# Mathematics Colloquium

## Evolution Flow and Ground States for Schrodinger Type Equation and Sytems

### When

April 27, 2022 | 11:00 a.m. - 12:00 p.m.

### Where

University Hall 4004

### Speaker

Vladimir Georgiev

### Abstract

The talk starts with an introduction to some basic models of Schrodinger type equations and their fractional generalizations. As an example the half - wave equation with nonlinear terms is considered.

We focus our study on the case of nonlinear half - wave equation. The results we present can be divided in the following main topics:

- existence, asymptotic properties of ground states and their linear stability/instability;
- existence or explosion phenomena of the evolution flow with large data below/above the ground state barrier for the corresponding Cauchy problem for the half – wave equation;
- blow - up for mass-critical half - wave equation.

## Deep Learning of Conjugate Mappings

### When

April 15, 2022 | 3:00 - 4:00 p.m.

### Where

University Hall 1005

### Speaker

Jason Bramburger

### Abstract

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding.

In this talk I present a method of discovering explicit Poincaré mappings using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. We illustrate with low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on the infinite-dimensional Kuramoto-Sivashinsky equation.

## UAB Mathematics Open House and Alumni Forum

**Featuring a presentation from:**

Dr. Stephen Odaibo

March 25, 2022

University Hall, 2005

1402 10th Avenue South

Birmingham AL 35294-1241

### Morning Session

9:15 - 9:30 a.m.: Welcome

9:30 - 11:10 a.m.: Faculty Presentations: research opportunities – Room 4004

11:15 a.m. - 12:30 p.m.: Lunch meeting with current graduate and fast track students – Room 4002

12:30 - 1:30 p.m.: Campus Tour

### Alumni Forum

1:30 - 2:30 p.m.: Poster session and table second ﬂoor lobby

2:30 - 3:30 p.m.: Talk by Dr. Stephen Odaibo – Room 2005

3:30 - 4 p.m.: Reception"

## Some recent results on regularity theory for linear parabolic equations with singular-degenerate coefficients

### When

March 4th, 2022 | 2:30 – 3:30 p.m.

### Where

University Hall 1005

### Speaker

Dr. Tuoc Phan

### Abstract

We discuss classes of linear parabolic equations in the upper-half space whose coefficients can be singular or degenerate as a power of the vertical variable. Two typical examples are the class of Caffarelli-Silvestre type extensional equations arising in the study of fractional Laplace equations or fractional heat equations, and a class of degenerate viscous Hamilton-Jacobi equations. The boundary conditions are either homogeneous Dirichlet or conormal one. Depending on the boundary conditions, generic weighted Sobolev spaces are found in which existence, uniqueness, and regularity estimates of solutions are proved. The main features in our results are that the coefficients may not be in the A_2 class of Muckenhoupt weights as previously studied in the available literature. The talk is based on several recent papers that are the joint work with Hongjie Dong (Brown University) and with Hung Vinh Tran (University of Wisconsin Madison).

## 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

### Speaker

Andrew Comech

### Abstract

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

- there are corresponding "virtual states" from a space "slightly weaker" than L^2;
- there is no limiting absorption principle in their vicinity (e.g. no weights such that the "sandwiched" resolvent is uniformly bounded);
- 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.