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Series: Geometry Topology Seminar

Series: CDSNS Colloquium

We will consider the
Frenkel-Kontorova models and their higher dimensional generalizations
and talk about the corresponding discrete weak KAM theory. The existence
of the discrete weak KAM solutions is related to the additive
eigenvalue problem in
ergodic optimization. In particular, I will show that the discrete weak
KAM solutions converge to the weak KAM solutions of the autonomous
Tonelli Hamilton-Jacobi equations as the time step goes to zero.

Series: Other Talks

The TraX project
is an inter-university effort, involving researchers from 8
universities, aimed at elucidating the geometric structures in phase
space which determine the speed and nature of chemical reactions and how
they are affected by external influences such as light pulses or noise.
The effort is highly interdisciplinary and it involves Mathematics
(Dynamical Systems), Numerical Computations, Physics, and Chemistry all
working together to understand experimental phenomena and make
predictions.
The project has been funded by the European Research Council,
Mathematics Division for 4 years and it will sponsor visits of European
scientists to GT and provide opportunities for graduate students to
collaborate in this area. http://traxkickoff.gatech.edu/

Series: ACO Student Seminar

The random to random shuffle on a deck of cards is given by at each
step choosing a random card from the deck, removing it, and replacing it
in a random location. We show an upper bound for the total variation
mixing time of the walk of 3/4n log(n) +cn steps. Together with matching
lower bound of Subag (2013), this shows the walk mixes with cutoff at
3/4n log(n) steps, answering a conjecture of Diaconis. We use the
diagonalization of the walk by Dieker and Saliola (2015), which relates
the eigenvalues to Young tableaux.
Joint work with Evita Nestorid.

Series: Algebra Seminar

Chai and Oort have asked
the following question: For any algebraically closed field $k$, and for
$g \geq 4$, does there exist an abelian variety over $k$ of dimension
$g$ not isogenous to a Jacobian? The answer in characteristic 0 is now
known to be yes.
We present a heuristic which suggests that for certain $g \geq 4$, the
answer in characteristic $p$ is no. We will also construct a proper
subvariety of $X(1)^n$ which intersects every isogeny class, thereby
answering a related question, also asked by Chai
and Oort. This is joint work with Jacob Tsimerman.

Series: Dissertation Defense

We present two distinct problems in the field of dynamical systems.I the first part, we cosider an atomic model of deposition of materials over a quasi-periodic medium, that is, a quasi-periodic version of the well-known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form.In the second part, we consider a simple model of chemical reaction and present a numerical method calculating the invariant manifolds and their stable/unstable bundles based on parameterization method.

Series: Dissertation Defense

Kinetic theory is the branch of mathematical physics that studies the motion of gas particles that undergo collisions. A central theme is the
study of systems out of equilibrium and approach of equilibrium, especially in the context of Boltzmann's equation. In this talk I will present Mark Kac's stochastic N-particle model, briefly show its connection to Boltzmann's equation, and present known and new results about the rate of approach to equilibrium, and about a finite-reservoir realization of an ideal thermostat.

Series: PDE Seminar

The Cucker-Smale system is a popular model of collective behavior of interacting agents, used, in particular, to model bird flocking and fish swarming. The underlying premise is the tendency for a local alignment of the bird (or fish, or ...) velocities. The Euler-Cucker-Smale system is an effective macroscopic PDE limit of such particle systems. It has the form of the pressureless Euler equations with a non-linear density-dependent alignment term. The alignment term is a non-linear version of the fractional Laplacian to a power alpha in (0,1). It is known that the corresponding Burgers' equation with a linear dissipation of this type develops shocks in a finite time. We show that nonlinearity enhances the dissipation, and the solutions stay globally regular for all alpha in (0,1): the dynamics is regularized due to the nonlinear nature of the alignment. This is a joint work with T. Do, A.Kiselev and C. Tan.

Series: Algebra Seminar

I will discuss the interplay between tangent lines of algebraic and tropical curves. By tropicalizing all the tangent lines
of a plane curve, we obtain the tropical dual curve, and a recipe
for computing the Newton polygon of the dual projective curve.
In the case of canonical curves, tangent lines are closely related
with various phenomena in algebraic geometry such as double covers, theta characteristics and Prym varieties. When degenerating
them in families, we discover analogous constructions in tropical
geometry, and links between quadratic forms, covers of graphs and
tropical bitangents.

Series: Geometry Topology Seminar

Alexandru Oancea:
Title: Symplectic homology for cobordisms
Abstract: Symplectic homology for a Liouville cobordism - possibly filled
at the negative end - generalizes simultaneously the symplectic homology of
Liouville domains and the Rabinowitz-Floer homology of their boundaries. I
will explain its definition, some of its properties, and give a sample
application which shows how it can be used in order to obstruct cobordisms
between contact manifolds. Based on joint work with Kai Cieliebak and Peter
Albers.
Basak Gürel:
Title: From Lusternik-Schnirelmann theory to Conley conjecture
Abstract: In this talk I will discuss a recent result showing that whenever
a closed symplectic manifold admits a Hamiltonian diffeomorphism with
finitely many simple periodic orbits, the manifold has a spherical homology
class of degree two with positive symplectic area and positive integral of
the first Chern class. This theorem encompasses all known cases of the
Conley conjecture (symplectic CY and negative monotone manifolds) and also
some new ones (e.g., weakly exact symplectic manifolds with non-vanishing
first Chern class). The proof hinges on a general Lusternik–Schnirelmann
type result that, under some natural additional conditions, the sequence of
mean spectral invariants for the iterations of a Hamiltonian diffeomorphism
never stabilizes. Based on joint work with Viktor Ginzburg.