# Pulse replication and slow absolute spectrum in the FitzHugh-Nagumo system

Paul Carter

UC Irvine

## Time:

Tuesday, November 16, 2021 - 1:00pm to 2:00pm

## Location:

Zoom

Traveling waves arise in partial differential equations in a broad range of applications. The notion of stability of a traveling wave solution concerns its resilience to small perturbations and can frequently be inferred from an eigenvalue problem obtained by linearizing the PDE about the solution. I will discuss these ideas in the context of the FitzHugh--Nagumo system, a simplified model of nerve impulse propagation. I will present existence and stability results for (multi)pulse solutions, and I will describe a phenomenon whereby unstable eigenvalues accumulate as a single pulse is continuously deformed into a double pulse.

# Shadowing in dynamical systems

Grisha Monakov

UC Irvine

## Time:

Tuesday, November 23, 2021 - 1:00pm to 2:00pm

## Location:

Zoom

We say that a dynamical system satisfies shadowing property if for any pseudotrajectory there exists an exact trajectory that is pointwise close to it. This property was introduced by Anosov in 1970th and plays an important role in the theory of dynamical systems. Shadowing property is known to have strong connections with hyperbolicity and structural stability. In this talk I will give an overview of classical results in shadowing theory and will present a new proof of Anosov shadowing lemma.

# Spectral estimates of dynamically-defined and amenable operator families

Alberto Takase

UC Irvine

## Time:

Tuesday, October 26, 2021 - 1:00pm to 2:00pm

## Location:

Zoom

Suppose that at each vertex of the Cayley graph of a finitely generated group G is a person holding a dollar. Everybody is told to pass their dollar bill to a neighbor. This can be done so that each person’s net worth increases if and only if the group G is non-amenable. Thus, one can think of non-amenable groups as those where Ponzi schemes can benefit everyone. The Cayley graph of the free group with two generators is an infinite 4-valent tree. If everyone passes their dollar towards the origin then everyone’s net worth increases! Because we live in a world where Ponzi schemes don't work, we restrict our attention to amenable groups such as the integer lattice. For dynamically-defined operator families, the Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the group is amenable. We prove that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $\frac{1}{2}$-Hölder continuous.

# A brief introduction to complex dynamics

Anton Gorodetski

UC Irvine

## Time:

Tuesday, October 5, 2021 - 1:00pm

## Location:

Zoom

We will go over the basic notions and objects in complex dynamics (normal families, Fatou and Julia sets, Mandelbrot set) and consider some examples. The talk is a precursor to the talk by Michael Yampolsky (Toronto University) that is scheduled on October 19.

# Chaotic dynamics meets computer science: a study of computability of Julia sets

## Speaker:

Michael Yampolsky

## Institution:

Toronto University

## Time:

Tuesday, October 19, 2021 - 1:00pm

## Location:

Zoom

Numerical simulation has played a key role in the study of dynamical systems, from modeling ecosystems to weather simulations. Archetypical examples of complex fractals generated by simple non-linear dynamical systems are Julia sets of quadratic polynomials. Computer-generated Julia sets are among the most familiar mathematical images, enjoyed both for their beauty and for the deep theory behind them. In a series of works with M. Braverman and others we have put to the test the modern paradigm of numerical simulation of chaotic dynamics, and asked whether images of Julia sets can always be computed if the parameters are known. My talk will describe some of the surprising results we have obtained, and several intriguing open problems.

# Hausdorff Dimension of the spectrum of the Thue-Morse Hamiltonian

William Wood

UC Irvine

## Time:

Tuesday, November 19, 2019 - 1:00pm to 2:00pm

## Location:

RH 440R

We will discuss a dynamical approach by Qinghui Liu and Yanhui Qu to study the spectrum of the Thue-Morse Hamiltonian, a Schrodinger operator on l^2. They use the trace map formalism to describe the spectrum, and study the corresponding polynomials to characterize subsets of their zeros that converge to a Cantor set. In doing this they establish bounds of the Hausdorff dimension. We will also discuss a possiblility to obtain similar bounds on the thickness of the spectrum.

# Renormalization: from holomorphic dynamics to two dimensions

Denis Gaidashev

## Institution:

Uppsala University

## Time:

Tuesday, November 12, 2019 - 1:00pm to 2:00pm

## Location:

RH 440R

I will describe recent advances in promoting renormalization techniques from one dimensional holomirphic setting to two dimensions, together with the set of questions that these techniques are aiming to address, such as rigidity and geometry of attractors.

# Cantor Sets, their Sums, and the Lagrange and Markov Spectra

William Wood

UC Irvine

## Time:

Tuesday, October 29, 2019 - 1:00pm to 2:00pm

## Location:

RH 440R

Cantor sets and their fractional dimensions are related to many unsolved conjectures. One of these conjectures, known as Palis' Conjecture, involves the dimension of the Minkowski sums of Cantor sets, and in this talk we will describe the details of the Palis Conjecture. In addition, we will use some of the tools involving Cantor sets to describe of the some properties of the Lagrange and Markov spectra, in particular, related to existence of the Hall's ray. The talk will be mostly based on the work done by Moreira in recent years.

# Ergodic Schrodinger operators in the infinite-measure setting

Jake Fillman

## Institution:

Texas State University

## Time:

Thursday, October 24, 2019 - 2:00pm to 3:00pm

## Location:

RH 340P

We develop the basic spectral theory of ergodic Schrodinger operators when the underlying dynamics are given by a conservative ergodic transformation of a \sigma-finite measure space. Some fundamental results, such as the Ishii--Pastur theorem carry over to the infinite-measure setting. We also discuss some examples in which straightforward analogs of results from the probability-measure case do not hold. We will discuss some examples and some interesting open problems.

The talk is based on a joint work with M. Boshernitzan, D. Damanik, and M. Lukic.

# Nonstationary low-dimensional dynamics

Victor Kleptsyn

IRM de Rennes

## Time:

Thursday, September 26, 2019 - 2:00pm to 3:00pm

## Location:

RH 340P

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.

An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products,
which is also of independent interest.

This is a joint project with A.Gorodetski.