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.
Friday, April 28, 2017 - 11:05 , Location: Skiles 006 , Ananth Shankar , Harvard University , Organizer: Padmavathi Srinivasan
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.
Thursday, April 27, 2017 - 10:00 , Location: Skiles 005 , Lei Zhang , Georgia Institute of Technology , Organizer: Lei Zhang
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.
Wednesday, April 26, 2017 - 09:00 , Location: Skiles 005 , Hagop Tossounian , Georgia Tech , firstname.lastname@example.org , Organizer: Hagop Tossounian
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.
Monday, April 24, 2017 - 15:05 , Location: Skiles 005 , Yoav Len , University of Waterloo , Organizer: Matt Baker
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.
Monday, April 24, 2017 - 14:30 , Location: UGA Room 303 , Alexandru Oancea and Basak Gurel , Jussieu and University of Central Florida , Organizer: Caitlin Leverson
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.