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Title: Polynomial interpolation on arbitrary varieties Speaker: Boris Shekhtman Time: 4:00pm–5:00pm Place: MS Teams

I will talk about the following problem: Let \(V_1,...,V_{n}\) be varieties in \(\mathbb{C}^{d}\) and let \(p_1,...,p_{n}\) 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.

Title: Quadratic differentials and associated extremal problems, Part III Speaker: E. Rakhmanov Time: 4:00pm–5:00pm Place: MS Teams

Title: Quadratic differentials and associated extremal problems, Part II Speaker: E. Rakhmanov Time: 4:00pm–5:00pm Place: MS Teams

Title: Quadratic differentials and associated extremal problems Speaker: E. Rakhmanov Time: 4:00pm–5:00pm Place: MS Teams

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 ...

Title: Finding solutions to a free boundary problem involving surface tension, Part III Speaker: Nathan Hayford Time: 4:00pm–5:00pm Place: MS Teams

Title: Finding solutions to a free boundary problem involving surface tension, Part II Speaker: Nathan Hayford Time: 4:00pm–5:00pm Place: MS Teams

Title: Finding solutions to a free boundary problem involving surface tension Speaker: Fudong Wang Time: 4:00pm–5:00pm Place: MS Teams

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.

Title: Why \((2,3)\)-cusp is impossible in Hele-Shaw droplets Speaker: Seung-Yeop Lee Time: 4:00pm–5:00pm Place: MS Teams

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