Clemson University Analysis Seminar Schedule

Spring 2022

TBA

Speaker:                John-Oliver MacLellan - University of Alabama
Time:                     Friday, January 28, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
Abstract:
TBA

TBA

Speaker:               Vishwa Dewage - Louisiana State University
Time:                     Friday, February 4, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
Abstract:
TBA

TBA

Speaker:                Ilia Zlotnikov - University of Stavanger
Time:                     Monday, Februrary 28, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
Abstract:
TBA

TBA

Speaker:               Raffael Hagger - Christian Albrechts University in Kiel
Time:                     Wednesday, March 9, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
Abstract:
TBA

TBA

Speaker:                Tapesh Yadav - University of Florida
Time:                     Friday, March 18, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
Abstract:
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TBA

Speaker:                Yiran Wang - Emory University
Time:                     Friday, April 8, 2022 - 11:15 am
Location:               https://clemson.zoom.us/j/93164718636
Recording:           
Abstract:
TBA

Fall 2021

Alpha-area Siegel-Veech constants for branched cyclic covers

Speaker:              Martin Schmoll - Clemson University
Time:                   Wednesday, September 8, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636            
Recording:         
https://clemson.zoom.us/rec/share/P3F-RhO_84Fih7HrfS_69Ix5jFPg-QCvwcxj6ytQNYNJt5cOQaKDxM2OA5SbCQ3f.J2pJjEXX1kh8tyW7
Abstract:            
We will present a relative formula for asymptotic quadratic growth rates of periodic cylinders for branched cyclic covers that seems to hold in more generality. We will talk about to which extend we can 
prove this formula and its inner structure. Background on flat surfaces is provided (mostly informally), as well as a brief introduction into the Siegel-Veech formula and its evaluation. An interesting feature   
of the formula for general surfaces of higher genus is that it is essentially the formula for branched 
torus covers. It is worth noting that the resulting formula has been predicted by computer experiments using an approach that does not involve the Siegel-Veech formula our proof relies on. 

This is ongoing research together with David Aulicino (Brooklyn College), Aaron Calderon (Yale) and Nick Salter (Notre Dame). 

A completeness and cyclicty problem for a semigroup of weighted composition operators

Speaker:              Chris Felder - Washington University in St. Louis
Time:                   Friday, September 17, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:         
https://clemson.zoom.us/rec/share/mrVKwMs1SUs60OQPYhaX918aRRkyQasVa9D7rgUGEKo8CBvFfexEmNxGs160waNn.uySmKnrFOAsh0gQlAbstract:            
This talk will be largely expository, following recent work of S. Waleed Noor. We will discuss a completeness problem in the classical Hardy space, along with a cyclicity problem for a semigroup of weighted composition operators. We will then talk about a least-squares approach to solving such problems. 

On the John-Nirenberg constant of BMO^p, 0<p<1

Speaker:              Brandon Sweeting - University of Alabama
Time:                   Friday, September 24, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          
https://clemson.zoom.us/rec/share/wY3lLpR6YaMOEfzV620_7Zl4DwDlB3XbzDTntNDz4CWFT34t5iNYEDzKfGzbwjHc.4nQ_nvaXBU_wQCrF
Abstract:            
We present sharp L^p lower bounds for logarithms of A_\infty weights as a means of estimating the John-Nirenberg constant of the space BMO^p, 0<p<1. The corresponding Bellman function solves the homogeneous Monge-Ampère equation, but the geometry of the solution goes beyond established theory due to the lack of regularity in the boundary condition. This is joint work with Leonid Slavin.

Stabilization rates for the damped wave equation with polynomial and oscillatory damping

Speaker:              Perry Kleinhenz - Michigan State University
Time:                   Wednesday, September 29, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          
https://clemson.zoom.us/rec/share/QJyTrBOTEZEAt1D1TpEa85IDK8KsmtRM25fnRJJrDpdDo2jM2eLobXXtd-YH6ScK.rLp2y4B0R6cLYZ6h
Abstract:            
In this talk I will discuss energy decay of solutions of the Damped wave equation. After giving an overview of classical results I'll focus on the torus with damping that does not satisfy the geometric control condition. In this setup properties of the damping at the boundary of its support determine the decay rate, however a general sharp rate is not known.

I will discuss damping which is 0 on a strip and vanishes either like a polynomial x^b or an oscillating exponential e^{-1/x} sin^2(1/x). Polynomial damping produces decay of the semigroup at exactly t^{-(b+2)/(b+3)}, while oscillating damping produces decay at least as fast as t^{-4/5+\delta} for any \delta>0. I will explain how these model cases are proved and how they direct further study of the general sharp rate.

An approach to universality using Weyl m-functions

Speaker:              Milivoje Lukić - Rice University
Time:                   Friday, October 8, 2021 - 11 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          https://clemson.zoom.us/rec/share/nxKeOonTCQVLcOuCmDRvhTHvJ3ZyBSTP5oOVvM2AP1dsOOnTjNnmCyCrEwYK_XxK.eEXyo8QiLKvEGYsl
Abstract:            
 In this talk, I will present joint work with Benjamin Eichinger and Brian Simanek: a new approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel--Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel--Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with 2x2 transfer matrices such as continuum Schrodinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.

 Random Interpolating Sequences in the Polydisc and the Unit Ball

Speaker:              Alberto Dayan - Norwegian University of Science and Technology
Time:                   Friday, October 15, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          
https://clemson.zoom.us/rec/share/K8w8WWFRCjd-1hdQsKiPv9CG5lMzgM9cil79hwWxvgpbZftAU06utMklcUTEBXxc.vi5Jde9tO2WJZB_Z
Abstract:            
A random sequence Z in the unit disc is determined by a sequence of deterministic radii and a sequence of i.i.d. random variables uniformly distributed on the unit circle. Chochran and Rudowicz found the 0-1 Kolmogorov law for Z to be interpolating, that is, the cut-off condition on the a-priori fixed radii in order for Z to be interpolating almost surely. In this talk, we will extend their work to random interpolating sequences for bounded analytic functions in the d-dimensional polydisc and for the Besov-Sobolev spaces on the unit ball. The case of the Besov-Sobolev spaces is more treatable, since such spaces have their interpolating sequences well understood and characterized in the deterministic setting. This is not the case for interpolating sequences in the polydisc: in this second case our necessary and sufficient conditions for almost sure interpolation do not coincide, and they are obtained by looking at the decay of related random Grammians off their diagonals.

This is a joint work with Brett Wick and Shengkun Wu

Weighted Inequalities on Spaces of Homogeneous Type

Speaker:              Manasa Vempati - Georgia Institute of Technology
Time:                   Wednesday, October 27, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          
https://clemson.zoom.us/rec/share/2vJIqKrOP3-ThONoMEEGYwB59xA9iD_46YyBMB5nlUCaJe4UMtAlLvW336jIP7Vr.9dFmp5Eq4u3hoxpu
Abstract:            
We will discuss the two weight inequalities for Calderon-Zygmund operators and commutators. We work in the setting of spaces of homogeneous type defined in the sense of Coifman and Weiss. Subject to the pair of weights  u and v satisfying a side condition, we will show a characterization for the boundedness of a Calderon-Zygmund operator T from L^2(u) to L^2(v) in terms of the A_2 condition and two testing conditions. We will also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type.

A wavepacket approach to the random Schrodinger equation

Speaker:              Felipe Hernandez - Stanford University
Time:                   Monday, November 1, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:         
https://clemson.zoom.us/rec/share/pWqhzsgJPH_c-dYqQ8iSLgwmxpWM582VYRiLhFJwRuZSyqzdqfrRK9n7vjv_JjCp.zDrVf12RVFeSWGuv
Abstract:            
I will discuss my work in progress on the Schrodinger equation perturbed by a random potential.  This equation is a simple model for wave propagation in random environments.  A key feature of solutions to this equation is a diffusion phenomenon, meaning that the mass of the wavefunctions weakly solves a heat equation.  A previous work of Erdos, Salmhofer, and Yau in 2007 derived this heat equation using a complex diagrammatic expansion.  I will explain in this talk how wavepacket decompositions can be used to give a more geometric understanding of the problem.

A strange limit of horocycle ergodic measures in a stratum of translation surfaces

Speaker:              Jon Chaika - University of Utah
Time:                   Friday, November 12, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          
https://clemson.zoom.us/rec/share/AFw2KmJyaJHgnIZLafLWxYWzeJYSS5NuNeaUHzy-I18T9lS9lUlXCQqUZHubgJpT.x1ujxcE5J3RK3T3G
Abstract:            
The main result of this talk is that in the space of unit area translation surfaces with one cone point there is a weak-star limit of measures on periodic horocycles that is fully supported in the 7-dimensional space but gives positive measure to a 3-dimensional submanifold. As a consequence we obtain a non-genericity result for the horocycle flow in this space. I will define the terminology. This is joint work with Osama Khalil and John Smillie.

Pressure Function on Compact Symbolic Systems

Speaker:              Tamara Kucherenko - CUNY
Time:                   Friday, November 19, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          
https://clemson.zoom.us/rec/share/buQ77jNb-zIw84sKYNE_oj8ekMs56UYcBirEVmjMpPouOzQXa4LMFSg-U35u8g9Q.dmrrwdnjpWnhq2Yb
Abstract:            
We study the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. In fact, we establish a multidimensional version of this result. As a consequence, we obtain that for a continuous observable the phase transitions can occur at a countable dense set of temperature values. We go further and show that one can vary the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number, finite or infinite.

Optimal Tomography with Local Data

Speaker:              Francis Chung - University of Kentucky
Time:                   Monday, November 29, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Recording:          https://clemson.zoom.us/rec/share/Y1_kuoNIa2InVkYzJ_Uw_gNmoH0cZ_Bm74PkIQoGYd2DAOxsUvKrcXkOjydf0Lni.NcgW0cND9iLtZlEx
Abstract:            
Optical tomography is the problem of reconstructing interior properties of an object from optical measurements at the boundary. The mathematical version of this problem is to reconstruct the coefficients of a PDE (in my case, a radiative transport equation) from measurements of the solutions at the boundary. In this talk I'll discuss an optical tomography problem with local data, in which measurements are restricted to a subset of the boundary. I plan to introduce the problem, discuss the wider context, and describe a recent result. 

Spring 2021

Translational tilings: structure and decidability

Speaker:              Rachel Greenfeld - University of California Los Angeles
Time:                   Wednesday, January 13, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636    
Abstract:            
Let F be a finite subset of the d-dimensional integer lattice. We say that F is a translational tile of Z^d if it is possible to cover Z^d by translates of F with no overlaps. Given a finite subset F of Z^d, could we determine whether F is a translational tile in finite time? Suppose that F does tile, what can be said about the structure of the tiling?  A well known argument of Wang shows that these two questions are closely related.  In the talk, I will introduce and demonstrate this relation and present some new results, joint with Terence Tao, on the rigidity of tiling structures in Z^2, and their applications to decidability. 

Thin elastic plates involving fractional rotational forces: semigroups regularity and stability

Speaker:              Louis Tebou - Florida International University
Time:                   Wednesday, January 20, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Recording:        https://clemson.zoom.us/rec/share/kaVTn2mlCtC7WDzmSPCW7fwFiHSAtZay4H65opC2b7tNH4H4xLid2csQLzQDvLzP.vBp1e7uhyDJbFV_g
Abstract:              
In this talk, finite thin elastic plates involving fractional rotational forces are considered. Using resolvent estimates, new regularity and stability results for the underlying semigroups are established. First, I examine a thermoelastic plate with fractional rotational forces, and prove Gevrey regularity, as well as exponential stability of the associated semigroup. Afterward, I analyze a mechanically damped plate, where the rotational forces and damping involve fractional powers of the Laplacian. For this latter model, it will be shown that the underlying semigroup is analytic, or of a certain Gevrey class, is exponentially or polynomially stable, depending on the relationship between the fractional exponents of the damping and rotational forces. The models considered are new and lie between the classical Euler-Bernoulli model and the Kirchhoff model for thin plates in either case. This is a joint-work with my colleagues Valentin Keyantuo (University of Puerto Rico, San Juan) and Mahamadi Warma (George Mason University)

Strongly Peaking Representations and Compressions of Operator Systems

Speaker:              Benjamin Passer- United States Naval Academy
Time:                   Wednesday, January 27, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Recording:          https://clemson.zoom.us/rec/share/wa7P_b2yuDqbtFZH_BCsHXDYWV38hV8t0D0NV5HCp17QwqsldYjBMejCtQclxwoh.554bcM6hPKMhHhJn
Abstract:              
 (Joint work with Ken Davidson.) Just as a compact convex set is generated by its extreme points, it is known that an operator system is completely normed by its boundary representations. We analyze a special class of boundary representations in order to study operator systems which are presented in a smallest possible way. Our results extend recent work in the study of free spectrahedra and matrix convex sets.

Uncertainty principle in harmonic analysis and its applications

Speaker:              Mishko Mitkovski - Clemson University
Time:                   Wednesday, February 3, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Recording:      https://clemson.zoom.us/rec/share/lzoo7jpalnDAwkae4CxEEp52h7VeEcouFIUp9BfsZvUMlqQmZ5tPFCkLjvw1BR9D.59_HuUfcHkWGtltv
Abstract:                
I will present several new forms of the uncertainty principle. These new forms can be viewed as extensions/sharpening of some classical uncertainty principles, in a sense that we impose restrictions on the Fourier support and deduce sampling inequalities. Many of our results were inspired by some control and damping problems for linear PDE’s and I will try to present some of these applications. This is a collaborative work with W. Green, B. Jaye, and H. Li.

Data-Dependent Distances for Unsupervised and Active Learning

Speaker:              James Murphy - Tufts University
Time:                   Wednesday, February 10, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Recording:      https://clemson.zoom.us/rec/share/yhVScfi9xLhaX5BE63nsG2dfG8z20tn_YCR2IfZzFOsNpoaZ7GTiPRBlrvteJscQ.4j7kQukwXmB6ypHv
Abstract:              
Approaches to unsupervised clustering and active learning with data-dependent distances are proposed.  By considering metrics derived from data-driven graphs, robustness to noise and class geometry is achieved.  The proposed algorithms enjoy theoretical guarantees on flexible data models, and also have quasilinear computational complexity in the number of data points.  Connections will be made to geometric analysis and percolation theory.  Applications to image processing and biological networks will be shown, demonstrating state-of-the-art empirical performance.

Weighted inequalities for Haar multipliers

Speaker:              Claire Huang - Washington University in St. Louis
Time:                   Friday, February 19, 2021 - 3:00 pm
Location:             https://clemson.zoom.us/j/93164718636 
Abstract:              
A measure on R^n is called “dyadic doubling” if the measure ratio of any dyadic parent and its dyadic child is uniformly bounded. This property enables a stopping time argument presented by Katz and Pereyra in their 1998 survey on the L^p boundedness of Haar multipliers. Highly inspired by these two authors’ work, this presentation focuses on extending their results to weighted spaces. In particular, we are interested in the L^p boundedness of Haar multipliers with respect to weights in dyadic Muckenhoupt classes, as they guarantee dyadic doubling, and the closely related weights in dyadic reverse Hölder classes.

Trace minmax functions and the radical Laguerre-Polya class

Speaker:              James Pascoe - University of Florida
Time:                   Monday, February 22, 2021 - 4:00 pm
Location:             https://clemson.zoom.us/j/93164718636
Recording:          https://clemson.zoom.us/rec/share/wbwTPQj1CjrP3sG8IeSwK1FAiYo0M1ekFKMVquNT6At4o4dUom9k6BNygP23CaAn.E5k5eIiqnRiHYWAa
Abstract:              
We classify functions f:(a,b) -> R which satisfy the inequality tr f(A)+f(C) >=  tr f(B)+f(D) when A<= B<= C are self-adjoint matrices, D= A+C-B, the so-called trace minmax functions. (Here A <= B if B-A is positive semidefinite, and f is evaluated via the functional calculus.) A function is trace minmax if and only if its derivative analytically continues to a self map of the upper half plane. The negative exponential of a trace minmax function g=e^{-f} satisfies the inequality det g(A)g(C) <=  det g(B)g(D) for A, B, C, D as above. We call such functions determinant isoperimetric. We show that determinant isoperimetric functions are in the ``radical" of the the Laguerre-Polya class. We derive an integral representation for such functions which is essentially a continuous version of the Hadamard factorization for functions in the the Laguerre-Polya class. We apply our results to give some equivalent formulations of the Riemann hypothesis. Finally, we discuss some recent joint work with Kelly Bickel and Meredith Sargent on relaxations of the above framework, and an avenue for obtaining zero-free regions for the Riemann zeta function.

On dual molecules and convolution-dominated operators

Speaker:              Jordy Timo van Velthoven - University of Vienna
Time:                   Wednesday, March 3, 2021 - 9:00 am
Location:             https://clemson.zoom.us/j/93164718636     
Abstract:              
This talk considers coherent systems arising from unitary representations. It is shown that frames and Riesz sequences can be obtained whose dual systems form molecules, ensuring that their elements satisfy appropriate size estimates as if they were obtained one from the other by the group action. As a consequence, the canonical expansions extend to associated Banach spaces. The main tool is a local holomorphic calculus for convolution-dominated operators, valid for groups with possibly exponential volume growth.

Control and Inverse Problems for a Strongly Coupled Hyperbolic System of PDEs

Speaker:              Jason Kurz - Clemson University
Time:                   Friday, March 12, 2021 - 3:00 pm
Location:             https://clemson.zoom.us/j/93164718636
Recording:      
https://clemson.zoom.us/rec/share/kqS-vXvbCN4rv2dIje3_LA161_b2tr26pxj4DhnDupWUvaNxbJxhCEpxPHO9Zyi_.qaPB0nTgFTPTF5k6
Abstract:              
In this presentation we consider an inverse and control problem for the Mindlin--Timoshenko plate system, which is a strongly coupled two dimensional system consisting of a wave equation and a system of isotropic elasticity, that arises in modeling plate vibrations especially at high frequencies and thicker plates. More precisely, we prove the global uniqueness of recovering the plate density from a single boundary measurement under appropriate geometrical assumptions. Secondly, we demonstrate the controllability of the system via an indirect control technique that proves a two-level indirect inverse observability estimate. Our approach for both problems relies on diagonalizing the principal part of the system and making it a system of wave-like equations.

Wavelet Representation of Singular Integral Operators

Speaker:              Tyler Williams - Washington University in St. Louis
Time:                   Friday, March 26, 2021 - 3:00 pm
Location:             https://clemson.zoom.us/j/93164718636
Recording:         https://clemson.zoom.us/rec/share/cH_Ks7eEVYxNvfL2GqsRR9qkE2jP2hXly1BrWak81I4RJv-UXY4QmUIqxflCiqZS.gZrUMlgZdQ6I6hcq
Abstract:              
We describe a new representation technique for one and multiple parameter singular integrals in terms of continuous model operators. Unlike the well established dyadic counterpart, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of T(1) theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. I present the one parameter case, where we obtain the Sobolev space analogue of the A2 theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where we obtain quantitative Ap estimates which are best known, and sharp in the range max{p, p’} ≥ 3. These estimates are beyond the reach of current dyadic methods.

Boundedness of commutators on weighted Hardy spaces

Speaker:              Marie-Jose Kuffner - Johns Hopkins University
Time:                   Friday, April 2, 2021 - 3:00 pm
Location:             https://clemson.zoom.us/j/93164718636   
Abstract:              
It is known that boundedness of the commutator [b,H] on weighted L^p-spaces for 1<p<\infty is characterized by b being in a certain BMO space adapted to the given weights. In this talk, we present the case p=1 and discuss the space that characterizes boundedness of [b,H] on the weighted Hardy space H^1(w) for certain A_p weights.

Inverse Problems for Nonlinear PDEs

Speaker:              Ting Zhou - Northeastern University
Time:                   Wednesday, April 7, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Abstract:              
In this talk, I will demonstrate the higher order linearization approach to solve several inverse boundary value problems for nonlinear PDEs modeling nonlinear electromagnetic optics including nonlinear time-harmonic Maxwell's equations with Kerr-type and second harmonic generation nonlinearity. The problem will be reduced to solving for the coefficient functions from their integrals against multiple linear solutions. We will focus our discussion on different choices of linear solutions. A similar problem for nonlinear magnetic Schrodinger equation will be considered as a comparison.

Tiling the integers with translates of one tile: the Coven-Meyerowitz tiling conditions for three prime factors

Speaker:              Itay Londner - University of British Columbia
Time:                   Monday, April 12, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Abstract:              
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behavior persists for all tilings with no restrictions on the number of prime factors of M. 
In an ongoing collaborations with Izabella Laba, we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce the main ingredients of the proof.

Extrapolation of compactness on weighted spaces

Speaker:              Stefanos Lappas - University of Helsinki
Time:                   Wednesday, April 21, 2021 - 11:15 am
Location:             https://clemson.zoom.us/j/93164718636
Recording:      
Abstract:              
The extrapolation theorem of Rubio de Francia is one of the most powerful tools in the theory of weighted norm inequalities: it allows one to deduce an inequality (often but not necessarily: the bounded of an operator) on all weighted L^p spaces with a range of p, by checking it just for one exponent p (but all relevant weights). My topic is an analogous method for extrapolation of compactness. In a relatively soft way, it recovers several recent results about compactness of operators on weighted spaces and also gives some new ones. This is a joint-work with Tuomas Hytönen.

Fall 2020

C-cyclical monotonicity in optimal transport and beyond (Colloquium)

Speaker:              Mathias Beiglböck - University of Vienna
Time:                   Wednesday, September 9, 2020 - 11 am
Location:             https://clemson.zoom.us/j/95434272959
Recording:      https://clemson.zoom.us/rec/share/3E3SCCsPG1A7l0oGXa-2HBWL3xOOhPNWDiA0JmanJUhRT5_ZDWMSzeq2MFFSWm1G.JyJq0Bde37P-ZmCJ
Abstract:              
A fundamental idea in optimal transport is that the optimality of a transport plan is reflected by the geometry of its support set. Often this is key to understanding the transport problem. On the level of support sets, the relevant notion is c-cyclical monotonicity. The relevance of this concept for the theory of optimal transport has been fully recognized by Gangbo and McCann, based on earlier work of Knott and Smith and Ruschendorf among others. Since then it has been understood through the work of various authors that analogous "monotonicity principles" are also highly useful in a number of related areas. The goal of this talk is to provide an introduction to these ideas and to sketch some of the recent developments.

Weighted Estimates for the Bergman and Szego Projections on Strongly Pseudoconvex Domains with Near Minimal Smoothness

Speaker:               Nathan Wagner - Washington University in St. Louis
Time:                    Wednesday, September 23, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Abstract:                
The Bergman and Szego projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Bekolle-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale

What is a quantum Markov chain? A beginner's look into non-commutative probability

Speaker:               Renato Feres - Washington University in St. Louis
Time:                    Wednesday, October 7, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/C0BOuqPmVH_XonaHVkbhLgdmvUWuXL4o-w98YjRVdNptwWzG6CWmROshYADba_Ni.CjY5jiWOYs_DnDdv
Abstract:               
For a mathematician, basic quantum theory may be viewed as a generalized (non-commutative) probability theory. The current mathematical research in non-commutative probability and stochastic processes has been greatly influenced by scientific developments in quantum communication and the physics of open quantum systems. In this presentation, I wish to explore a few of the more fundamental ideas in (discrete time,Markovian) quantum stochastic processes, guided by a class of classical Markov chains obtained from elementary mechanical systems.

A Calderón-Zygmund decomposition for von Neumann algebra valued functions

Speaker:               José Conde-Alonso - Universidad Autónoma de Madrid
Time:                    Wednesday, October 14, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/OHV_rAM4zZrEkf12YiSFcc52UcAnvwvbY66pCRL0DO_cxvOfaiODd8GTSiIJDtn-.RCyh--aoMwVjHtmr
Abstract:                
The classical Calderón-Zygmund decomposition is a fundamental tool that helps one study endpoint estimates near L1. In this talk, we shall study an extension of the decomposition to a particular operator valued setting where noncommutativity makes its appearance, allowing to get rid of the (usually necessary) UMD property of the Banach space where functions take values.

A two-dimensional inverse problem in magnetism

Speaker:               Elodie Pozzi - St. Louis University
Time:                    Wednesday, October 21, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/MaDJW8xxTgJLKpJfys0fUXtEC6intIfpE4us9X_me02GlZRUeclTHhV4HlHb_bM.4U-MSK6_TABqH_dW
Abstract:                
Inverse problems have known a recent development in many fields like signal processing, medical imaging and more recently paleomagnetism. Broadly speaking, an inverse problem consists in reconstructing from a set of measurements the original source. We consider a two dimensional inverse problem in magnetism to estimate the net moment represented by the mean value of a function supported on an interval K of the real line from the partial knowledge of the magnetism on another interval S located on the parallel line to K at height h. We will see how this question can be rephrased using complex analysis, harmonic analysis and operator theory. To estimate the mean value, we will construct and solve a constrained approximation problem. This talk is based on a joint work with Juliette Leblond, INRIA, France.

Commutators and bounded mean oscillation

Speaker:               Brett Wick - Washington University in St. Louis
Time:                    Wednesday, October 28, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/SDJIEF5xgmX7ttXn6IEMKB8xNJooOobt12NRcGQL9KF0RTx4yPguvY_235ztEOmF.86UFHEL9Z9qKNqq0
Abstract:                
We will discuss some recent results about commutators of certain Calderon-Zygmund operators and BMO spaces and how these generate bounded operators on Lebesgue spaces.  Results on the Heisenberg group, pseudoconvex domains with $C^2$ boundary, and other examples will be explained.  This talk is based on joint collaborative work.

A different approach to endpoint weak-type estimates for Calderón-Zygmund operators

Speaker:               Cody Stockdale - Clemson University
Time:                    Wednesday, November 4, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/_MSiI8Pn7rwMLIQGjlIxkyXm2pr5ZLWnOG0CzA9HoVCRaUgksAfZ1PIpKVo0khSB.6k1gsehO8UDcKdmH
Abstract:                
The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.

Sharp trace inequalities involving differential operators

Speaker:               Franz Gmeineder - University of Bonn
Time:                    Wednesday, November 11, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/IP6Y_lxOixwQycAw6KGUdWzYluh7nrdnNdTvr-Xyk_iRIyLplaoUa0Np7P7VKRhL.Mi-xJXtD_e30JR7z
Abstract:                
Essentially by the non-availability of singular integrals on L^1, one cannot estimate the L^1-norms of the full k-th order gradients against those of k-th order differential expressions in general. This is known as Ornstein's Non-Inequality. Over the past years, however, a theory has been developed that allows to estimate lower order quantities by the L^1-norms of differential expressions in a sharp way. In this talk, which is joint work with Lars Diening (Bielefeld), we present sharp conditions on differential operators to retrieve the well-known boundary  trace estimates from the Sobolev case.

Sampling, density and equidistribution

Speaker:               José Luis Romero - University of Vienna
Time:                    Tuesday, November 17, 2020 - 9 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/vqYYCwBo_y73KFRfkfcOYEXGpCjmgi2I0GBWVxFZfGq9mmoqvtXexW6HzrX5-eMN.01qm-YZwG-FIl1QY
Abstract:                
The sampling problem concerns the reconstruction of every function within a given class from their values observed only at certain points (samples). A density theorem gives necessary or sufficient conditions for such reconstruction in terms of an adequate notion of density of the set of samples. The most classical density theorems, due to Shannon and Beurling, involve bandlimited functions (that is, functions whose Fourier transforms are supported on the unit interval) and provide a precise geometric characterization of all configurations of points that lead to reconstruction. I will present modern variants of these results and their applications in other fields of analysis.

Spectral geometry of CR manifolds

Speaker:               Yunus Zeytuncu - University of Michigan Dearborn
Time:                    Monday, November 23, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           https://clemson.zoom.us/rec/share/cFUx3KvrBGMBSloIdJkycDF1jeDsFGS4tNs62sNotnaoenmZJmDDdo7g66vkn7J9.jHLh3JjsAllXKgqG
Abstract:                
In this talk, we look at the spectrum of the Kohn Laplacian on spheres \mathbb{S}^{2n-1} and the Rossi sphere \mathbb{S}^3_t. We relate spectral calculations to the geometry of the manifolds. In the first part, we use the lower bounds on the essential spectrum to understand the embeddability of the Rossi sphere. In the second part, we compute the leading coefficient in the asymptotic expansion of the counting function on \mathbb{S}^{2n-1} by a Tauberian argument, and we relate it to the volume of the spheres.

A unified method for commutators of Calderón-Zygmund operators

Speaker:               Kabe Moen - University of Alabama
Time:                    Wednesday, December 2, 2020 - 11:15 am
Location:              https://clemson.zoom.us/j/93496025081?pwd=TlYrRmVwSU9YY0orcnFnV09TZUhJUT09
Recording:           
https://clemson.zoom.us/rec/share/yoO3oytvYa2Jr-3YXfnLLtcRVfiBm9m5FO0FQQoSJaX3kFobSotDJyp8ctmoKncX.AKheI2nzSRE9TcAO
Abstract:                
We present a unified method to obtain weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones.  We also look at some new two weight bump conditions for commutators.