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Mathematics & Statistics

(Leader: Prof. Dmitry Khavinson)

Due to the MAA Suncoast Regional Meeting, there will be no seminar this week.

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Bellman Function for the Dyadic Maximal Operator

Alex Stokolos

DePaul University

4:00pm-5:00pm

PHY 108

**Abstract**

We found the Bellman function for the dyadic maximal operator as the solution of a Bellman PDE of Monge-Ampere type. This function has been previously found by A. Melas in a different way, but it is our PDE-based approach that is of principal interest here. Clear and replicable, it holds promise as a unifying template for past and current Bellman function investigations. This is a joint work with L. Slavin and V. Vasunin.

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**Speaker**

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Looking for singularities of the solutions to the Dirichlet Problem, Part II

Dmitry Khavinson

4:00pm-5:00pm

PHY 108

**Title**

**Speaker**

**Time**

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Looking for singularities of the solutions to the Dirichlet Problem

Dmitry Khavinson

4:00pm-5:00pm

PHY 108

**Abstract**

The classical Dirichlet Problem (DP) reduces to matching a given data function on the boundary of a domain with a function harmonic in the whole domain. When the boundary is smooth the solution exists and unique. Yet, even for the best imaginable data functions (e.g., polynomials) the solutions mysteriously develop singularities outside the initial domain where the DP was initially posed.

Questions like: (i) for which domains it never happens, (ii) how is the appearance of singularities related to the geometry of the initial domain, (iii) is there a data that produces ALL possible singularities at once turn out to be excruciatingly difficult.

In this talk we shall discuss some recent developments in understanding this problem.

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Fixed points of conjugated Blaschke products and gravitational lensing

L. Kuznia

4:00pm-5:00pm

PHY 108

**Abstract**

In 2006, Khavinson and Neumann showed that for a rational function of degree \(n > 1\), \(\#\left\{z:r(z)=z^*\right\}\le 5n-5\), where \(^*\) denotes complex conjugation. This resolved a conjecture from astrophysics regarding the number of images created by a gravitational lens consisting of \(n\) point masses (this conjecture was put forth by Rhie in 2001). In this talk, we will examine the situation when \(r(z)\) is a Blaschke product. After obtaining a sharp bound for this case, we point out a corollary regarding proper anti-analytic maps, which has interesting ramifications in gravitational lensing.

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On simultaneous block diagonalization of cyclic sequences of commuting matrices, Part II

Tom McKinley

4:00pm-5:00pm

PHY 108

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**Time**

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On simultaneous block diagonalization of cyclic sequences of commuting matrices

Tom McKinley

4:00pm-5:00pm

PHY 108

**Abstract**

I would like to present our results pertaining to the simultaneous block-diagonalization of cyclic \(d\)-tuples of commuting matrices, i.e., \(d\) commuting cyclic matrices \(A_1,\dotsc,A_d\) in block-diagonal form. Also, their connection to the ideal projectors will be discussed as well. In particular, we extend Hans-Stetters theorem characterizing Lagrange projectors.

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Nested interpolation and ideal extensions

Boris Shekhtman

4:00pm-5:00pm

PHY 108

**Abstract**

I will present a counterexample to a conjecture of Tomas Sauer that every ideal that complements polynomials of degree n can be extended to an ideal that complements polynomials of degree \(n-1\). To do so I will need to explain Macaley duality (inverse systems). Time permitting, I will also outline some devastating consequences of this example to the theory of divided differences in approximation theory and dualizing module in algebraic geometry.