On the smooth realization problem in ergodic theory

Speaker: 

Benjamin Weiss

Institution: 

Hebrew University

Time: 

Monday, December 2, 2019 - 4:00pm

Host: 

Matthew D. Foreman (he/him/his)

Location: 

NSII 1201

The outstanding open problem in the interface between smooth dynamics and ergodic theory is whether or not every finite entropy abstract ergodic transformation is isomorphic to a smooth diffeomorphism preserving volume element on a compact manifold. While the problem was essentially formulated by von Neumann in 1932 there has been very little progress and it is open even for very basic examples such as odometers.  I will discuss some recent work on the problem (joint with Matt Foreman) of two kinds. On the one hand we provide a host of new examples that can be realized, while on the other hand we show that the isomorphism problem for smooth diffeomorphisms preserving Lebesgue measure on the torus is as complex as the general abstract isomorphism problem for ergodic transformations.

 

Mathematical Modeling of Prion Aggregate Dynamics within a Growing Yeast Population

Speaker: 

Suzanne Sindi

Institution: 

UC Merced

Time: 

Thursday, April 18, 2019 - 4:00pm to 5:00pm

Host: 

Alessandra Pantano (she/her/hers)

Location: 

RH 306

Prion proteins are responsible for a variety of neurodegenerative diseases in mammals such as Creutzfeldt-Jakob disease in humans and "mad-cow" disease in cattle. While these diseases are fatal to mammals, a host of harmless phenotypes have been associated with prion proteins in S. cerevisiae, making yeast an ideal model organism for prion diseases.

Most mathematical approaches to modeling prion dynamics have focused on either the protein dynamics in isolation, absent from a changing cellular environment, or modeling prion dynamics in a population of cells by considering the "average" behavior. However, such models have been unable to recapitulate in vivo properties of yeast prion strains including experimentally observed rates of prion loss.

My group develops physiologically relevant mathematical models by considering both the prion aggregates and their yeast host. We then validate our model and infer parameters through carefully designed in vivo experiments. In this talk, I will present two recent results. First, we adapt the nucleated polymerization model for aggregate dynamics to a stochastic context to consider a rate limiting event in the establishment of prion disease: the rst the successful amplication of an aggregate. We then develop a multi-scale aggregate and generation structured population model to study the amplication of prion aggregates in a growing population of cells. In both cases, we gain new insights into prion phenotypes in yeast and quantify how common experimentally observed outcomes depend on population heterogeneity.

 

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: 

Isaac M. Goldbring

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: 

Hongkai Zhao

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: 

Paata Ivanisvili

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: 

Jesse Wolfson

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

Speaker Link: 

http://www.damtp.cam.ac.uk/user/tf227/

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).

The Sperner property

Speaker: 

Richard Stanley

Institution: 

MIT

Time: 

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

Host: 

Michael C. Cranston

Location: 

RH 306

In 1927 Emanuel Sperner proved that if $S_1,\dots,S_m$ are distinct 
subsets of an $n$-element set such that we never have $S_i\subset S_j$, 
then $m\leq \binom{n}{\lfloor n/2\rfloor}$. Moreover, equality is achieved 
by taking all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This 
result spawned a host of generalizations, most conveniently stated in the 
language of partially ordered sets. We will survey some of the highlights 
of this subject, including the use of linear algebra and the cohomology of 
certain complex projective varieties. An application is a proof of a 
conjecture of Erd\H{o}s and Moser, namely, for all integers $n\geq 1$ and 
real numbers $\alpha\geq 0$, the number of subsets with element sum 
$\alpha$ of an $n$-element set of positive real numbers cannot exceed the 
number of subsets of $\{1,2,\dots,n\}$ whose elements sum to $\lfloor 
\frac 12\binom n2\rfloor$. We will conclude by discussing two recent 
proofs of a 1984 conjecture of Anders Bj\"orner on the weak Bruhat order 
of the symmetric group $S_n$.

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