The goal of MWAA is to facilitate interactions between mathematicians from the midwestern United States working in approximation theory, mathematical physics, potential theory, complex analysis, and related fields by bringing them together in an informal way for a two day workshop. A major objective is to expose attending graduate students to different areas of analysis. We hope these meetings will help to strengthen the partnership and increase the collaboration between analysts in the midwest. We would appreciate if you could help us to disseminate information about the workshop by printing and posting our poster at your department.
This year's workshop is in part supported by National Science Foundation grants DMS2331073. It will take place on IUPUI campus. The talks will be held at IT252 and the coffee breaks and poster presentations will take place in IT109.
If you are planning to drive to campus, please, look for visitors' parking (follow the links to Inlow Hall of Informatics
or University Library
). You will have to pay for parking, but we shall reimburse you later. Please, consider carpooling.
Accommodations of speakers are arranged by the organizers. All other participants are expected to book their hotel themselves. There are many hotels next to IUPUI campus. Accomodations found through Airbnb will be reimbursed as well. There will be a dinner on Saturday night at The Rathskeller, see menu. The participants will need to cover their own bill (around $50 with gratuity and without drinks), which will be reimbursed later together with travel and lodging to those seeking financial support.
The program and the list of submitted abstracts can be found below or here. If you have any further questions, please email the organizers at mwaa@iu.edu.
More picture from the workshop can be found here.
3:304:00pm
4:005:00pm
CalderónZygmund operators, which arise naturally in partial differential equations and complex analysis, are integral operators that are associated to a kernel possessing a singularity on the diagonal. Understanding the mapping properties of these operators on various function spaces is an important area of current research. There is a proof strategy that exists, “T1 Theorems”, that utilizes natural necessary testing conditions to provide sufficient conditions to verify the boundedness of the CalderónZygmund operator. Additional tools in the proof are the use of dyadic harmonic analysis techniques. In this talk we'll discuss some recent results about the behavior of CalderónZygmund operators on weighted spaces in various settings and outline some of the main ideas behind the proof and provide some motivation to consider such questions.
9:009:40am
Victor Alves de Souza: The Rational Weighted PolyaTchebotarev Problem; Chad Berner: Framelike Fourier Expansions for Finite Borel Measures; Shreedhar Bhat: \( p \)-Skwarczynski Distance; Nick Castillo: Global Rational Approximations of Functions with Factorially Divergent Asymptotic Series; Ana Colovic: Garsia Norm of Hankel Operators in Weighted Hardy Spaces; Abdullah Helal: Proper Holomorphic Mappings Between Ball Complements; German Mora Saenz: Spectral Stability of Traveling Waves in a Thin-Layer Two-Fluid Couette Flow; Kenta Miyahara: Connection Formulae for the Radial Toda Equations; Achinta Nandi: Boundary Behavior of Geodesics in Asymptotically Symmetric Spaces; Tomas Rodriguez: Compactness of Toeplitz operators with Continuous Symbols on Pseudoconvex Domains in \( \mathbb{C}^n \)
09:4010:20am
Structured determinants and their large size asymptotics are at the core of important questions in various areas, including random matrix theory and statistical mechanics. In this talk we describe the general framework for obtaining Strong Szegő Limit Theorems for (multi)bordered, semiframed, and (multi)framed Toeplitz determinants. These determinants appear in the study of ensembles of nonintersecting paths, entanglement entropy for disjoint subsystems in the XX spin chain, and offdiagonal correlations in the twodimensional Ising model.
10:3011:10am
Joint work with AK. Gallagher and P. Gupta
Wiegerinck proved that the Bergman space over any domain in the complex plane is either trivial or infinite dimensional. In this talk I will discuss various generalizations and open questions related to this theorem. I will survey the case of the complex plane being replaced by \( \mathbb C^n \) as well as a domain in a compact Riemann Surface.
11:20am12:00pm
Joint work with Matthew Satter
The Painlevé functions are a family of ordinary differential equations with myriad applications to mathematical physics and probability. The rational solutions of these equations have drawn attention for the remarkable geometric structure of their zeros and poles. We study the family of rational solutions of the PainlevéV equation built from the socalled Umemura polynomials. We derive a new RiemannHilbert representation and use it to obtain the boundary of the pole region and the largedegree behavior in the polefree region.
12:002:00pm
2:002:40pm
Joint work with B.R Choe, H. Koo, and W. Smith
The composition operator is a well-known linear operator acting boundedly on several analytic function spaces. There is a profusion of remarkable results on the composition operator on the Hardy space \( H^p \) and the Bergman space \( A^p \) of the unit disk \( \mathbb{D} \), for instance. Less is known on domains with less regular boundary. In this talk, we give an overview of some results on less smooth domains. We will give a new boundedness result on Carleson domains, known as domains \( \Omega \) (bounded or unbounded) such that for any Carleson measure \( \mu \), the \( L^1(\Omega,d\mu) \)-norm of a function in \( H^1 \) is less than \( C(\mu)\Vert f\Vert_{L^1(\partial\Omega)} \). We will provide some simple general domains for which the composition operator is bounded.
2:503:30pm
Joint work with Carlos Cabrelli
Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert space, \(\{f_k\}_{k \in I} \subset H\), be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, \(\{T^n \varphi\}_{n=0}^{\infty} \subset H\), to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the HardyHilbert space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame.
3:304:00pm
4:004:40pm
The Nagata Conjecture governs the minimal degree required for a plane algebraic curve to pass through a collection of \( r \) general points in the projective plane \( \mathbb P^2 \) with prescribed multiplicities. The "SHGH" Conjecture governs the dimension of the linear space of these polynomials. We formulate transcendental versions of these conjectures in term of pluripotential theory and we are making some progress.
4:505:30pm
Joint work with Nets Katz and Joshua Zahl
The \( SL_2 \) Kakeya problem is a special case of the three-dimensional Kakeya conjecture, and it is surprisingly connected to the restricted projection problem in geometric measure theory. In this talk, I will first survey the Kakeya conjecture. Then I will discuss a solution to the \( SL_2 \) Kakeya problem and its connection to the restricted projection theorem.
9:0009:40am
See above
09:4010:20am
Multivariate polynomial splines represent the simplest example of piecewise functions. They are wellknown tools in computeraided design and manufacturing, image processing and numerical PDEs. They have been studied extensively in approximation theory and numerical analysis. Since splines are piecewise polynomials of fixed degree and smoothness defined over suitable partitions of domains in \( \mathbb R^n \), they form vector spaces. Spline analysis usually includes determining the dimension of such spaces, finding bases that can be used to solve interpolation problems, and computing approximation order. In this talk we will first address a lesserknown aspect of multivariate splines: intrinsic supersmoothness. In the definition of a spline, there is a prescribed value for the global smoothness of the piecewise function. Intrinsic supersmoothness is a phenomenon that describes additional nonprescribed smoothness across certain faces that appears in almost all multivariate splines except in the univariate setting. We show that the geometry of the underlying partition determines the associated supersmoothness. We demonstrate how intrinsic supersmoothness can be used to compute dimensions of splines. We next show how the concept of intrinsic supersmoothness generalizes to nonpolynomial piecewise functions.
10:3011:10am
Suppose that \( K \subset \mathbb C^d \) is a compact set. For data given on \( K \) (with random errors) it is possible, by means of polynomial regression, to predict (or extrapolate) a value at a point \( z_0 \) exterior to \( K \). An optimal prediction measure is the probability measure on \( K \) which describes the data distribution on \( K \) for which the predicted value has least variance. We will discuss this problem and its relation to another classical approximation problem, give some examples, and discuss some conjectures and open problems.
11:10am12:00pm
We give a brief introduction to RiemannHilbert problems (RHPs) and describe the ideas of the DeiftZhou steepest descent method applied to a particular \( 2\times2 \) matrix RHP. This particular RHP appears when one studies large gap asymptotics for the Bessel kernel determinant.