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Mathematics & Statistics

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On Mergelyan sets and Farrell sets

Arthur Danielyan

4:00pm–5:00pm

MS Teams

**Abstract**

The concepts of Mergelyan sets and Farrell sets have been introduced by Lee Rubel in 1973 for his known joint approximation problem in complex domain and studied by many authors. We present the relevant classical results and consider some questions on Mergelyan sets and Farrell sets for \(Hp\) (\(1 \le p < \infty\)) spaces of analytic functions in the unit disc for both the weak topology and the norm topology.

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Optimal Polynomial Approximants and Design of Digital Filters, Part II

Catherine Bénéteau

4:00pm–5:00pm

MS Teams

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Optimal Polynomial Approximants and Design of Digital Filters

Catherine Bénéteau

4:00pm–5:00pm

MS Teams

**Abstract**

In this talk, I will discuss the notion of optimal polynomial approximants, which are polynomials that approximate, in some sense to be discussed, inverses of functions in certain Hilbert spaces of analytic functions. In the last 10 years, a number of papers have appeared examining the zeros of these polynomials, rates of convergence, efficient algorithms for their computation, and connections to orthogonal polynomials and reproducing kernels, among other topics. On the other hand, in the ‘70s, researchers in engineering and applied mathematics introduced least squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space \(H^2\) the optimal polynomial approximants and the least squares inverses are identical. In this talk, I will survey known results and some open problems related to the zeros of optimal polynomial approximants and implications for the design of ideal digital filters. This talk is based on a survey paper that is joint with Ray Centner. Don't worry, I will start from the beginning.

No seminar this week.

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Polynomial interpolation on arbitrary varieties, Part II

Boris Shekhtman

4:00pm–5:00pm

MS Teams

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Polynomial interpolation on arbitrary varieties

Boris Shekhtman

4:00pm–5:00pm

MS Teams

**Abstract**

I will talk about the following problem: Let \(V_1,\dotsc,V_{n}\) be varieties in \(\mathbb{C}^{d}\) and let \(p_1,\dotsc,p_{n}\) be given polynomials of \(d\) variables. When can we find one polynomial \(p\) such that \(p=p_{j}\) on each variety \(V_{j}\)? This, to the best of my knowledge, is the first extension of classical interpolation problem when \(V_{j}\) are chosen to be points. All results, in my opinion, are cute and proofs are very simple.

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Quadratic differentials and associated extremal problems, Part III

E. Rakhmanov

4:00pm–5:00pm

MS Teams

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Quadratic differentials and associated extremal problems, Part II

E. Rakhmanov

4:00pm–5:00pm

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Quadratic differentials and associated extremal problems

E. Rakhmanov

4:00pm–5:00pm

MS Teams

**Abstract**

It is well known that each rational guadratic differential on the sphere without recurrent trajectories solves a number of extremal problems which may be formulated in different terms: electrostatic, metric or other.

I will discuss a few examples having in mind to outline some kind of classification for problems associated with a closed QD. It is known that it is not possible. (Too many problems as the last talks confirmed.) Anyway ...

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Finding solutions to a free boundary problem involving surface tension, Part III

Nathan Hayford

4:00pm–5:00pm

MS Teams

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Finding solutions to a free boundary problem involving surface tension, Part II

Nathan Hayford

4:00pm–5:00pm

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Finding solutions to a free boundary problem involving surface tension

Fudong Wang

4:00pm–5:00pm

MS Teams

**Abstract**

We will discuss how to construct explicit solutions to a free boundary problem considered by E. B. McLeod (1955), and later by P. R. Garabedian (1965). A one-parameter family of droplets to a free boundary problem with prescribed poles will be investigated following the scheme introduced in the paper by Khavinson-Solynin-Vassilev (2005). This is joint work with N. Hayford.

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Why \((2,3)\)-cusp is impossible in Hele-Shaw droplets

Seung-Yeop Lee

4:00pm–5:00pm

MS Teams

**Abstract**

Sakai classified all the possible cusp singularities that can happen in Hele-Shaw droplets. We will go over the proof.