Abstracts

Titles and Abstracts

Afrah Abdou, Jeddah University

Title: Fixed Points Results for Multivalued Contractive Mappings with CLR property

Abstract: In this work, we define a new property which contains the properties (CLR) and (owc) for four single and multi-valued maps and give some new common fixed point results. Our results extend and improve some results given by some authors.

Murat Akman, University of Connecticut

Title: Perturbations of elliptic operators on non-smooth domains

Abstract: In this talk, we study perturbations of elliptic operators on domains with rough boundaries. In particular, we focus on the following problem: suppose that we have ``good estimates'' for the Dirichlet problem for a uniformly elliptic operator L<sub>0</sub>, under what optimal conditions, are those good estimates transferred to the Dirichlet problem for uniformly elliptic operator L which is a ``perturbation'' of L₀?

We prove that if discrepancy between L₀ and L satisfies certain smallness assumption then the elliptic measure 𝜔_L corresponding to L is in the reverse Hölder class with exponent 2 with respect to the elliptic measure 𝜔_L0 corresponding to L₀ when the domain is 1-sided NTA satisfying the capacity density condition. Our work extends classical results of Fefferman, Kenig, and Pipher in Lipschitz domains, and Milakis, Pipher, and Toro in chord-arc domains to 1-sided NTA domains satisfying the CDC.

This is a joint work in progress with Steve Hofmann, José María Martell, and Tatiana Toro.

Stephen Farnham, Syracuse University

Title: The Convergence of Blaschke Expansions

Abstract: In this talk, the notion of a Blaschke Expansion for a function in H²(D) (The Hardy space on the unit complex disc) will be discussed, as well as new results on the bounds for convergence of arbitrary H²(D) functions using Blaschke Expansions.

Taryn Flock, University of Massachusetts

Title: A nonlinear Brascamp-Lieb inequality

Abstract: Brascamp-Lieb inequality generalizes many inequalities in analysis, including the Hölder, Loomis-Whitney, and Young's convolution inequalities. The focus of the talk will be a nonlinear generalisation of the classical Brascamp-Lieb inequality in a general setting. As a corollary, we show that the best constant in Young's convolution inequality in a small neighbourhood of the identity of a general Lie group, approaches the euclidean constant as the size of the neighbourhood approaches zero.

A first step in this analysis is understanding the regularity of the sharp constant in the Brascamp-Lieb constant. This work has other applications including a mutlilinear Kakeya-type inequality which is used in Bourgain, Demeter's, Guth's proof of the Vinogradov mean value theorem. The Brascamp-Leib constant also makes a surprise appearance in operator scaling, a generalization of the well-known matrix scaling algorithm from computer science. Time permitting, we will discuss these connections as well. (Joint work with Jon Bennett, Neal Bez, Stefan Buschenhenke, Michael Cowling, and Sanghyuk Lee)

Bryan Goldberg, University at Albany SUNY

Title: Complex Dynamics on the Projective Spectrum of the Infinite Dihedral Group

Abstract: Using the self-similarity of the infinite dihedral group (D_∞) in Joint Spectrum and the Infinite Dihedral Group, Grigorchuk and Yang defined a mapping F: C³ → C³ where F(z) = (z₀(z₀²-z₁²-z₂²), z₁²z₂, z₂(z₀² -z₂²)). After establishing some background on F(z) we'll use complex dynamics to establish some properties of this mapping. We'll use equivalent projective space and look at F: P² → P² to discuss some results including the Fatou and Julia sets of F(z) restricted to the projective spectrum. We'll conclude by examing connections between spectral theory and dynamics in this particular situation. This is joint work with Rongwei Yang.

Kris Hollingsworth, University of Delaware

Title: Constructing Discrete Frames from Continuous Wavelet Transforms

Abstract: A discrete frame for L²(R^d) is a sequence {e_j}_{j∈ J} in L²(R^d) together with real constants 0<A≤B< ∞ such that

A\|f\|₂² ≤ ∑_{j∈ J} | <f, e_j> |² ≤ B\|f\|₂²,

for all f \in L²(R^d). We begin by surveying recent work on the discretization problem. Next we present our method of sampling continuous frames arising from square integrable representations. This involves discretizing the ambient space through use of a ``tiling system" and discretizing an integral with tools from harmonic analysis. We will illustrate this method with an explicit construction for L²(R⁴). This talk is based on joint work with Mahya Ghandehari, Nathaniel Kim, and Paige Shumskas.

Hyun-Kyoung Kwon, University at Albany SUNY

Title: A subclass of the Cowen-Douglas class

Abstract: We consider a subclass of the Cowen-Douglas class where the problems of unitary equivalence and similarity become a bit more manageable. This talk is based on joint work with Kui Ji.

Jungang Li, University of Connecticut

Title: The Chang-Marshall Type Inequality For Sobolev Functions In Any Higher Dimension

Abstract: In this talk, we establish a Change-Marshall type inequality for Sobolev functions on bounded domains in any higher dimension. This extends the celebrated Chang-Marshall inequality for holomorphic functions on the planar disk.

Amy Peterson, University of Connecticut

Title: The Gaussian Limit for High Dimensional Spherical Means

Abstract: Given a function f we can associate to an affine subspace A the integral of f over A. For A, an affine subspace in l² of finite codimension and A_N its intersection with R^N, we create a circle S_AN which is the intersection of A_N with the sphere S^N-1(√N). We show that, in the large-N limit, the spherical integral of f over S_AN converges to a Gaussian integral of f in infinite dimensions.

Vyron Vellis, UConn

Title: Fractional rectifiability

Abstract: Given a bounded set E⊂ Rⁿ, when is it possible to construct a nice map (Holder, Lipschitz) from the unit interval into Rⁿ so that E is contained in its image? In this talk we discuss an extension of Peter Jones' traveling salesman construction, which provides a sufficient condition for E to be contained in a (1/s)-Hölder curve, s ≥ 1. The original result, corresponding to the case s=1, identified subsets of rectifiable curves. When s>1, (1/s)-Hölder curves are more exotic objects than rectifiable curves that include snowflake curves and space-filling curves as basic examples. This talk is based on a joint work with Matthew Badger and a joint work with Matthew Badger and Lisa Naples.

Dominick Villano, University of Pennsylvania

Title: Radon-like transforms in intermediate dimension

Abstract: The mapping properties of Radon-like transforms are governed by the geometry of sub-manifolds in Euclidean space. In general, much more is known when these sub-manifolds are curves or hypersurfaces. In this talk, I will describe a technique that leverages the one dimensional theory to produce bounds for all dimensions.

Kazuo Yamazaki, University of Rochester

Title: Recent developments in stochastic analysis on fluid dynamics PDE

Abstract: I plan to survey recent developments in particular directions of research on fluid dynamics PDE forced by random noise. In particular, the system of equations should include the Navier-Stokes equations, magnetohydrodynamics system, Hall-magnetohydrodynamics system, as well as possibly KPZ equation and more. The directions of research to be discussed should include issues such as well-posedness and ergodicity.

Rongwei Yang, University at Albany SUNY

Title: Projective spectrum and its applications in group theory and complex geometry

Abstract: For a tuple A=(A₁, A₂, ..., A_n) of elements in a unital algebra B over C, its projective spectrum P(A) is the collection of z ∈ Cⁿ such that A(z)=z₁A₁+z₂A₂+...+z_nA_n is not invertible in B.

When the tuple A is a generating tuple for a group G with respect to some representation π, the projective spectrum reflects the structure of G as well as the properties of π.

Interesting connections are found with classical Dedekind-Frobenius theory of group determinants and more recent work on group of intermediate growth. In a different direction, the Maurer-Cartan type 1-form 𝜔_A(z)=A^-1(z)dA(z) reveals much topological information about the complement P^c(A)= Cⁿ \ P(A). Moreover, a natural Hermitian metric can be defined on P^c(A) through the operator-valued (1, 1)-form

𝜔^*_A ⋀ 𝜔_A and it relates projective spectrum to Kähler geometry and Yang-Mills equations.

Zhen Zeng, University of Pennsylvania

Title: On the power decay property of multilinear oscillatory integrals

Abstract: In the breakthrough paper of Christ, Li, Tao and Thiele, they initial the study of the conditions on the polynomial phase P and the projections {π_i} to ensure the power decay estimate of the corresponding multilinear oscillatory integrals. In this talk, I will extend their results and tools to the cases that have not been explored. It turns out that \lambda-uniformity introduced by Christ, Li, Tao and Thiele can be generalized to deal with the trilinear case. I will also introduce the concept of preservation of direct sum decomposition to the case where the ambient space can be "separated" into two subspaces. This concept can be used to solve the case where every projection preserves this separation structure.

Xiaodan Zhou, Worchester Polytechnic Institute

Title: Strong Comparison Principle for p-harmonic functions in Carnot-Caratheodory spaces

Abstract: We extend Bony's propagation of support argument to C¹ solutions of the non-homogeneous sub-elliptic p-Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf. This is joint work with Luca Capogna.