Midwestern Workshop on Asymptotic Analysis

3:30—4:00pm

Coffee

4:00—5:00pm

Multivariate Bounded Rational Functions

Joint work with Greg Knese, James Pascoe, and Alan Sola

One–variable 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 two–variable domains like the bi–upper half–plane \( \mathbb H^2 \) 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:

1. Given a polynomial \( p \), which numerators \( q \) make \( q/p \) bounded on the domain (or at least near a given boundary zero of \( p \))?
2. What types of behaviors do bounded rational functions exhibit near boundary singularities?

[slides]

9:00—9:40am

Poster Display and Coffee

Chad Berner: Operator Orbit Frames and Frame–like Expansions; Abdullah Al Helal: Rational \(m-\)fold Sphere Maps; Tomas Lasic Latimer: The Riemann–Hilbert Problem and its Applications to Differential and Difference Equations; Kenta Miyahara: Asymptotics of Real Solutions of \( \sinh \)-Gordon Equation; Andres Quintero–Santander: On Stationary Measures for a Family Generated by Hénon–type Map

09:40—10:20am

Weighted Bergman Kernels, Weak–Type Estimates, and Schur's Test on Non-Smooth Domains in \( \mathbb C^n \)

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.

[slides]

10:30—11:10am

Riesz Energy with an External Field: Dimensionality of Minimizers

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.

[slides]

11:20am—12:00pm

A Hot Spots Theorem for the Zaremba Eigenvalue Problem with Small Dirichlet Region

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 Dirichlet–Neumann) 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.

12:00—2:00pm

Lunch Break

2:00—2:40pm

Complex Analytic Approach to Spectral Problems for Differential Operators

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 Krein–de Branges theory. Then I will present an explicit algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally–finite periodic spectral measures, as well as an extension of their work to certain classes of non–periodic spectral measures. Finally, I will talk about some recent developments on direct spectral problems.

2:50—3:30pm

A Mahler–type Inequality for Bergman Kernels

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 Santalò–type inequalities for Bergman kernels.

[slides]

3:30—4:00pm

Coffee Break

4:00—4:40pm

A Brief Introduction to the Circle Method and Sparse Domination

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 Hi–Low 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.

[slides]

4:50—5:30pm

On a Problem of E. Meckes for the Unitary Eigenvalue Process on an Arc

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 \( e^{i \theta/2} \).

[slides]

9:00—09:40am

Poster Display and Coffee

Chad Berner: Operator Orbit Frames and Frame–like Expansions; Abdullah Al Helal: Rational \(m-\)fold Sphere Maps; Tomas Lasic Latimer: The Riemann–Hilbert Problem and its Applications to Differential and Difference Equations; Kenta Miyahara: Asymptotics of Real Solutions of \( \sinh \)-Gordon Equation; Andres Quintero–Santander: On Stationary Measures for a Family Generated by Hénon–type Map

09:40—10:20am

Asymptotics for some Singular Monge–Ampere Equations

Given a psh function \( u \) in the Cegrell class and a smooth, non–negative function \( g \), it is know that one can always solve the Monge–Ampere equation \( MA(u_g) = g^n MA(u) \), with some form of Dirichlet boundary values, by work of Ahag–Cegrell–Czyz–Pham. 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.

10:30—11:10am

Fourier Restriction and Well–Approximable Numbers

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 Mockenhaupt–Mitsis–Bak–Seeger 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 Mockenhaupt–Mitsis–Bak–Seeger Fourier restriction theorem for dimension \( d=1 \) and parameter range \( 0 < a,b \leq d \) and \( b\leq 2a \). Previous constructions by Hambrook–Łaba and Chen required randomness and only covered the range \( 0 < b \leq a \leq d=1 \).

[slides]