CBMS’20 Register

Lectures

The conference will feature ten lectures delivered by Dr. Svitlana Mayboroda 

Besides these lectures, we plan several 45-minute long lectures. We also plan a poster session (with posters printed at FSU for free!). 

The list of speakers includes:

  • Matthew Badger, University of Connecticut
  • Ariel Barton, University of Arkansas
  • Blair Davey, Montana State University
  • Guy David, Université de Paris Sud
  • Max Engelstein, University of Minnesota, Twin-Cities
  • John Garnett, UCLA
  • Joseph Feneuil, Université Paris-Saclay 
  • Linhan Li, University of Minnesota, Twin-Cities
  • Xavier Tolsa, Universitat Autonoma de Barcelona
  • Zihui Zhao, University of Chicago

Schedule

Zoom link (Passcode: CBMS2022)

 

 

 

All talks will be in HCB 214
The coffee will be served in HCB 220 at 9:30am
Time Monday Tuesday Wednesday Thursday Friday

10-10:45

S.M.

S.M.

S.M.

S.M.

S.M.

 

10:45-11:00

Questions

Questions

Questions

Questions

Questions

 

11:00-11:15

Break

Break

Break

Break

Break

 

11:15-12:00

S.M.

S.M.

S.M.

S.M.

S.M.

 

12:00-12:15

Questions

Questions

Questions

Questions

Questions

 

12:15-2:15

Lunch

Lunch

 

Lunch

Lunch

 

2:15-3:00

David

Tolsa

 

Badger

Li

 

3:00-3:15

Questions

Questions

 

Questions

Questions

 

3:15-3:30

Break

Break

 

Break

Break

 

3:30-4:15

Engelstein

Garnett

 

Davey

Feneuil

 

4:15-4:30

Questions

Questions

 

Questions

Questions

 

4:30-4:45

Break

Break

 

 

 

 

4:45-5:30

Barton

Zhao

 

 

 

 
6:00       Conference Dinner    

 

Preliminary Materials

Some preliminary materials can be found here

Abstracts of Talks

Matthew Badger, University of Connecticut

Title: Bourgain's constants for harmonic and caloric measures

Abstract: Bourgain’s constant is the largest number bn such that the dimension of harmonic measure on any n-dimensional domain is at most n − bn. Jones and Wolff (1988) proved that b2 = 1. When n ≥ 3, Bourgain (1987) proved that bn > 0 and Wolff (1995) proved that bn < 1. In recent work, joint with Alyssa Genschaw, we prove that Bourgain’s theorem on the dimension of harmonic measure persists in the setting of the heat equation. Further, refining Bourgain’s original outline, we prove asymptotic lower bounds on Bourgain’s constant in high dimensions and explicit lower bounds on Bourgain’s constant in dimensions 3 and 4.

 

Ariel Barton, University of Arkansas

Title: Higher order elliptic partial differential equations with lower order terms

Abstract: Operators of the form Lu = ∇ ⋅ (Au), that is, second order linear differential operators in divergence form, are by now very well understood. Two important generalizations are higher order operators Lu = ∇m ⋅ (Amu), for m ≥ 2 an integer, and operators with lower order terms Lu = ∇ ⋅ (Au) + ∇ ⋅ (Bu) + C ⋅ ∇u + Du. In this talk we discuss some first attempts to combine the two generalizations and study higher order operators with lower order terms.

 

Blair Davey, Montana State University

Title: Exponential Decay Estimates for Fundamental Matrices of Generalized Schrödinger Systems

Abstract: In this work, we consider the behavior of the fundamental matrices associated to systems of generalized Schrödinger operators. The Schrödinger operators that we deal with have leading coefficients that are bounded and uniformly elliptic, and they have zeroth-order potentials that belong to certain reverse Hölder matrix classes. By exploring the properties of reverse H\"older matrices and building on the ideas of Shen, Mayboroda and Poggi, we establish the existence of fundamental matrices and prove upper and lower exponential decay estimates for them. This talk covers joint work with Josh Isralowitz.

 

Guy David, Université de Paris Sud

Title: An elliptic operator on the four-corner Cantor set, with elliptic measure proportional to H1

Abstract: The lecture is about the construction, with S. Mayboroda, of an elliptic operator L =  − div(a(x)∇) on the complement of a one-dimensional Cantor set E in the plane, so that the corresponding elliptic measure on E is the product of the length measure H1 by a function which is bounded and bounded from below. The main idea is to build the Green function by hand, deduce the function a, and use the fractal invariance to prove it is elliptic. I’ll try to explain why we care about such examples.

 

Max Engelstein, University of Minnesota, Twin-Cities

Title: Harmonic analysis tools for free boundary problems

Abstract: In this talk we will review some recent and not so recent work in which we apply tools and ideas from harmonic analysis and geometric measure theory to study (almost-)minimizers to free boundary problems of Alt-Caffarelli-Friedman type. In particular we will show how the regularized distances of David-Feneuil-Mayboroda can be used to produce (counter-)examples regarding the behavior of cusp points in two-phase free boundary problems. This is joint work with Guy David, Mariana Smit Vega Garcia and Tatiana Toro.

 

Joseph Feneuil, Université Paris-Saclay 

Title: Generalized Carleson perturbations of elliptic operators.

Abstract: Let L0=div A0  be a uniformly elliptic operator on a domain Ω, and we assume that the elliptic measure ωL0 is A-absolutely continuous with respect to a doubling measure σ on the boundary Ω. We are interested in perturbations L1 of L0 that preserve the A-absolute continuity of the elliptic measure w.r.t. σ.

We shall discuss the literature on “Carleson perturbations of elliptic operators”, from the initial work of Fefferman, Kenig and Pipher in 1991, to the recent article of Bruno Poggi and the speaker. In the later paper, we proposed an extended notion of Carleson perturbations, and we worked on a wide range of domains and elliptic operators by assuming only a suitable elliptic theory.

 

John Garnett, UCLA

Title: Carleson measure estimates for bounded harmonic functions, without Ahlfors regularity assumptions.

 

Linhan Li, University of Minnesota, Twin-Cities

Title: Green function estimates and elliptic measures
Abstract: We are interested in the relations among an elliptic operator on a domain, the geometry of the domain, and the boundary behavior of the Green function. In this talk, I'll talk about estimates for the Green function under some optimal condition on the elliptic operator, and their applications to elliptic measures. I'll also talk about analogous results on sets with lower-dimensional boundaries. 

 

Xavier Tolsa, ICREA - Universitat Autònoma de Barcelona - CRM

Title: The regularity problem for the Laplace equation and boundary Poincaré inequalities in rough domains

Abstract: Given a bounded domain Ω ⊂ ℝn, one says that the Lp-regularity problem is solvable for the Laplace equation in Ω if, given any continuous function f defined in Ω and the harmonic extension u of f to Ω, the non-tangential maximal function of the gradient of u can be controlled in Lp norm by the tangential derivative of f in Ω. Up to quite recently this was only known to hold for Lipschitz domains (in some range of p’s). In my talk I will explain a recent result with Mihalis Mourgoglou where we show that the Lp-regularity is also solvable in more general domains, such as 2-sided chord-arc domains. In the solution of this problem, the Poincaré inequality in the boundary of the domain plays an important role. I will also discuss this issue and a related joint result with Olli Tapiola where we show that the boundaries of 2-sided chord-arc domains support 1-Poincaré inequalities.

 

Zihuy Zhao, University of Chicago

Title: Boundary unique continuation and the estimate of the singular set
Abstract: Unique continuation property is a fundamental property of harmonic functions, as well as solutions to a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes to infinite order at a point, it must be zero everywhere. In the same spirit, we can use the local growth rate of harmonic functions to deduce global information, such as estimating the size of the singular set for elliptic PDEs. This is joint work with Carlos Kenig.

Host University and Local Arrangements

Conference location

 The main part of the conference will be held at the HCB building.

Housing

There are several hotels within walking distance from campus. There are also plenty of Airbnb and Vrbo options.

The following hotels gave a discounted rate for the conference participants:

Aloft Tallahassee Downtown Reserve

Hampton Inn & Suites Reservation Link Reserve

Dining

Seminole Cafe and the Suwanee room, two buffet-style cafeterias, are located on Campus at five to ten minutes walking distance from the Department of Mathematics. Both places offer very good pricing and choice of food (in particular, vegetarian), and serve Breakfast, Lunch, and Dinner. Moreover, there are many independent dining options within 1 mile from campus, serving a wide variety of food. We particularly encourage the participants to check out the restaurants located on and around Gaines Street.

Internet Access

Florida State University uses the eduroam network, and the participants will be able to join it. In addition, guest accounts for the FSU network will be created for the participants, which will provide high-speed internet access. Access to printers and computer lab will be provided as well, in case the participants will need to download, copy or print papers or other materials.

Sponsors

CBMS’22 is supported by

FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS & SCIENCES

DMS-1933361

DMS-1933361