# Research

## Analysis

### Friday, April 27, 2018

Title: Boundary behavior of optimal approximants
Speaker: Daniel Seco, Consejo Superior de Investigaciones Científicas
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We compute the Taylor coefficients of $$p_nf-1$$, where $$p_n$$ denotes the optimal approximant of degree $$n$$ to $$1/f$$ in a Hilbert space of analytic functions over the unit disc $$D$$, and $$f$$ is a polynomial of degree $$d$$ with $$d$$ simple zeros. As an application, we show that the sequence $$p_nf-1$$ is uniformly bounded and, if $$f$$ has no zeros inside the disc, the values of $$p_nf-1$$ converge locally uniformly towards $$0$$ at every point of the boundary except the zeros of $$f$$.

### Friday, April 20, 2018

Title: Stahl's and GRS theorems on a closed Riemann Surface, II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, April 13, 2018

Title: Stahl's and GRS theorems on a closed Riemann Surface
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The theorems (mentioned above) on zero distribution of complex orthogonal polynomials remain valid (in more or less the same form) for orthogonal polynomials on a closed Riemann Surface $$R$$.

Polynomials on $$R$$ are meromorphic functions on $$R$$ whose poles are located at a single fixed point on $$R$$: such functions are, in particular, algebraic. To prove such generalization we have to discuss some basics of logarithmic potential theory on a Riemann surface.

### Friday, April 6, 2018

Title: The lemniscate tree of a random polynomial
Speaker: Erik Lundberg, Florida Atlantic University
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We investigate the landscape given by considering the modulus $$|p(z)|$$ of a complex polynomial. In the spirit of Arnold's program of enumerating algebraic Morse functions, F. Catanese and M. Paluszny classified (1991) the landscapes generated by complex polynomials, and they enumerated all possible topological equivalence classes (the equivalence being up to diffeomorphism of the domain and range) using a combinatorial scheme based on a certain class of labeled binary trees that we refer to as “lemniscate trees”. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the various singular level sets $$|p(z)|=t$$. The level sets are referred to as lemniscates, and a generic polynomial of degree $$n$$ has $$n-1$$ singular lemniscates (each passes through a critical point of $$p$$). In this talk, we investigate the global structure of the landscape by addressing the question “How many branches appear in a typical lemniscate tree?” We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class and second for the lemniscate tree arising from a random polynomial generated by i.i.d. zeros. This is joint work with Boris Hanin and Michael Epstein.

### Friday, March 30, 2018

Title: Sharp estimates for Hörmander-type operators with positive-definite phase
Speaker: Marina Iliopoulou, University of California, Berkeley
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In the heart of harmonic analysis lies the restriction problem: the study of Fourier transforms of functions that are defined on curved surfaces. The problem came to life in the late 60s, when Stein observed that such Fourier transforms have better behaviour than if the surfaces were flat. Soon after, Hörmander conjectured that oscillatory integral operators with more general phase functions should also demonstrate similar agreeable behaviour. Surprisingly, 20 years later Bourgain disproved Hörmander's conjecture. However, under additional assumptions on the phase function one can expect better estimates than the sharp ones by Bourgain. In this talk, we present such better estimates in the sharp range, under the assumption that the phase function is positive definite. This is joint work with Larry Guth and Jonathan Hickman, and builds on recent work of Guth that improved on the restriction problem via the polynomial method.

### Friday, March 23, 2018

Title: What is an “inner function”? Part II
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, March 16, 2018

Spring Break — no seminar this week.

### Friday, March 2, 2018

Title: What is an “inner function”?
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The concept of an inner function has been a focal point of function theoretic operator theory since the celebrated Beurling theorem characterizing invariant (with respect to the unilateral shift) subspaces in Hardy spaces in the unit disk.

In the 1990s, it was extended to Bergman spaces where however Beurling's theorem fails. Since the 1960s, many plausible ways of extending the notion of an inner function were pursued in the context of Hardy, Bergman and other spaces of analytic functions in a more general setting than the unit disk: multiply connected domains, Riemann surfaces and several variables. Yet, the “big picture” is far from clear.

Note: This talk is based on a joint research project with C. Bénéteau, M. Fleeman, C. Liaw and A. Sola.

### Friday, February 23, 2018

Title: Projective connections and extremal solutions for optimization problems in spaces of analytic functions III
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, February 16, 2018

Title: Projective connections and extremal solutions for optimization problems in spaces of analytic functions II
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, February 9, 2018

Title: Projective connections and extremal solutions for optimization problems in spaces of analytic functions
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

An unexpected outcome of the recent proof [1] of the longstanding conjecture that disks and annuli are the only domains where the uniform distance from $$\bar{z}$$ to analytic functions achieves its lower bound, is a new insight into projective connections and the classification of quadratic differentials. I will define projective connections and discuss their relation to problems of approximation with analytic functions, as well as the way in which they generalize such optimization problems. If time allows, we will also consider the implications of degenerate representations of conformal transformations associated with a given projective connection.

1. Ar. Abanov, C. Beneteau, D. Khavinson, and R. Teodorescu, “A free boundary problem associated with the isoperimetric inequality,” to appear in Journal d'Analyse, 2018.

### Friday, January 26, 2018

Title: Polynomial inequalities on Jordan curves
Speaker: Vilmos Totik
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We review some of the classical polynomial inequalities (due to Riesz, Bernstein and Markov), and give their sharp forms on Jordan curves and arcs. If time permits, the same problem for rational functions will also be mentioned.

### Friday, January 19, 2018

Title: Scaling limits in conformal random growth models: the singular case
Speaker: Alan Sola, Stockholm University
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

This talk will be about small-particle limits in conformal mapping version of Laplacian random growth. After a brief overview of the subject, I will discuss ongoing joint work with Turner and Viklund, where we focus on the case of highly localizing feedback.