49th Annual Spring Lecture Series
Harmonic analysis, partial differential equations, and geometric measure theory
May 3–5, 2024
Please register here! Partial travel funding for selected participants will be supported by a grant from the NSF.
This conference is in Cooperation with AWM and supports the Welcoming Environment Statement of the Association for Women in Mathematics.
Principal Speaker: Svitlana Mayboroda
McKnight Presidential Professor, University of MinnesotaPublic Lecture | Change of Perspective in Mathematics
May 3, 2024 (5:30 pm CDT)
Public Lecturer: Michael Orrison
Professor of Mathematics, Harvey Mudd CollegeNote: No registration is necessary for the public lecture.
Women in Mathematics Panel
Friday, May 3, 2024 (5:00 pm CDT)
Invited Speakers
Matthew Badger (University of Connecticut)
Blair Davey (Montana State University)
Guy David (Université de Paris-Sud (Orsay))
Joseph Feneuil (Australian National University)
Silvia Ghinassi (University of Washington)
Steve Hofmann (University of Missouri)
José María Martell (Instituto de Ciencias Matemáticas (ICMAT))
Zihui Zhao (Johns Hopkins University)
Organizers
Ariel Barton (aeb019@uark.edu)
Associate Professor of Mathematical Sciences, University of ArkansasZachary Bradshaw (zb002@uark.edu)
Associate Professor of Mathematical Sciences, University of ArkansasTentative Schedule of Talks
Friday, May 3th |
Saturday, May 4th
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Sunday, May 5th
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8:30 | Registration—Coffee/Tea SCEN 407 | Coffee/Tea Hillside 208 | Coffee/Tea Hillside 208 |
8:55 | Opening remarks | ||
9:00 | Guy David Université de Paris-Sud (Orsay) SCEN 407 |
Silvia Ghinassi University of Washington Hillside 208 | Short talks by early career Mathematicians Hillside 208 |
10:00 | Svitlana Mayboroda University of Minnesota and ETH Zurich (Lecture #1) SCEN 407 |
Svitlana Mayboroda University of Minnesota and ETH Zurich (Lecture #3) Hillside 208 |
Svitlana Mayboroda University of Minnesota and ETH Zurich (Lecture #5) Hillside 208 |
11:00 | Blair Davey Montana State University SCEN 407 |
Steve Hofmann University of Missouri Hillside 208 |
Joseph Feneuil Australian National University Hillside 208 |
12:00 | Lunch | Lunch | Shuttle to Airport |
2:30 | Svitlana Mayboroda University of Minnesota and ETH Zurich (Lecture #2) Hillside 208 |
José María Martell Instituto de Ciencias Matemáticas (ICMAT) Hillside 208 | |
3:30 | Matthew Badger University of Connecticut Hillside 208 |
Svitlana Mayboroda University of Minnesota and ETH Zurich (Lecture #4) Hillside 208 | |
4:30 | Zihui Zhao Johns Hopkins University Hillside 208 | ||
5:00 | Women in Mathematics panel Hillside 208 | ||
5:30 | Public Lecture Michael Orrison, Harvey Mudd College Hillside 208 | Walk to dinner |
Abstracts of Talks
Matthew Badger, University of Connecticut
Title: Nodal Domains of Homogeneous Caloric Polynomials
Abstract: With a view towards confirming the existence of singular strata in Mourgoglou and Puliatti's two-phase free boundary regularity theorem for caloric measure, we identify the minimum number of nodal domains of homogeneous caloric polynomials (hcps) in Rn+1 of degree d. We also provide estimates on the maximum number of nodal domains for all n and d. I'll survey the techniques that go into the proofs of the theorems, particularly the construction of hcps that realize the minimum number of nodal domains. This is joint work with Cole Jeznach.
Blair Davey, Montana State University
Title: Fractional parabolic theory as a high-dimensional limit of fractional elliptic theory
Abstract: Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we discuss how a high-dimensional limiting technique can be used to prove theorems about solutions to the fractional heat equation (or its Caffarelli-Silvestre extension problem) from their elliptic analogues. This talk covers joint work with Mariana Smit Vega Garcia.
Guy David, Université de Paris-Sud (Orsay)
Title: Counterexamples involving elliptic measure and Canto sets
Abstract: We’ll describe examples of Cantor sets or snowflakes and elliptic operators for which the elliptic measure on the given set is equivalent to the natural Hausdorff measure. This is joint work with Jeznach, Julia, Mayboroda, and Perstneva.
Joseph Feneuil, Australian National University
Title: Green functions, smooth distances, and uniform rectifiability
Abstract: The past 10 years have seen considerable achievements at the intersection of harmonic analysis, PDE, and geometric measure theory. One now better understands the relationship between the geometry of the boundary of a domain and the regularity of harmonic/elliptic solutions inside the domain. For instance, it was proved that the uniform rectifiability (UR) of a codimension 1 set is characterized by the A∞-absolute continuity of its harmonic measure with respect to the surface measure - or equivalently the solvability of a Lp Dirichlet problem in the complement.
In this talk, I will show that another characterization of UR sets of codimension 1 can be obtained by comparing the Green function G with some regularized version of the distance to the boundary. Moreover, I will obtain a characterization of any UR set of any codimension by an estimate on ∇|∇G|. Those are joint works with Guy David, Linhan Li, and Svitlana Mayboroda.
Steve Hofmann, University of Missouri
Title: A problem of free boundary type for caloric measure
Abstract:
For an open set Ω⊂Rd with an Ahlfors regular boundary, solvability
of the Dirichlet problem for Laplace’s equation, with boundary data in
Lp for some p<∞, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic
measure with respect to surface measure on ∂Ω. A similar statement
is true in the caloric setting. Thus, it is of interest to find geometric
criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in
the harmonic case. In this talk, we shall discuss recent progress in the
caloric setting, in which we show that quantitative absolute continuity
of caloric measure, with respect to “surface measure” on the parabolic
Ahlfors regular (lateral) boundary Σ, implies parabolic uniform rectifiability of Σ. We observe that this result may be viewed as the solution
of a certain 1-phase free boundary problem.
This is joint work with S. Bortz, J. M. Martell and K. Nyström.
José María Martell, Instituto de Ciencias Matemáticas (ICMAT)
Title: The Dirichlet problem with data in Hölder spaces in rough domains
Abstract: We consider the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. We work in open sets satisfying the capacity density condition (a quantitative version of the Wiener), without any further topological assumptions such as connectivity, and show that the Dirichlet boundary value problem is well-posed for boundary data in Hölder spaces with small enough exponent if Ω is either bounded, or unbounded with unbounded boundary. However, when Ω is unbounded with bounded boundary (e.g., the complement of a compact set), we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition.
Zihui Zhao, Johns Hopkins University
Title: Boundary unique continuation of harmonic functions
Abstract: Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.