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Mathematics & Statistics

(Leader: Prof. Dmitry Khavinson )

**Subject**

**Time**

**Place**

Open Problem Session

4:00pm-5:00pm

PHY 120

**Title**

**Speaker**

Some questions in two-dimensional potential theory

Razvan Teodorescu

**Abstract**

Let \(\Omega\subset\mathbb{C}\) be a domain of finite area \(A\), bounded by a piecewise continuous Jordan curve \(\Gamma=\partial\Omega\), and such that the *exterior* harmonic moments \(t_k=\int\limits_{\mathbb{C}/\Omega}x^{-k}dA(z)\), \(k\ge 1\), define a holomorphic function \(V(z)=\sum\limits_{k=1}^da_k\log\,\left(z-z_k\right)\), \(a_k>0\), which has the expansion \(\sum\limits_kt_kz^k=V(z)\) near the origin.

For \(N\in\mathbb{N}\), we consider the space \(\mathcal{M}_N\) of \(N\)-point atomic measures with support in \(\Omega\), of total mass \(1\), and the uniform measure on \(\Omega\), \(\mu_\Omega\), also normalized to \(1\). THe problems below are all related to approximations of \(\mu_\Omega\) through elements in \(\mathcal{M}_N\), which are optimal in some sense:

**The moment problem.**Denote by \(\{m_n\}_{n=1}^\infty\) the sequence of harmonic moments of \(\mu_\Omega\), \(m_n=\int\limits_\Omega z^nd\mu_\Omega\), and by \(\left\{m_n^{(N)}\right\}_{n=1}^\infty\) the moments of the measure \(\nu_N\in\mathcal{M}_N\). Which measure \(\nu_N\) minimizes the \(\ell_p\)-distance between the two sets of moments? For which values of \(p\ge 1\) can we prove the convergence as \(N\to\infty\)?**The entropy problem.**Which measure \(\nu_N\in\mathcal{M}_n\) minimizes the Kullback-Leibler divergence? (This is defined by the expression below.) $$ K\left(\mu_\Omega,\nu_N\right)\equiv\int_\Omega\int_\Omega\log\,|z-\zeta|\,d\mu_\Omega(z)d\nu_N(\zeta). $$**The “variational” problem.**Denote by \(\mathcal{A}(\Omega)\) the space of analytic functions on \(\Omega\), \(L_1\)-integrable with \(\mu_\Omega\). Then find: $$ \inf_{\nu_N\in\mathcal{M}_N}\sup_{f\in\mathcal{A}}\left|\int_\Omega f(z)d\mu_\Omega-\int_\Omega f(z)d\nu_N\right|. $$

For the first two problems, we have conjectures but no definitive proofs:

- If \(\Omega\) is a generalized quadrature domain, then let \(p_n^{(N)}(z)\) be the orthogonal polynomials with the measure \(d\Phi^{(N)}=e^{-N\left[|z|^2-\Re V(z)\right]}dA(z)\), and \(\xi_N\) the distribution of zeros of the polynomial \(p_N^{(N)}(z)\). Then \(\xi_N\) solves the moment problem in the \(\ell_2\) sense.
- The “equilibrium distribution” of eigenvalues of \(N\times N\) normal random matrices with measure \(d\Phi_N\), \(\rho_N(z)\), solves the entropy problem in the large \(N\)-limit.

For the third problem we don't really have a conjecture yet.

**Title**

**Speaker**

H. Bohr's theorem for polynomials

Dmitry Khavinson

**Abstract**

The abstract for this talk can be found here.

**Title**

**Speaker**

Solutions of a transcendental equation and gravitational lensing

Erik Lundberg

**Abstract**

As indicated by the most recent Nagle lecture and the most recent issue of the AMS *Notices*, gravitational lensing is a current topic of interest.

The following problems come directly from gravitational lensing. Solutions of the following equations represent images lensed by a galaxy. Solutions of the equations can be seen as (i) fixed points of an anti-analytic function, (ii) zeros of a harmonic map, (iii) equilibrium points of an electrostatic field from logarithmic potential theory (involving an extra external field).

**Previously Solved Problem:** Given an upper bound for the number of solutions to the following equation, where \(k\) is a real parameter and \(w\) is a complex parameter
\begin{equation}
\arcsin\left(\frac{k}{\bar{z}+\bar{w}}\right)=z,
\tag{1}
\end{equation}
where we take the principal branch of \(\arcsin\).

Solutions describe images lensed by an elliptical galaxy with a physically-motivated density.

**Open Problem (i):** Give an upper bound for the number of solutions to the following equation, where \(k\) is a real parameter and \(\gamma\) and \(w\) are complex parameters
\begin{equation}
\arcsin\left(\frac{k}{\bar{z}+\bar{w}}\right)+\gamma\bar{z}=z,
\tag{2}
\end{equation}
where we take the principal branch of \(\arcsin\).

With this linear perturbation, an empirical investigation suggests a sharp upper bound of \(8\), but for a rigorous proof, the two approaches ([Khavinson and Lundberg] and [Bergweiler and Eremenko]) used to study Eq. (1) seem to break down unless \(\gamma\) is real.

Solutions describe images lensed by a galaxy with an additional distant “tidal” force. \(\gamma\) real means the force is alligned with one of the axes of the elliptical galaxy.

**Open Problem (ii):** Are there choices of parameters for which the following equation has six solutions?
\begin{equation}
c\left(k\arcsin\left(\frac{k}{\bar{z}+\bar{w}}\right)-\bar{z}-\bar{w}+\sqrt{k^2-\left(\bar{z}-\bar{w}\right)^2}\right)=z.
\tag{3}
\end{equation}

One could also ask about the number of possible zeros, but just an example with six would be interesting to astronomers. This equation comes from adjusting the density of the galaxy modeled by Eq. (1) so that it does not have a “sharp edge” (so that the density does not vanish abruptly across its support). An affirmative answer would perhaps lead to reevaluation of the mainstream lensing models. If there is an example with six images then finding it should be feasible, but proving a bound for the number of solutions to Eq. (3) appears to be a much more difficult case than Eq. (1).

**Title**

**Speaker**

**Time**

**Place**

On Hermite-Padé approximants

E. A. Rakhmanov

4:00pm-5:00pm

PHY 120

**Abstract**

Some old and new results in the field will be discussed. In particular, we will mention fundamental connections with potential theory and theory of algebraic functions.

**Title**

**Speaker**

**Time**

**Place**

Quadrature domains: equivalent definitions, applications, and approximation

Erik Lundberg

4:00pm-5:00pm

PHY 120

**Abstract**

A domain is called a “quadrature domain” if it admits a quadrature formula expressing integration of an analytic function as a finite sum of weighted point evaluations of the function and its derivatives. In this talk, I will discuss equivalent definitions including such notions as the Cauchy transform of a domain, the Schwarz function of the boundary, and the conformal map from the disk. Then we will see how quadrature domains arise in applications, where it is often important to find an approximate quadrature formula for a given domain. I will mention some approaches for doing this.

**Title**

**Speaker**

**Time**

**Place**

Complete List of Exact Solutions for Laplacian Growth with Arbitrary Sources

Mark Mineev-Weinstein

Max Planck Institute

2:00pm-3:00pm

PHY 108

**Abstract**

We establish that an arbitrary interface in the Laplacian growth can be represented as results of continua of evolutions of the initial circles, which correspond to various distribution of sources presenting in the viscous domain. Also, we worked on the inverse problem of recovery of the singularities of the Schwarz function of the moving interface from an arbitrary distribution of sources.

**Title**

**Speaker**

**Time**

**Place**

An extremal problem for non-vanishing functions in the Bergman space

Catherine Bénéteau

4:00pm-5:00pm

PHY 120

**Abstract**

In this talk, I will discuss the problem of finding the minimum norm of a non-vanishing Bergman space function whose first two Taylor coefficients are specified. In 2005, Aharonov, Bénéteau, Khavinson, and Shapiro proved that the solution to this problem exists, is unique, and is bounded. In addition, the authors conjectured the exact form of the extremal solution, and proved this conjecture under the additional assumption that the singular part of the extremal function has only a point mass. In this talk, I will discuss these results and some recent progress on this problem due to T. Sheil-Small.

**Title**

**Speaker**

**Time**

**Place**

Quantum graphs, torsional rigidity and isospectrality

Patrick McDonald

New College of Florida

Sarasota, FL

4:00pm-5:00pm

PHY 120

**Abstract**

Originally developed by chemists as a model for studying the structure and reactivity of molecules, quantum graphs have become useful tools for studying a variety of physical and mathematical phenomena. The associated spectral theory is highly developed and includes a number of beautiful results, including exact trace formulae and inverse spectral theorems. Starting with the fundamental definitions, we will develop a collection of geometric invariants for quantum graphs and investigate their relationship to Dirichlet spectrum. In particular, we will construct quantum graph analogs of well known isospectral planar domains and show that our invariants distinguish the corresponding quantum graph isospectral pairs.

**Title**

**Speaker**

**Time**

**Place**

A Sub-Riemannian Maximum Principle and its application to the \(p\)-Laplacian in Carnot groups

Tom Bieske

4:00pm-5:00pm

PHY 120

**Abstract**

We prove a sub-Riemannian maximum principle for semicontinuous functions. We apply this principle to Carnot groups to provide a “sub-Riemannian” proof of the uniqueness of viscosity infinite harmonic functions and to establish the equivalence of weak solutions and viscosity solutions to the \(p\)-Laplace equation.

**Title**

**Speaker**

**Time**

**Place**

Ping pong balayage and convexity of equilibrium measures

David P. Benko

University of South Alabama

*and*

Peter D. Dragnev

Indiana University-Purdue University, Ft. Wayne

4:00pm-5:00pm

PHY 120

**Abstract**

In this work we prove that the equilibrium measure of a compact subset on the real line has essentially convex density, namely if the compact set contains an interval, its equilibrium measure is absolutely continuous and has a convex density. We will also investigate Riesz kernels. Other related results will be also presented. Applications to external field problems and constrained energy problems are given.

**Title**

**Speaker**

**Time**

**Place**

A seminorm problem for power series: solution and applications

Arcadii Grinshpan

4:00pm-5:00pm

PHY 120

**Abstract**

We will consider a seminorm product problem for power series and present its solution based on the recursive properties of binomial coefficients. This result will lead us to multiparameter generalizations of the classical Hoelder inequalities (both discrete and integral versions) and to further applications involving special functions, convolutions, integral transforms, univalent functions, fractional operators. Some examples and applications will be discussed in the talk. The proofs of the basic theorems are given in the article Weighted inequalities and negative binomials, Adv. in Appl. Math. 45(4) (2010), 564-606.

**Title**

**Speaker**

**Time**

**Place**

On non-existence of certain error formulas for multivariate interpolation

Boris Shekhtman

4:00pm-5:00pm

PHY 120

**Abstract**

I will show that certain well-known error (remainder) formulas for Hermite interpolation fail to extend to multivariate setting; thus solving a problem posed by Carl de Boor.

Time permitting, I will attempt (probably unsuccessfully) to explain the relevant proofs.

**Title**

**Speaker**

**Time**

**Place**

Minimal norms of polynomials, Part II

Vilmos Totik

4:00pm-5:00pm

PHY 120

**Title**

**Speaker**

**Time**

**Place**

Minimal norms of polynomials

Vilmos Totik

4:00pm-5:00pm

PHY 120

**Abstract**

Consider a compact set \(E\) on the complex plane and the associated monic polynomials of degree \(n\) with minimal supremum norms on \(E\) (Chebyshev polynomials). If their minimal norm is \(t_n\), then it is classical that \(\left\{t_n^{1/n}\right\}\) has a limit which equals the transfinite diameter \(\tau(E)\) of \(E\) (and also equals the logarithmic capacity of \(E\)). It is always the case that \(t_n\ge\tau(E)^n\). In this talk we are interested in how close one can get with \(t_n\) to the lower bound \(\tau(E)^n\), and some recent results concerning disconnected sets, sets consisting of finitely many intervals or finitely many Jordan curves will be mentioned. Another topic that we shall discuss is that if \(P_n(z)=z^n+\dotsb\) are monic polynomials which have \(k_n\) zeros on \(E\) or on some part of it, then how close its norm can be to the theoretical lower bound \(\tau(E)^n\) (say, for \(E\) the unit circle or some families of Jordan curves). The unit circle of this latter problem is coming from Turán's power-sum method in number theory. One of the results is the following: if \(P_n\) has \(n|J|/2\pi+k_n\) zeros on a subarc \(J\) of the unit circle \(E\), then \(\|P_n\|_E\ge c\,\exp\left(ck_n^2/n\right)\), and this estimate is exact.