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Series: Graph Theory Seminar

We study the uniqueness of optimal configurations in extremal
combinatorics. An empirical experience suggests that optimal solutions to
extremal graph theory problems can be made asymptotically unique by
introducing additional constraints. Lovasz conjectured that this phenomenon
is true in general: every finite feasible set of subgraph density
constraints can be extended further by a finite set of density constraints
such that the resulting set is satisfied by an asymptotically unique graph.
We will present a counterexample to this conjecture and discuss related
results.
The talk is based on joint work with Andrzej Grzesik and Laszlo Miklos
Lovasz.

Series: Dissertation Defense

We first discuss the construction of whiskered invariant tori for fibered holomorphic dynamics using a Nash-Moser iteration. The results are in a-posteriori form. The iterative procedure we present has numerical applications (it lends itself to efficient numerical implementations) since it is not based on transformation theory but rather in applying corrections which ameliorate notably the curse of dimensionality. Then we will discuss results on compensated domains in a Banach space.

Series: CDSNS Colloquium

Consider an affine skew product of the complex plane. \begin{equation}\begin{cases} \omega \mapsto \theta+\omega,\\ z \mapsto =a(\theta \mu)z+c, \end{cases}\end{equation}where $\theta \in \mathbb{T}$, $z\in \mathbb{C}$, $\omega$ is Diophantine, and $\mu$ and $c$ are real parameters. In this talk we show that, under suitable conditions, the affine skew product has an invariant curve that undergoes a fractalization process when $\mu$ goes to a critical value. The main hypothesis needed is the lack of reducibility of the system. A characterization of reducibility of linear skew-products on the complex plane is provided. We also include a linear and topological classification of these systems. Join work with: N\'uria Fagella, \`Angel Jorba and Joan Carles Tatjer

Series: GT-MAP Seminars

This workshop is sponsored by College of Science, School of Mathematics, GT-MAP and NSF.

The goal of this workshop is to bring together experts in various aspects of optimal transport and related topics on graphs (e.g., PDE/Numerics, Computational and Analytic/Probabilistic aspects).

Series: Geometry Topology Seminar

We will discuss a relation between some notions in three-dimensional topology and four-dimensional aspects of knot theory.

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.