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 DMS2426785 and DMS2331073. It will take place on IU Bloomington campus. The talks will be held at Swain Hall East 105 with coffee breaks set up at Rawles Hall lounge.
If you are planning to drive to campus, parking will be free on any campus lot during the weekend. The closest ones are the Atwater parking garage, Henderson parking garage, and the big outdoor lots between Atwater St. and Third St. (between Fess St. and Woodlawn).
We should be able to cover local traveling and lodging expenses for all the participants.
Accommodation of speakers is arranged by the organizers. All other participants are expected to book their hotel themselves. Nearby hotels include Courtyard by Marriott (0.7 miles from workshop), Hyatt Place (0.8 miles from workshop), Hilton Garden Inn (0.9 miles from workshop), Travelodge by Wyndham (1.3 miles from workshop), Hampton Inn (1.7 miles from workshop), Comfort Inn (2.2 miles from workshop), Holiday Inn (2.3 miles from workshop). There will be a dinner on Saturday night at Samira Restaurant. The participants will need to cover their own bill , which will be reimbursed later together with travel and lodging to those seeking financial support.
Anyone interested in participating should complete the registration form below. All attendees are encouraged to present a poster.
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.
3:304:00pm
4:005:00pm
Joint work with Greg Knese, James Pascoe, and Alan Sola
Onevariable rational functions \( q/p \) that are bounded on a domain \( U \) are easy to describe; after cancelling common factors, \( p \) cannot have zeros on the closure of \( U \). In contrast, even on nice twovariable domains like the biupper halfplane \( \mathbb H^2 \) or the unit bidisk \( \mathbb D^2 \), the multivariate situation is surprisingly complicated. The denominator \( p \) of a bounded rational function must still be stable (i.e., have no zeros on the domain), but it can now have boundary zeros. This leads to a host of questions such as:
9:009:40am
Chad Berner; Abdullah Al Helal; Kenta Miyahara; Ljupcho Petrov; Andres QuinteroSantander
09:4010:20am
We will discuss tools for studying the Bergman kernel and projection, a fundamental singular integral operator in complex analysis, on generalized non-smooth domains in \( \mathbb C^2 \) and \( \mathbb C^3 \). To obtain the weak-type regularity and a sharp range of \( L^p \) boundedness for the Bergman projection, we use proper holomorphic mappings and apply Schur's test using asymptotic results on the polydisk. In particular, we show that in our non-smooth setting, the Bergman projection satisfies a weak-type estimate at the upper endpoint of \( L^p \) boundedness but not at the lower endpoint.
10:3011:10am
We will discuss the minimization of Riesz energies with external fields $$ I_{s,V}(\mu) = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \Big( \frac{1}{s}\| x- y \|^{-s} + V(x) + V(y) \Big) d \mu(x) d\mu(y). $$ We are interested in how the choices of \( s \) (the strength of repulsion between electrons
) and \( V \) (the external field) affect the structure of the equilibrium measure, particularly the dimension of its support. We will focus on radially symmetric external fields of the form \( V(x) = \gamma \|x\|^{\alpha} \) for \( \alpha, \gamma > 0 \) (these act as an attractive sink at the origin), and will classify exactly when the support is the uniform measure on a sphere.
11:20am12:00pm
The hot spots conjecture of Rauch states that a second Neumann eigenfunction of the Laplacian on a simply connected domain in Euclidean space has no interior extrema. In the past year or so, several researchers have considered a corresponding question about the first Zaremba (i.e., mixed DirichletNeumann) eigenfunction. We will present a new theorem showing that on convex domains with connected and sufficiently small Dirichlet region, the first mixed eigenfunction indeed has no interior critical points.
12:002:00pm
2:002:40pm
Joint work with Richard S. Laugesen
The Riesz \( p \)capacity of a compact set in Euclidean space is defined in terms of an energy optimization problem with pairwise interaction kernel \( |x-y|^p \). In this talk, I will present properties of capacity as a function of \( p \), namely that capacity is leftcontinuous with respect to \( p \) and is right-continuous for sets satisfying an additional hypothesis. Moreover, diameter and volume are recovered in the endpoint limits.
2:503:30pm
Joint work with B. Berndtsson and Y. Rubinstein
By applying techniques from convex geometry, we establish sharp bounds on the Bergman kernels of tube domains.
In 2012, Nazarov discovered that the Mahler volume of a convex body is bounded from below by the Bergman kernel of the tube domain over the body. He suggested that finding an optimal lower bound on such Bergman kernels could lead to a resolution of the celebrated Mahler's conjecture (1930's). In 2014, Błocki conjectured that this optimal bound is obtained by a cube. We prove Błocki's conjecture in dimension \( n=2 \) using techniques inspired by shadow systems in convex geometry. We also explain why Nazarov's and Błocki's approach is a complex approach to an \( L^1 \)Mahler conjecture
rather than the original Mahler conjecture, and we describe how this gap might be bridged. Lastly, we use symmetrization techniques to obtain sharp upper bounds in all dimensions, establishing Santalòtype inequalities for Bergman kernels.
3:304:00pm
4:004:40pm
This talk is an exposition of certain important techniques in discrete harmonic analysis and is aimed at an audience that is familiar with the material of an average, introductory, graduate harmonic analysis course. We will introduce sparse domination and its significance for maximal inequalities, as well as the circle method and HiLow decomposition for obtaining \( \ell^p \)bounds for operators. The techniques are going to be examined in the context of discrete averaging operators, and we will be referencing joint works with Michael Lacey, Hamed Mousavi and Yaghoub Rahimi.
4:505:30pm
Given a random unitary \( n\times n \) matrix and an arc \( [0, \theta] \), \( 0<\theta< 2 \pi \), on the unit circle, we consider an eigenvalue counting function \( \mathcal{N}_{\theta} :=\#\{j:0<\theta_j<\theta\} \) and explore some of the consequences of the determinantal structure of the eigenvalue processes for \( \mathcal{N}_{\theta} \). Specifically, we study the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process and relate the question to the following energy problem on the unit circle, which is of independent interest. Namely, for given \( \theta \) and \( q \), \( 0 < q < 1 \), we determine the function $$J(q) =\inf \{I(\mu): \mu \in \mathcal{P}(S^{1}), \mu(A_{\theta}) = q\},$$ where \( I(\mu):= \iint \log\frac{1}{|z - \zeta|} d\mu(z) d\mu(\zeta) \) is the logarithmic energy of a probability measure \( \mu \) supported on the unit circle and \( A_{\theta} \) is the arc from \( e^{-i \theta/2} \) to \( e^{i \theta/2} \).
9:0009:40am
See above
09:4010:20am
Given a psh function \( u \) in the Cegrell class and a smooth, nonnegative function \( g \), it is know that one can always solve the MongeAmpere equation \( MA(u_g) = g^n MA(u) \), with some form of Dirichlet boundary values, by work of AhagCegrellCzyzPham. Left unsaid in their work is how \( u_g \) compares with \( u \) near the polar set \( {u = -\infty} \). We present a simple condition on \( u \) which allows us to show that \( u_g \) behaves (to leading order) like \( gu \) away from the boundary of \( { g > 0} \). Our results also apply to complex Hessian equations.
10:3011:10am
Joint work with Robert Fraser and Kyle Hambrook
Suppose that \( \mu \) is a Borel probability measure on \( \mathbb{R}^d \) such that \( \mu(B(x,r)) \lesssim r^{a} \) for all \( x \in \mathbb{R}^d \) and all \( r > 0 \) and \( |\widehat{\mu}(\xi)| \lesssim (1+|\xi|)^{-b/2} \) for all \( \xi \in \mathbb{R}^d \). The MockenhauptMitsisBakSeeger Fourier restriction theorem says that for each \( p \geq (4d-4a+2b)/b \), $$\|\widehat{f\mu}\|_{L^p(\mathbb{R}^d)} \lesssim_p \|f\|_{L^2(\mu)}$$ holds for all \( f \in L^2(\mu) \). We use a deterministic construction to prove the optimality of range of \( p \) in the MockenhauptMitsisBakSeeger Fourier restriction theorem for dimension \( d=1 \) and parameter range \( 0 < a,b \leq d \) and \( b\leq 2a \). Previous constructions by HambrookŁaba and Chen required randomness and only covered the range \( 0 < b \leq a \leq d=1 \).
11:10am12:00pm
Joint work with Alexei Poltoratski
This talk will be about applications of complex function theory to spectral problems for canonical systems, which constitute a broad class of second order differential equations. I will start with the basics of Kreinde Branges theory. Then I will present an explicit algorithm for inverse spectral problems developed by Makarov and Poltoratski for locallyfinite periodic spectral measures, as well as an extension of their work to certain classes of nonperiodic spectral measures. Finally, I will talk about some recent developments on direct spectral problems.