# Research

## Classical Analysis

### Friday, March 5, 2021

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

#### Abstract

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

### Friday, February 19, 2021

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

### Friday, February 12, 2021

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

### Friday, February 5, 2021

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

### Friday, January 29, 2021

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

### Friday, January 22, 2021

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

### Friday, January 15, 2021

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

#### Abstract

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