# Mathematics Colloquium

## Limiting absorption principle and virtual levels of operators in Banach spaces

### When

November 12, 2021 | 3:30 - 4:30 p.m.

### Where

University Hall 1005

Andrew Comech

### Abstract

Virtual levels, also known as threshold resonances, admit several equivalent characterizations:

1. there are corresponding "virtual states" from a space "slightly weaker" than L^2;
2. there is no limiting absorption principle in their vicinity (e.g. no weights such that the "sandwiched" resolvent is uniformly bounded);
3. an arbitrarily small perturbation can produce an eigenvalue.

We develop a general approach to virtual levels in Banach spaces and provide applications to Schroedinger operators with nonselfadjoint potentials and in any dimension, deriving optimal estimates on the resolvent.

## Renormalization in Conformal Dynamics

### When

October 15, 2021 | 3:30 - 4:30 p.m.

### Where

University Hall 1005

### Speaker

Dr. Nikita Selinger

### Abstract

Renormalization is a technique that allows us to study a dynamical system on small scales by restricting an iterate of the original system to a small part of the original domain. Successful renormalization theories produce rigidity statements: the geometry of an infinitely renormalizable system is completely determined by its combinatorial properties. Studying dynamics of the renormalization operator itself yields understanding of the structure not only of individual dynamical systems but of the parameter space of the class of dynamical systems considered.

I will discuss various renormalization schemes in real and complex discrete dynamical systems and their applications.

## Siegel capture polynomials in parameter spaces

### When

April 19, 2019 | 2:30 - 3:30 p.m.

### Speaker

Dr. Lex Oversteegen (joint with Alexander Blokh, Arnaud Cheritat, Toulouse, Lex Oversteegen, and Vladlen Timorin, Moscow)

### Abstract

We consider the set of cubic polynomials $f$ with a marked fixed point. If $f$ has a Siegel disk at the marked fixed point, and if this disk contains an eventual image of a critical point, we call $f$ a \emph{IS-capture polynomial}. We study the location of IS-capture polynomials in the parameter space of all marked cubic polynomials modulo affine conjugacy. In particular, it is shown that any IS-capture is on the boundary of a unique bounded hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets).

## Slices of parameter space of cubic polynomials

### When

April 12, 2019 | 2:30 - 3:30 p.m.

### Speaker

Dr. Alexander Blokh (joint with Lex Oversteegen and Vladlen Timorin, Moscow)

### Abstract

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the \emph{main cubioid} in this parameter space. The \emph{main cubioid} is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials $z^2+c$ for $c$ in the filled main cardioid.

## A class of Schrodinger operators with convergent perturbation series

### When

April 5, 2019 | 2:30 - 3:30 p.m.

### Speaker

Dr. Ilya Kachkovskiy, Michigan State University

### Abstract

Rayleigh-Schrodinger perturbation series is one of the main tools of analyzing eigenvalues and eigenvectors of operators in quantum mechanics. The first part of the talk is expository and should be accessible to students with working knowledge of linear algebra: I will explain a way of representing all terms of the series in terms of graphs with certain structure (similar representations appear in physical literature in various forms). The second part of talk is based on joint work in progress with L. Parnovski and R. Shterenberg. We show that, for a class of lattice Schrodinger operators with unbounded quasiperiodic potentials, one can establish convergence of these series (which is surprising because the eigenvalues are not isolated). The proof is based on the careful analysis of the graphical structure of terms in order to identify cancellations between terms that contain small denominators. The result implies Anderson localization for a class of Maryland-type models on higher-dimensional lattices.

## The story of the little ell one norm and its friends

### When

March 29, 2019 | 2:30 - 3:30 p.m.

### Speaker

Dr. Carmeliza Navasca

### Abstract

The popularity of sparse ell one norm optimization problem was due to Emmanuel Candes and Terrence Tao via compressed sensing. I will start by introducing the little ell one norm and its minimization. Then, I will describe how and why these sparse optimization problems are useful in solving today’s challenging problems in data science and machine learning. Numerical examples in foreground and background separation in surveillance videos, matrix and tensor completion as well as deep neural network for image classification are included. In this talk, one can observe the interplay of (multi)linear algebra, optimization and numerical analysis with applications in computer science.

This is joint work with Xiaofei Wang (former postdoc at UAB, now Prof at Normal University, China), Ramin Karim Goudarzi, Fatou Sanogo, Ali Fry (former Fast-Track) and Da Yan (CS Prof at UAB).