## Seminars and Colloquia by Series

### Approximation of invariant manifolds for Parabolic PDEs over irregular domains

Series
CDSNS Colloquium
Time
Friday, May 13, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jorge GonzalezGeorgia Tech

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.

### An army of one: stable solitary states in the second-order Kuramoto model

Series
CDSNS Colloquium
Time
Friday, May 6, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
Igor BelykhGeorgia State University

Symmetries are  fundamental concepts in modern physics and biology. Spontaneous symmetry breaking often leads to fascinating  dynamical patterns such as  chimera states in which structurally and dynamically identical oscillators  split into coherent and incoherent clusters.  Solitary states in which one oscillator separates from the coherent cluster and oscillates with a different frequency represent  “weak” chimeras. While a rigorous stability analysis of a “strong” chimera with a multi-oscillator incoherent cluster  is typically out of reach for finite-size networks, solitary states offer a unique test bed for the development of stability approaches to large chimeras. In this talk, we will present such an approach and study the stability of solitary states in Kuramoto networks of identical 2D phase oscillators with inertia and a phase-lagged coupling.   We will derive asymptotic stability conditions for such solitary states as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our analysis for the emergence of rotatory chimeras and splay states. This is a joint work with V. Munyaev, M. Bolotov, L. Smirnov, and G. Osipov.

### Back to boundaries in billiards

Series
CDSNS Colloquium
Time
Friday, April 29, 2022 - 13:00 for
Location
Speaker
Yaofeng SuSoM, GT

Abstract: This talk has 4 or 5 parts

1. I will start with a physical toy model to introduce billiards/open billiards, which describe the dynamics of a particle in a compact manifold/in a particular open area of this manifold.

2. One of the main questions of open billiards is Poisson approximations. It describes the asymptotic behavior of the dynamics in statistical distributions.  I will define it for billiards systems.

3. The main result is that such approximations hold for a billiard system that has arbitrarily slow chaos.

4. I will sketch the idea of the proof.

5. If time permits, I will talk about the connection between this work and riemann hypothesis.

This is a joint work with Prof. Leonid Bunimovich.

### Quasi periodic motions of the generalized SQG equations

Series
CDSNS Colloquium
Time
Friday, April 22, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
Jaemin ParkUniversity of Barcelona

In this talk, we study the existence of quasi periodic solutions to the generalized Surface Quasi-Geostropic (gSQG) equations. Despite its similar structure with the 2D Euler equation, the global existence/finite time singularity formation of gSQG equations have been open for a long time. Exploiting its Hamiltonian structure, we are able to construct a quasi periodic solutions with the initial date that are sufficiently close to its steady states. This is a joint work with Javier Gomez-Serrano and Alex Ionescu.

### On local rigidity of linear abelian actions on the torus

Series
CDSNS Colloquium
Time
Friday, April 15, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Remote via Zoom
Speaker

In which cases and ways can one perturb the action on the torus of a commuting pair of $SL(n, \mathbb Z)$ matrices?

Two famous manifestations of local rigidity in this context are: 1) KAM-rigidity of simultaneously Diophantine torus translations (Moser) and 2) smooth rigidity of hyperbolic or partially hyperbolic higher rank actions (Damjanovic and Katok). To complete the study of local rigidity of affine $\mathbb Z^k$ actions on the torus one needs to address the case of actions with parabolic generators. In this talk, I will review the two different mechanisms behind the rigidity phenomena in 1) and 2) above, and show how blending them with parabolic cohomological stability and polynomial growth allows to address the rigidity problem in the parabolic case.

This is joint work with Danijela Damjanovic and Maria Saprykina.

### Stiffness and rigidity in random dynamics

Series
CDSNS Colloquium
Time
Friday, April 8, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Aaron BrownNorthwestern University

Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M.  Define the random walk on M by selecting either f or g (i.i.d.) at each iterate.  A number of questions arise in this setting:

1. What are the closed subsets of M invariant under both f and g?
2. What are the stationary measures on M for the random walk.  In particular, are the stationary measures invariant under f and g?

Conjecturally, for a generic pair of f and g we should be able to answer the above.  I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions.  Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.

### On mix-norms and the rate of decay of correlations

Series
CDSNS Colloquium
Time
Friday, March 18, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker

Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm --- a negative Sobolev norm --- and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q-recurrent; otherwise it is q-transient. We use this classification to study several examples and raise questions for future investigations.

### On Herman positive metric entropy conjecture

Series
CDSNS Colloquium
Time
Friday, March 11, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Dmitry TuraevImperial College

Consider any area-preserving map of R2 which has an elliptic periodic orbit. We show that arbitrarily close to this map (in the C-infinity topology) there exists an area-preserving map which has a "chaotic island" - an open set where every point has positive maximal Lyapunov exponent. The result implies that the naturally sound conjectures that relate the observed chaotic behavior in non-hyperbolic conservative systems with the positivity of the metric entropy need a rethinking.

### An analytic study of intermittency and multifractality through Riemann's non differentiable function

Series
CDSNS Colloquium
Time
Friday, March 4, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Victor Vilaça Da RochaGeorgia Tech

Different ways have been introduced to define intermittency in the theory of turbulence, like for example the non-gaussianity, the lack of self-similarity or the deviation of the theory of turbulence by Kolmogorov from 1941.

The usual tool to measure intermittency is the flatness, a measure of the variation of the velocity at small scale, using structure functions in the spatial domain, or high-pass filters in the frequency domain. However, these two approaches give different results in some experiences.

The goal here is to study and compare these two methods and show that the result depends on the regularity of the studied function. For that purpose, we use Riemann's non-differentiable functions. To motivate this choice, we'll present the link between this function, the vortex filament equation, and the multifractal formalism.
This is a work in collaboration with Daniel Eceizabarrena (University of Massachusetts Amherst) and Alexandre Boritchev (University of Lyon)

### Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds

Series
CDSNS Colloquium
Time
Friday, February 25, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jonathan DeWittU Chicago

Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,...,R_m$ generate the isometry group of $M$. Let $f_1,...,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from $S^n$ to real, complex, and quaternionic projective spaces.