Model theoretic and nonstandard methods in algebraic geometry and combinatorics

Speaker: 

Anand Pillay

Institution: 

University of Notre Dame

Time: 

Thursday, October 24, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

I will discuss some model theoretic methods with a "nonstandard" flavour.  I will touch on a second generation proof of function field Mordell-Lang in positive characteristic (2016). And then talk about recent and current work on arithmetic regularity lemmas from combinatorics. 

Optimal transport for seismic imaging

Speaker: 

Bjorn Engquist

Institution: 

University of Texas, Austin

Time: 

Thursday, June 6, 2019 - 4:00pm

Host: 

Location: 

RH 306

In Full Waveform Inversion seismic imaging is formulated as PDE constrained minimization where the miss-match between measured and computed signals plays an important role. The purpose is to find geophysical properties, such as wave velocity and location of reflecting sub layers, which are represented by the coefficients in the PDE. We propose using optimal transport and the Wasserstein metric for this miss-match in order to reduce the risk of only finding local minima in the PDE constrained minimization. The optimal transport can be given by the gradient of the solution to a Monge–Ampère equation. Analysis of convexity properties and numerical examples comparing these new techniques with the classical L2 miss-match will be presented.

Modeling stripe and mutated skin pattern formation on zebrafish

Speaker: 

Alexandria Volkening

Institution: 

Ohio State University

Time: 

Thursday, February 7, 2019 - 10:00am to 11:00am

Location: 

Nat Sci II 1201

Wild-type zebrafish (Danio rerio) feature black and yellow stripes across their body and fins, but mutants display a range of altered patterns, including spots and labyrinth curves. All these patterns form due to the interactions of pigment cells, which sort out through movement, birth, competition, and transitions in cellular shape during early development. The diversity of patterns on zebrafish makes it a useful organism for helping elucidate how genes, cell behavior, and visible animal characteristics are related, and this is the motivation for my work. Using an agent-based approach to describe pigment cells, I couple deterministic cell migration by ODEs with stochastic rules for updating population size on growing domains. Our model suggests the unknown cellular signals behind newly observed cell behaviors and makes experimentally-testable predictions about how various Danio fish may be related evolutionarily. I will also discuss the associated non-local continuum limit of the agent-based model and highlight several future directions for this project.

Zeros of harmonic functions and Laplace eigenfunctions

Speaker: 

Alexander Logunov

Institution: 

Princeton University

Time: 

Monday, January 28, 2019 - 3:00pm

Host: 

Location: 

RH 306

We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. A curious object is Laplace eigenfunctions on two-dimensional sphere, which are restrictions of homogeneous harmonic polynomials of three variables onto 2-dimensional sphere. They are called spherical harmonics. Zero sets of such functions are unions of smooth curves with equiangular intersections. Topology of zero set could be quite complicated, but the total length of the zero set of any spherical harmonic of degree n is comparable to n. Though the Laplace eigenfunctions are known for ages, we still don't understand them well enough (even the spherical harmonics). 

Fibers of maps to totally nonnegative spaces

Speaker: 

Patricia Hersh

Institution: 

North Carolina State University

Time: 

Thursday, March 14, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The space of totally nonnegative real matrices, namely the real n by n matrices with all minors nonnegative, intersected with the ``unipotent radical'' of upper triangular matrices with 1's on the diagonal carries important information related to Lusztig's theory of canonical bases in representation theory.   This space of matrices (and generalizations of it beyond type A) is naturally stratified according to which minors are positive and which are 0, with the resulting stratified space described combinatorially by a well known partially ordered set called the Bruhat order.   I will tell the story of these spaces and in particular of a map from a simplex to these spaces that has recently been used to better understand them.  The fibers of this map encode exactly the nonnegative real relations amongst exponentiated Chevalley generators of a Lie algebra.   This talk will especially focus on recent joint work with Jim Davis and Ezra Miller uncovering overall combinatorial and topological structure governing these fibers.  Plenty of background, examples, and pictures will be provided along the way. 

Asymptotics: the unified transform, a new approach to the Lindelöf Hypothesis, and the ultra-relativistic limit of the Minkowskian approximation of general relativity

Speaker: 

Athanassios S. Fokas

Institution: 

University of Cambridge/USC

Time: 

Thursday, April 11, 2019 - 4:00pm to 5:00pm

Location: 

RH 306

Employing standard, as well as novel techniques of asymptotics, three different problems will be discussed: (i) The computation of the large time asymptotics of initial-boundary value problems via the unified transform (also known as the Fokas Method, www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large t-asymptotics to all orders of the Riemann zeta function [2], and the introduction of a new approach to the Lindelöf Hypothesis [3]. (iii) The proof that the ultra-relativistic limit of the Minkowskian approximation of general relativity [4] yields a force with characteristics of the strong force, including confinement and asymptotic freedom [5].

[1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925 (2015).
J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with t-Periodic Data: II. Perturbative Results, Proc. R. Soc. A 471, 20140926 (2015).
[2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear).
[3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis, Transactions of Mathematics and its Applications (to appear).
[4] L. Blanchet and A.S. Fokas, Equations of Motion of Self-Gravitating N-Body Systems in the First Post-Minkowskian
Approximation, Phys. Rev. D 98, 084005 (2018).
[5] A.S. Fokas, Super Relativistic Gravity has Properties Associated with the Strong Force, Eur. Phys. J. C (to appear).

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