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.

TBA

Speaker: 

Bryna Kra

Institution: 

Northwestern University

Time: 

Thursday, October 15, 2020 - 4:00pm to 5:00pm

Location: 

RH 306

TBA

Speaker: 

Karen Smith

Institution: 

University of Michigan

Time: 

Tuesday, October 20, 2020 - 4:00pm to 5:00pm

Location: 

RH 306

TBA

Speaker: 

Valentino Tosatti

Institution: 

Northwestern University

Time: 

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

Location: 

RH 306

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.

 

Pages

Subscribe to RSS - Colloquium