Titles and Abstracts

Murat Akman, University of Connecticut

Title: A Minkowski problem for nonlinear capacity

Abstract: In this talk, we study a Minkowski problem for a certain measure associated with a compact convex set $E$ with nonempty interior and its $\mathcal{A}-$harmonic capacitary function in the complement of $E$. Here $\mathcal{A}-$harmonic PDE is a non-linear elliptic PDE whose structure is modeled on the p-Laplace equation. If $\mu_E$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure $\mu$ on $\mathbb{S}^{n-1}$, find necessary and sufficient conditions for which there exists $E$ as above with $\mu_E =\mu$. We show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem for volume as well as in the work of Jerison for electrostatic capacity. We also show that this problem has a unique solution up to translation when $p \neq n -1$ and translation and dilation when $p = n- 1$.

Title: Weighted Regularity

Abstract: Regularity of a compact set is important in Pluripotential theory, Approximation theory and Separately analytic functions. Recently Sadullaev introduced weighted regularity notions. In this talk, we will discuss the Weighted local and global Pluriregularity, and supports of weighted equilibrium measures.

We will present positive and negative answers to few open problems in this area.

Title: Generalized rectifiability of measures and the identification problem

Abstract: One goal of geometric measure theory is to understand how measures in Euclidean space interact with canonical lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to any distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets), but all of them can be encoded using the same framework. In this talk, I will describe the framework for generalized rectifiability of measures, review a selection of classical results in the context of this framework, and survey recent advances on the problem of identifying Radon measures that are carried by Lipschitz or Hölder images of Euclidean subspaces.

Title: On some vector-valued functions associated with submanifolds of the unit ball

Abstract: Suppose $D$ is a bounded domain in $C^n$ that covers a compact complex manifold, $X$. In recent work with N. Alluhaibi we constructed vector-valued Poincare series associated to submanifolds of $D$ and studied asymptotics of the inner products of these vector-valued functions when $D$ is the unit ball. After a quick review of previous work, I will state further results in this direction, and will also present a somewhat different construction, that uses a closed geodesic in $X$ (so, a submanifold of $X$, not a submanifold of $D$).

Almaz Butaev, Concordia U.

Title: Some refinements of the embedding of critical Sobolev spaces into BMO

Abstract: In 2004, Van Schaftinen showed that the inequalities established by Bourgain and Brezis give rise to new function spaces that refine the classical embedding $\mathring{W}^{1,n}(\mathbb{R}^n)\subset \rm{BMO}(\mathbb{R}^n)$.

In this talk, we discuss the non-homogeneous analogs of these function spaces on $\mathbb{R}^n$ and their generalizations to Riemannian manifolds with bounded geometry.

Title: Semilinear evolution equation on resistance spaces

Abstract: We will describe a class of semilinear parabolic evolution problems on resistance spaces (in the sense of Kigami) endowed with a possibly energy singular volume measure. The nonlinear operator governing the equation is a fractal Dirichlet Laplacian plus a certain nonlinear divergence term. Problems of this type occur in the study of hydrodynamic limits of exclusion processes on fractals. We formulate the equation in terms of the first-order calculus for Dirichlet forms, and establish existence, uniqueness, and basic regularity properties using monotone operator methods.

This is a joint work with Michael Hinz (Bielefeld) and Alexander Teplyaev (UConn).

Title: The $L^p$ estimate of an integral solution for $\bar \partial$ on product spaces

Abstract: Following the Cauchy-Fantappi\'{e} formalism, we construct a generating form and obtain an integral solution for the $\bar \partial$-equation on the product space. In particular, we study the $L^p$ estimate of the solution in dimension 2 and apply it to the Hartogs triangle.

Title: Steklov eigenproblems and representations of electrostatics approximations of vector fields.

Abstract: This talk will describe different representations for solutions of Laplacian boundary value problems on bounded regions. Null Dirichlet, Neumann and Robin boundary conditions are considered and the results hold for weak solutions in relevant subspace of Hilbert-Sobolev space associated with the problem. The solutions are represented by orthogonal series using the harmonic Steklov eigenfunctions. Results about the accuracy of these solutions will be discussed.

Galia Dafni, Concordia U.

Title: Some results on the space JN_p

Abstract: In their paper introducing the class of functions of bounded mean oscillation (BMO),

John and Nirenberg also considered functions satisfying a weaker condition (later known

as JN_p) and showed that such functions lie in weak L^p. The space JN_p contains

L^p, but it was not known that it was strictly larger. In one dimension, we show that JN_p and L^p coincide in the case of monotone functions, but they are distinct spaces, as shown by an example. We also characterize JN_p as the dual of a new space defined in terms of atomic decomposition. This is joint work with Tuomas Hytönen, Riikka Korte, and Hong Yue.

Title: FIOs with singularities in imaging problems

Abstract: We consider the composition calculus of Fourier integral operators (FIOs) which appear in the Synthetic Aperture Radar (SAR) imaging when the emitter and receiver are moving on different platforms. We use microlocal analysis to study the singularities of the operator F, and the properties of the normal operator F*F which is used to reconstruct the image. We show that the normal operator belongs to a class of distributions associated to two cleanly intersecting Lagrangians I^{p,l}(Delta, Lambda) and we characterize the strength of the artifact given by Lambda.

Taryn Flock, UMass Amherst

Title: Continuity of the Brascamp--Lieb constant and applications

Abstract: Brascamp-Lieb inequality generalizes many inequalities in analysis, including the Hölder, Loomis-Whitney, and Young's convolution inequalities. Sharp constants for such inequalities have a long history and have only been determined in a few cases. We investigate the stability and regularity of the sharp constant as a function of the implicit parameters. The focus of the talk will be a continuity result with several applications including a sharp nonlinear Brascamp-Lieb inequality for so-called "simple data" and a mutlilinear Kakeya-type inequality with applications to Bourgain and Demeter's decoupling method. This is joint work with Jonathan Bennett, Neal Bez, Stefan Buschenhenke, Michael Cowling and Sanghyuk Lee.

Mahya Ghandehari, U. of Delaware

Title: Derivatives on Fourier algebras

Abstract: A major trend in non-commutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of non-abelian groups.The Fourier algebra, Rajchman algebra and Fourier-Steiltjes algebra, which are associated with the regular representation, the $C_0$ universal representation and the universal representation of the ambient group respectively, are important examples of such function spaces. These function algebras encode the properties of the group in various ways; for instance the non-existence of derivations on such algebras indicates their lack of analytic properties, which in turn translates into forms of either commutativity or discreteness for the group itself. In this talk, we study some Banach algebra properties of these function algebras. In particular, we present explicit constructions of continuous derivations on the Fourier algebras of two important matrix groups, namely the group of ${\mathbb R}$-affine transformations and the Heisenberg group. Using the structure theory of Lie groups, we extend our results to semisimple Lie groups and nilpotent Lie groups. If time permits, we will discuss weighted versions of the Fourier algebra, called the Beurling-Fourier algebra, and some of their properties.

Title: An affine invariant for convex bodies via complex analysis

Abstract: In this talk, we will quantitatively compare two families of sets associated to a real convex body that arise from different considerations. One is its set of convex floating bodies, first introduced in the study of the Blaschke surface area measure. The other comes from the reproducing kernel of a space of holomorphic functions associated to the convex body. Based on this comparison, we will define a new affine invariant, compute it for some planar examples, and discuss further directions.

Eyvindur Ari Palsson, Virginia Tech

Title: Dimensional lower bounds for Falconer type incidence theorems

Abstract: In this talk, we will look at the problem of how large the Hausdorff dimension of E, a subset of d-dimensional Euclidean space, must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k+1 points from E) to have positive Lebesgue measure. This generalizes the k=1 case, the well-known Falconer distance problem and a major open problem in geometric measure theory. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. Two results will be presented. The first is a dimensional lower threshold of (d+1)/2 on incidence theorems for $k$-simplices, under certain restrictions on the dimension d, by generalizing an example of Mattila. The second is also a dimensional lower threshold of (d+1)/2 on incidence theorems in every dimension greater than $3$, however this time the restriction is that we can only deal with the case k=2. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

Bob Strichartz, Cornell University