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**Title**

**Speaker**

**Time**

**Place**

Optimal Approximants, Non-Vanishing Approximation, and Universality, Part III

Catherine Bénéteau

4:00pm-5:00pm

CMC 130

**Title**

**Speaker**

**Time**

**Place**

Optimal Approximants, Non-Vanishing Approximation, and Universality, Part II

Catherine Bénéteau

4:00pm-5:00pm

CMC 130

**Title**

**Speaker**

**Time**

**Place**

Optimal Approximants, Non-Vanishing Approximation, and Universality

Catherine Bénéteau

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk, I will examine the universality properties of optimal polynomial approximants of inverses of functions in the Hardy space \(H^2\) of the disk. Along the way, I will prove a double approximation property for non-vanishing functions in \(H^2\). Namely, let \(E\) be a closed subset of the unit circle having measure 0. Then if \(g\) is any function in \(H^2\) that doesn't vanish in the open unit disk, and if \(f\) is any continuous function on \(E\), there exists a sequence of polynomials that do not vanish in the closed unit disk and that simultaneously approximate \(g\) in \(H^2\) and \(f\) in \(C(E)\). This talk is based on recent work with Oleg Ivrii, Myrto Manolaki, and Daniel Seco.

This week's talk is joint with the Colloquium.

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**Speaker**

**Time**

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Offbeat Approximation Problems in \(L^1\) metric, Part II

Dima Khavinson

4:00pm-5:00pm

CMC 130

No seminar this week due to Spring Break.

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Dynamic Programming Principles for Nonlinear Elliptic Equations and Related Topics

Juan Manfredi

University of Pittsburgh

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk, we will further explore the topics highlighted in the colloquium talk. In addition providing a more detailed discussion, we will consider generalizations and further topics.

No seminar this week due to the Graduate Open House.

**Title**

**Speaker**

**Time**

**Place**

Offbeat Approximation Problems in \(L^1\) metric

Dima Khavinson

4:00pm-5:00pm

CMC 130

**Abstract**

Suppose a continuous on the closed unit disk function \(\omega\), \(||\omega||_L^oo =1\),can be approximated by analytic functions in \(L^1\) \((dA)\)-norm within \(\epsilon\), here \(dA\) stands for the area measure.

**Question**. Can we approximate it (in \(L^1(dA)\)) within \(\operatorname{Const}(\epsilon)\) by analytic functions with the \(L^\infty\) norm at most 1? The answer is unknown. (The constant, of course, not depending on \(\omega\).) The problem is the “\(\epsilon\)” version of the celebrated Hoffman-Wermer Theorem for Uniform Algebras amd was posed by J. Wermer in 1980.

The answer of the analogous question in \(L^1(d\theta)\), where \(d\theta\) is the Lebesgue measure on the unit circle, is “No” (DK-H. Shapiro-F. Perez-Gonzalez, 1998), and “Yes”, if we replace \(O(\epsilon)\) by \(O((\epsilon)\log(1/\epsilon))\). Yet, the precise asymptotics are unknown in that case either.

No seminar this week.

**Title**

**Speaker**

**Time**

**Place**

Weight knot: what the Hecke algebras can teach us about ODE monodromies?, Part II

Razvan Teodorescu

4:00pm-5:00pm

CMC 130

**Title**

**Speaker**

**Time**

**Place**

Weight knot: what the Hecke algebras can teach us about ODE monodromies?

Razvan Teodorescu

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk, I will attempt to present the conclusion to the series of seminars given by N. Hayford in Fall 2018, on a duality problem for ODEs on Riemann surfaces.

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Analytic capacity and a conjecture of T. Richards

Vilmos Totik

4:00pm-5:00pm

CMC 130

**Abstract**

The talk will prove a conjecture of T. Richards on an extension of the Gauss-Lucas theorem about the critical points of polynomials. The proof uses some results of X. Tolsa on analytic capacities (also to be introduced).

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Planar orthogonal polynomials with multiple logarithmic singularities in the potential

Seung-Yeop Lee

4:00pm-5:00pm

CMC 130

**Abstract**

We consider the large degree asymptotic behavior of the planar orthogonal polynomials when the orthogonality measure is given by the exponent of the sum of logarithmic singularities whose locations can be arbitrary. The limiting locus of the roots of the orthogonal polynomials is given in terms of the locus of the roots for the single pole case, that has been known previously. The Riemann-Hilbert formulation makes possible both the numerical support and the proof of the fact. This is a joint work with Meng Yang (UC Louvain, Belgium).