Multi-parameter paraproducts: Box condition versus Chang--Fefferman condition for weighted estimates

Speaker: 

Alexander Volberg

Institution: 

Michigan State University

Time: 

Thursday, February 13, 2020 - 4:00pm

Location: 

RH 306

Paraproducts are building blocks of many singular integral operators and the main instrument in proving ``Leibniz rule" for fractional derivatives (Kato--Ponce). Multi-parameter paraproducts are tools to prove more complicated Leibniz rules that are also widely used in well posedness questions for various PDEs. Alternatively, multi-parameter paraproducts appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgue spaces with respect to a measure in the polydisc.

Those Carleson embedding theorems often serve as a first building block for interpolation in complex space and also for corona type results. The embedding of spaces of holomorphic functions on n-polydisc can be reduced to problem (without loss of information) of  boundedness of weighted dyadic n-parameter paraproducts.

We  find the necessary and sufficient  condition for this boundedness in n-parameter case, when n is 1, 2, or 3.  The answer is quite unexpected and seemingly goes against the well known difference between box and Chang--Fefferman condition that was given by Carleson quilts example of 1974.

Surjectivity of random integral matrices on integral vectors

Speaker: 

Melanie Wood

Institution: 

UC Berkeley

Time: 

Thursday, January 30, 2020 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

A random nxm matrix gives a random linear transformation from \Z^m to \Z^n (between vectors with integral coordinates).  Asking for the probability that such a map is injective is a question of the non-vanishing of determinants.  In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question.  We show that when m=n+u, for u at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function.  This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields.  This talk is on joint work with Hoi Nguyen.

Kolmogorov, Onsager and a stochastic model for turbulence

Speaker: 

Susan Friedlander

Institution: 

USC

Time: 

Thursday, February 20, 2020 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

We will briefly review Kolmogorov’s (41) theory of homogeneous turbulence and Onsager’s ( 49)
conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of
vanishing viscosity. Although over the past 60 years there is a vast body of literature related to this subject, at present
there is no rigorous mathematical proof that the solutions to the Navier-Stokes equations yield
Kolmogorov’s laws. For this reason various models have been introduced that are more tractable but capture
some of the essential features of the Navier-Stokes equations themselves. We will discuss one such
stochastically driven dyadic model for turbulent energy cascades. We will describe how results for stochastic PDEs
can be used to prove that this dyadic model is consistent with Kolmogorov’s theory and Onsager’s conjecture.

This is joint work with Vlad Vicol and Nathan Glatt-Holtz.

Deep Learning and Multigrid Methods

Speaker: 

Jinchao Xu

Institution: 

Pennsylvania State University

Time: 

Monday, January 6, 2020 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In this talk, I will first give an introduction to some models and algorithms from two different fields: (1) machine learning, including logistic regression, support vector machine and deep neural networks, and (2) numerical PDEs, including finite element and multigrid methods.  I will then explore mathematical relationships between these models and algorithms and demonstrate how such relationships can be used to understand, study and improve the model structures, mathematical properties and relevant training algorithms for deep neural networks. In particular, I will demonstrate how a new convolutional neural network known as MgNet, can be derived by making very minor modifications of a classic geometric multigrid method for the Poisson equation and then explore the theoretical and practical potentials of MgNet.

This is a joint talk of Applied and Compuational Math Seminar.

$\overline{\partial}$-methods in complex analysis

Speaker: 

Emil Straube

Institution: 

Texas A&M

Time: 

Thursday, February 27, 2020 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Patching together locally defined analytic functions to obtain a globally well defined analytic function is notoriously difficult, as the use of cutoff functions destroys analyticity. Analytic functions are solutions of thehomogeneous Cauchy--Riemann equations. Therefore, if we understand (solvability of) the inhomogeneous Cauchy--Riemann equations, we can hope to produce a correction term to restore analyticity, in such away that the desired local behavior survives. In this colloquium, I will illustrate this philosophy with several examples, including one from the classical one variable theory. In dimension more than one, one is led to consider the Cauchy--Riemann operator not just on functions, but also on forms. I will indicate the central role these considerations play in several complex variables. 

On the smooth realization problem in ergodic theory

Speaker: 

Benjamin Weiss

Institution: 

Hebrew University

Time: 

Monday, December 2, 2019 - 4:00pm

Host: 

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: 

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: 

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.

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