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 , email@example.com , 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.
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.
Monday, April 24, 2017 - 14:05 , Location: Skiles 005 , Prof. George Mohler , IUPUI Computer Science , Organizer: Martin Short
In this talk we focus on classification problems where noisy sensor measurements collected over a time window must be classified into one or more categories. For example, mobile phone health and insurance apps take as input time series from the accelerometer, gyroscope and GPS radio of the phone and output predictions as to whether the user is still, walking, running, biking, driving etc. Standard approaches to this problem consist of first engineering features from statistics of the data (or a transform) over a window and then training a discriminative classifier. For two applications we show how these features can instead be learned in an end-to-end modeling framework with the advantages of increased accuracy and decreased modeling and training time. The first application is reconstructing unobserved neural connections from Calcium fluorescence time series and we introduce a novel convolutional neural network architecture with an inverse covariance layer to solve the problem. The second application is driving detection on mobile phones with applications to car telematics and insurance.
Series: CDSNS Colloquium
One dimensional discrete Schrödinger operators arise naturally in modeling the motion of quantum particles in a disordered medium. The medium is described by potentials which may naturally be generated by certain ergodic dynamics. We will begin with two classic models where the potentials are periodic sequences and i.i.d. random variables (Anderson Model). Then we will move on to quasi-periodic potentials, of which the randomness is between periodic and i.i.d models and the phenomena may become more subtle, e.g. a metal-insulator type of transition may occur. We will show how the dynamical object, the Lyapunov exponent, plays a key role in the spectral analysis of these types of operators.
Series: Combinatorics Seminar
Various parameters of many models of random rooted trees are fairly well understood if they relate to a near-root part of the tree or to global tree structure. In recent years there has been a growing interest in the analysis of the random tree fringe, that is, the part of the tree that is close to the leaves. Distance from the closest leaf can be viewed as the protection level of a vertex, or the seniority of a vertex within a network. In this talk we will review a few recent results of this kind for a number of tree varieties, as well as indicate the challenges one encounters when trying to generalize the existing results. One tree variety, that of decreasing binary trees, will be related to permutations, another one, phylogenetic trees, is frequent in applications in molecular biology.
Friday, April 21, 2017 - 15:00 , Location: Skiles 254 , Adrian P. Bustamante , Georgia Tech , Organizer:
A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present an efficient numerical algorithm, and its implementation, which approximate the conjugations given by the Theorem
Series: ACO Student Seminar
Beginning with Szemerédi’s regularity lemma, the theory of graph decomposition and graph limits has greatly increased our understanding of large dense graphs and provided a framework for graph approximation. Unfortunately, much of this work does not meaningfully extend to non-dense graphs. We present preliminary work towards our goal of creating tools for approximating graphs of intermediate degree (average degree o(n) and not bounded). We give a new random graph model that produces a graph of desired size and density that approximates the number of small closed walks of a given sparse graph (i.e., small moments of its eigenspectrum). We show how our model can be applied to approximate the hypercube graph. This is joint work with Santosh Vempala.