formulation of PDEs as a language to define models. We will explain the variational formulations for simple problems and then show how they can be extended to simulate fluid flow. The variational formulation also provides a firm theoretical foundation for understanding PDEs. We argue that combining the theory with practical coding provides a way to teach PDEs, their numerical solution, and associated modeling without requiring extensive mathematical prerequisites. As proof, this talk will require no background in PDEs or finite elements, only multi-variate calculus.
March 1 3:30 PM BSB D109
Susan Abernathy (Angelo-State University) Title: Knot and Tangle Embedding (AWM Lecture)
We introduce some basic concepts from not theory, beginning with the definition of a mathematical knot. This leads to a fundamental question: how do we tell two knows apart? We'll discuss several examples of knot invariants, which can help anser this question. We also consider mathematical tangles and how these relate to knots via embedding. We discuss one particularly interesting example of a tangle in a solid torus and ask whether it can be closed up to obtain an unknot.
March 15 3:30 PM SDRICH 207
Maddie Locus Dawsey (Emory University) Title: Densities of subsets of prime numbers
Thanks to the Prime Number Theorem, a lot is known about the distribution of prime numbers. In particular, one can ask whether the methods from analytic number theory yield formulas for densities of subsets of prime numbers. One famous instance of this is the strong form of Dirichlet’s famous theorem on primes in arithmetic progressions. Here, we give new formulations of Dirichlet’s Theorem, and more generally the Chebotarev Density Theorem, by offering completely new formulas expressing densities of suitable subsets of prime numbers. Our work can be thought of as the non-abelian extension of a theorem of Alladi from the 1970s.
April 12 4:00 PM SDRICH 207
Alpar Meszaros (UCLA) Title: Nonlinear cross-diffusion systems: an optimal transport approach
In this talk I will present a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in the space of Borel probability measures equipped with the so-called Wasserstein distance arising in the Monge-Kantorovich optimal transport problem. This leads first to a notion of discrete-time solutions. The continuum time limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, where the densities are guaranteed to be segregated, a stable interface appears between the two densities, and a stronger convergence result, in particular derivation of a standard weak solution to the system, is available. I will also present the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension the problem leads to a two-phase Hele-Shaw type flow. The talk is based on recent joint work with Inwon Kim (UCLA).
April 17 2:00 PM SDRICH 207
Barry Simon (Caltech) Title: Heinävarra’s Proof of the Dobsch-Donoghue Theorem on the Local Form of Loewner Theory
In 1934, Loewner proved a remarkable and deep theorem about matrix monotone functions. Recently, the young Finnish mathematician, Otte Heinävarra settled a 10 year old conjecture and found a 2 page proof of a theorem in Loewner theory whose only prior proof was 35 pages. I will describe his proof and use that as an excuse to discuss matrix monotone and matrix convex functions including, if time allows, my own recent proof of Loewner’s original theorem.
April 18 3:00 PM SDRICH 344
Barry Simon (Caltech) Title: Tales of Our Forefathers I (BLSM Public Lecture)
This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Among the mathematicians with vignettes in this talk are Euler, Weierstrass, Fejer, Banach, Kolmogorov, Weyl, Cotlar, Fourier, Paley, Hausdorff, and Schur.
April 18 4:30 PM SDRICH 344
Barry Simon (Caltech) Title: Tales of Our Forefathers II (BLSM Lecture 2)
This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. This talk will focus on the lives of 22 mathematicians including Gauss, Poincare, von Neumann, Loewner, Landau and Noether. This talk doesn't require the first "Tales" talk.
April 19 2:00 PM SDRICH 207
Barry Simon (Caltech) Title: Spectral Theory Sum Rules, Meromorphic Herglotz Functions and Large Deviations
After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's theorem as a sum rule for OPUC and the Killip-Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip-Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.
April 26 3:30 PM SDRICH 207
Gregory Berkolaiko (Texas A&M University) Title: Nodal count distribution of graph eigenfunctions
We start by reviewing the notion of “quantum graph”, its eigenfunctions and the problem of counting the number of their zeros. The n^{th} eigenfunction has n-1 zeros (like an interval in Sturm-Liouville theory) plus some extra zeros that can be attributed to the graph's topology; we call the latter number the "nodal surplus". When the graph is composed of two or more blocks separated by bridges, we propose a way to define a “local nodal surplus” of a given block. Since the eigenfunction index n has no local meaning, the local nodal surplus has to be defined in an indirect way via the nodal-magnetic theorem of Berkolaiko, Colin de Verdière and Weyand.
We will discuss the properties of the local nodal surplus and their consequences. In particular, its symmetry properties allow us to prove the long-standing conjecture that the nodal surplus distribution for graphs with β disjoint loops is binomial with parameters (β,1/2).
The talk is based on joint work with Lior Alon and Ram Band, arXiv:1709.10413 (Commun Math Phys, 2018).
The Applied PDE Seminar meets in SDRICH 213 on Tuesdays from 3:30 to 4:30 PM. For further information, please contact Jameson_Graber@baylor.edu.
Mathematical Physics Seminar
The Mathematical Physics Seminar meets in SDRICH 333 on Wednesdays from 2:00 - 3:00 PM. For further information contact Jon_Harrison@baylor.edu. Further information on talks in this seminar can be found by going to Mathematical Physics Seminar.
Representation Theory Seminar
The Representation Theory Seminar meets in SDRICH 219 on Wednesday from 11:00-2:00 PM. For further information contact: Markus_Hunziker@baylor.edu
Topology and Dynamics Seminar
The Topology and Dynamics Seminar meets in SDRICH 333 on Wednesdays at 3:30-4:30 PM For further information contact Jonathan_Meddaugh@baylor.edu .