# Research

## Classical Analysis

### Friday, March 13, 2020

Title: A Discussion of Dynamic Mode Decompositions and Reproducing Kernel Hilbert Spaces, Part II
Speaker: Joel Rosenfeldn
Time: 4:00pm–5:00pm
Place: CMC 130

This week's seminar is CANCELLED.

### Friday, March 6, 2020

Title: A Discussion of Dynamic Mode Decompositions and Reproducing Kernel Hilbert Spaces
Speaker: Joel Rosenfeldn
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We will talk about the dynamic mode decomposition (DMD) method for extracting fundamental governing principles from dynamic (time-series) data. The DMD approach provides a model free method for system identification through finite rank representations of various operators. We will discuss the connections of DMD with Koopman (or composition) operators for discrete time systems as well as with Liouville operators for continuous time systems. Each of these operators will be posed over reproducing kernel Hilbert spaces, where we will leverage identities using kernel functions and newly introduced occupation kernels.

### Friday, February 28, 2020

Title: Density and uniform distribution theorems under periodic perturbation
Speaker: Vilmos Totik
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In connection with asymptotics of orthogonal polynomials Peter Yuditskii asked if the sequence $$na + \sin(nb+c)$$ is dense in $$[0,1] \!\!\mod 1$$, where $$a$$ is an irrational number, $$b = a\pi$$ and $$c$$ is a fixed number. This talk will give an overview of Weyl's equidistribution theory, and then discuss the $$\!\!\mod 1$$ density and uniform distribution of sequences of the form $$na+f(n)$$ (or $$P(n)+f(Q(n))$$) where $$f$$ is a continuous periodic (or almost periodic) function (and $$P,Q$$ are polynomials). The vector-valued case will also be considered. It turns out that density is always true (under natural assumptions, say that $$a$$ is irrational) irrespective of $$f$$, but for uniform distribution one needs certainly rational independence among the parameters.

### Friday, February 21, 2020

Title: The density of shifted Eisenstein polynomials
Speaker: Giacomo Micheli
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In this talk we first explain how to compute the density of a subset of the integers using the data arising from the measures of its $$p$$-adic closures for all $$p$$'s, exploiting a local-to-global principle by B. Poonen and M. Stoll. Let now $$Z$$ be the ring of rational integers. A polynomial $$f$$ in $$Z[x]$$ is said to be Eisenstein if there exists a prime $$p$$ such that $$p$$ does not divide the leading coefficient of $$f$$, $$p$$ divides all the other coefficients of $$f$$, but $$p^2$$ does not divide the constant term of f. A standard fact is that an Eisenstein polynomial is irreducible. Testing irreducibility in this way is very fast compared with other methods, therefore it makes sense to ask the following: how likely is that, if we fix a polynomial $$f$$ “at random”, $$f(x+i)$$ will be Eisenstein for some prime $$p$$ and some integer $$i$$?

We provide an exact answer for this question, solving a problem posed by R. Heyman and I. Shparlinski.

The talk is meant to be self contained: in particular, no prior knowledge of $$p$$-adic numbers is required.

### Friday, February 7, 2020

Title: Equilibrium problems on a Riemann surface, Part II
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, January 31, 2020

No seminar this week due to Analysis Workshop in Orlando.

### Friday, January 24, 2020

Title: Equilibrium problems on a Riemann surface
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Equilibrium problems for logarithmic potential in plane (on the sphere) constitute one of the central fields of classical complex analysis with a large circle of applications.

Extension to a closed Riemann surface is rather straightforward in sense of definitions and statements of basic facts. A difficult problem is a description of associated geometry. We will discuss a few particular situations.