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Mathematics & Statistics

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On the Maximal Relative Projection Constants

Lesɫaw Skrzypek

4:00pm-5:00pm

CMC 130

**Abstract**

We will investigate the maximal \(m\)-dimensional spaces with regard to the relative projection constant and demonstrate how this notion translates to the problem of maximizing the sum of the m largest eigenvalues. Surprisingly (or not) this problem exhibits interesting connections to other fields like graph theory (Seidel matrices, Taylor graphs and the properties of their adjacency matrix) and equiangular tight frames. Based on our approach we will reprove the famous Kadec–Snobar inequality and its refinements. We will also apply Random Matrix techniques (semicircle law) to partially reverse the Kadec–Snobar inequality.

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Spectral properties of single layer operators

Seyed Zoalroshd

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk, we discuss the spectral properties of single layer operators. We give some criteria for Schatten class membership and also also injectivity of layer potentials. Regarding eigenfunctions, we show that polynomial eigenfunctions of layer potentials can, to some extent, determine the underlying curve. If time permits, some open problems will be discussed.

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Approximating \(z\)-bar in the Bergman Space

Matt Fleeman

4:00pm-5:00pm

CMC 130

**Abstract**

We consider the problem of approximating \(z\)-bar in the Bergman Space. We characterize the best approximation in terms of the solution of the Dirichlet problem with data \(|z|^2\) and give some examples. Finally we obtain an “isoperimetric sandwich” for the distance from \(z\)-bar to the Bergman space which yields the celebrated St. Venant inequality.

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Valence of harmonic polynomials and lemniscates

Seung-Yeop Lee

4:00pm-5:00pm

CMC 130

**Abstract**

The maximal valence of harmonic polynomials is an open problem. In a (failed) attempt to solve the problem, we consider the critical lemniscates of the harmonic polynomials. It is an elementary fact (by Sheil-Small) that a convex component of the lemniscate can have only one root of the harmonic polynomial. We explain the necessary and sufficient condition that a lemniscate can contain more than one root. This is a work with Khavinson and Saez.

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Polar Coordinates in spaces of Controlled Geometry

Tom Bieske

4:00pm-5:00pm

CMC 130

**Abstract**

We examine when polar coordinates can be constructed for various geometric versions of \(R^n\). We explore the consequences of having these polar coordinates and discuss the open problem of finding the largest class of spaces where polar coordinates can be constructed.

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Fatou's interpolation theorem implies the Rudin-Carleson theorem

Arthur Danielyan

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk it will be shown that the classical Rudin-Carleson interpolation theorem is a direct corollary of Fatou's much older interpolation theorem (of 1906). Some related questions will also be considered.

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Ergodic decomposition of group actions on rooted trees, Part II

Dima Savchuk

4:00pm-5:00pm

CMC 130

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Ergodic decomposition of group actions on rooted trees

Dima Savchuk

4:00pm-5:00pm

CMC 130

**Abstract**

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the case of groups generated by finite automata. Few examples, including the lamplighter group, will be considered in order to demonstrate applications of the theorem.

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Questions related to ideal interpolation, Part II

Boris Shekhtman

4:00pm-5:00pm

CMC 130

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Questions related to ideal interpolation

Boris Shekhtman

4:00pm-5:00pm

CMC 130

**Abstract**

In an effort to solicit comments from the audience, I will start with Jenya’s remark that the Newton series converges to the Taylor series, and discuss its generalization to polynomials of several variables. In particular I will try to outline some open and solved problems related to this subject and their connections to irreducibility of certain affine varieties, numerical approximation of differential operators, properties of sequences of commuting matrices and resolution of singularities.

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Stahl's theorem on convergence of Padé Approximants and beyond, Part II

E. A. Rakhmanov

4:00pm-5:00pm

CMC 130

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Stahl's theorem on convergence of Padé Approximants and beyond

E. A. Rakhmanov

4:00pm-5:00pm

CMC 130

**Abstract**

To state the theorem one needs to start with a discussion of several key concepts of complex analysis and approximation theory, such as Pade Approximations, analytic (multivalued) functions with “small” set of branch points, minimal capacity sets, quadratic differential, etc.

Generalizations to, say, Hermite-Padé Approximants are often discussed, but nothing really general is known. It would require going deep into Riemann surfaces and/or equilibrium problems and/or boundary value problems to formulate even a conjecture on poles distribution in rather simple situations.

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Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants, Part II

Alan Sola

4:00pm-5:00pm

CMC 130

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Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

Alan Sola

4:00pm-5:00pm

CMC 130

**Abstract**

In joint work with Bénéteau, Khavinson, Liaw, and Seco, we consider certain polynomial subspaces of Hilbert function spaces over the disk generated by a fixed function, and study the properties of certain families of polynomials associated with such subspaces. The location of zeros, and how it depends on both the given function and the Hilbert space, is the main focus of our work.