Series: Math Physics Seminar
We will talk about discrete versions of the Bethe-Sommerfeld conjecture. Namely, we study the spectra of multi-dimensional periodic Schrödinger operators on various discrete lattices with sufficiently small potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we characterize those energies at which gaps may open, and we give sharp arithmetic criteria on the periods that ensure no gaps open. We will also provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero. This talk is based on a joint work with Svetlana Jitomirskaya and another work with Jake Fillman.
Wednesday, September 5, 2018 - 14:00 , Location: Skiles 005 , Anubhav Mukherjee , GaTech , Organizer: Anubhav Mukherjee
This is the second lecture of the series on h-principle. We will introduce jet bundle and it's various properties. This played a big role in the devloping modern geometry and topology. And using this we will prove Whitney embedding theorem. Only basic knowledge of calculus is required.
Series: Analysis Seminar
I will discuss some free probability inequalities on the circle which can be seen in two different ways, one is via random matrix approximation, and another one by itself. I will show what I believe to be the key of these new forms, namely the fact that the circle acts on itself. For instance the Poincare inequality has a certain form which reflects this aspect. I will also briefly show how a transportation inequality can be discussed and how the standard Wasserstein distance can be modified to introduce this interesting phenomena. I will end the talk with a conjecture and some supporting evidence in the classical world of functional inequalities.
Series: High Dimensional Seminar
The concentration of Lipschitz functions around their expectation is a classical topic and continues to be very active. In these talks, we will discuss some recent progress in detail, including: A tight log-Sobolev inequality for isotropic logconcave densities A unified and improved large deviation inequality for convex bodies An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. (Warning: the talk will involve elementary calculus on the board, sometimes at an excruciatingly slow pace). Joint work with Yin Tat Lee.
Series: PDE Seminar
We consider a class of nonlinear, degenerate drift-diffusion equations in R^d. By a scaling argument, it is widely believed that solutions are uniformly Holder continuous given L^p-bound on the drift vector field for p>d. We show the loss of such regularity in finite time for p≤d, by a series of examples with divergence free vector fields. We use a barriers argument.
Series: School of Mathematics Colloquium
The colloquium will be the second lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. For further information please see http://math.gatech.edu/events/triad-distinguished-lecture-series-sara-van-de-geer-0.
Series: Stochastics Seminar
This talk concerns the description and analysis of a variational framework for empirical risk minimization. In its most general form the framework concerns a two-stage estimation procedure in which (i) the trajectory of an observed (but unknown) dynamical system is fit to a trajectory from a known reference dynamical system by minimizing average per-state loss, and (ii) a parameter estimate is obtained from the initial state of the best fit reference trajectory. I will show that the empirical risk of the best fit trajectory converges almost surely to a constant that can be expressed in variational form as the minimal expected loss over dynamically invariant couplings (joinings) of the observed and reference systems. Moreover, the family of joinings minimizing the expected loss fully characterizes the asymptotic behavior of the estimated parameters. I will illustrate the breadth of the variational framework through applications to the well-studied problems of maximum likelihood estimation and non-linear regression, as well as the analysis of system identification from quantized trajectories subject to noise, a problem in which the models themselves exhibit dynamical behavior across time.
Series: Graph Theory Seminar
Vertex minors are a weakening of the notion of induced subgraphs that benefit from many additional nice properties. In particular, there is a vertex minor version of Menger's Theorem proven by Oum. This theorem gives rise to a natural analog of branch-width called rank-width. Similarly to the Grid Theorem of Robertson and Seymour, we prove that a class of graphs has unbounded rank-width if and only if it contains all "comparability grids'' as vertex minors. This is joint work with Jim Geelen, O-joung Kwon, and Paul Wollan.