Monday, September 22, 2008 - 16:00 , Location: Room 322, Boyd Graduate Studies UGA , Michael Usher , Department of Mathematics, University of Georgia , Organizer: John Etnyre
Based on work of Schwarz and Oh, information coming from a filtration in Hamiltonian Floer homology can be used to construct "spectral invariants" for paths of Hamiltonian diffeomorphisms of symplectic manifolds. I will show how these invariants can be used to provide a unified approach to proving various old and new results in symplectic topology, including the non-degeneracy of the Hofer metric and some of its variants; a sharp version of an inequality between the Hofer-Zehnder capacity and the displacement energy; and a generalization of Gromov's non-squeezing theorem.
Monday, September 22, 2008 - 14:30 , Location: Room 322, Boyd Graduate Studies UGA , Stavros Garoufalidis , School of Mathematics, Georgia Tech , Organizer: John Etnyre
I will discuss a relation between the HOMFLY polynomial of a knot, its extension for a closed 3-manifold, a special function, the trilogarithm, and zeta(3). Technically, this means that we consider perturbative U(N) Chern-Simons theory around the trivial flat connection, for all N, in an ambient 3-manifold. This is rigorous, and joint with Marcos Marino and Thang Le.
Series: Analysis Seminar
The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The proof requires powerful tools from algebraic geometry. In this talk I will talk about our recent result of these inequalities that are indeed valid for self-adjoint operators of an arbitrary finite factors. Since in this setting there is no readily available machinery from algebraic geometry, we are forced to look for an analysts friendly proof. A (complete) matricial form of our result is known to imply an affirmative answer to the Connes' embedding problem. Geometers in town especially welcome!
Monday, September 22, 2008 - 13:00 , Location: Skiles 255 , Dongbin Xiu , Division of Applied Math, Purdue University , Organizer: Haomin Zhou
There has been growing interest in developing numerical methods for stochastic computations. This is motivated by the need to conduct uncertainty quantification in simulations, where uncertainty is ubiquitous and exists in parameter values, initial and boundary conditions, geometry, etc. In order to obtain simulation results with high fidelity, it is imperative to conduct stochastic computations to incorporate uncertainty from the beginning of the simulations. In this talk we review and discuss a class of fast numerical algorithms based on generalized polynomial chaos (gPC) expansion.The methods are highly efficient, compared to other traditional In addition to the forward stochastic problem solvers, we also discuss gPC-based methods for addressing "modeling uncertainty", i.e., deficiency in mathematical models, and solving inverse problems such as parameter estimation. ones, and suitable for stochastic simulations of complex systems.
Series: Combinatorics Seminar
The Balog-Szemeredi-Gowers theorem is a widely used tool in additive combinatorics, and it says, roughly, that if one has a set A such that the sumset A+A is "concentrated on few values," in the sense that these values v each get close to n representations as v = a+b, with a,b in A, then there is a large subset A' of A such that the sumset A'+A' is "small" -- i.e. it has size a small multiple of n. Later, Sudakov, Szemeredi and Vu generalized this result to handle multiple sums A_1 + ... + A_k. In the present talk we will present a refinement of this result of Sudakov, Szemeredi and Vu, where we get better control on the growth of sums A'+...+A'. This is joint work with Ernie Croot.
Friday, September 19, 2008 - 14:00 , Location: Skiles 269 , John Etnyre , School of Mathematics, Georgia Tech , Organizer: John Etnyre
This will be an introduction to Legendrian knots (these are interesting knots that blend topological and geometric concepts) and a powerful invariant of Legendrian knots in R^3 called contact homology. On the first pass this invariant is combinatorial and has a lot of interesting algebraic structure. In a future talk (probably a few weeks from now), I will explain more about the analytic side of the theory as well as deeper algebraic aspects. This talk should be accessible anyone interested in topology and geometry.
Series: Stochastics Seminar
I will discuss some recent (but modest) results showing the existence and slow mixing of a stationary chain of Hamiltonian oscillators subject to a heat bath. Surprisingly, even these simple results require some delicate stochastic averaging. This is joint work with Martin Hairer.
Series: Graph Theory Seminar
Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. A graph is called pebbleable if for each vertex v there is a sequence of pebbling moves so that at least one pebble can be placed on vertex v. The pebbling number of a graph G is the smallest integer k such that G is pebbleable given any configuration of k pebbles on G. We improve on the bound of Bukh by showing that the pebbling number of a graph of diameter 3 on n vertices is at most the floor of 3n/2 + 2, and this bound is best possible. We give an alternative proof that the pebbling number of a graph of diameter 2 on n vertices is at most n + 1. This is joint work with Noah Streib and Carl Yerger.
Series: School of Mathematics Colloquium
It has been found about ten years ago that most of the real networks are not random ones in the Erdos-Renyi sense but have different topology (structure of the graph of interactions between the elements of a network). This finding generated a steady flux of papers analyzing structural aspects of networks. However, real networks are rather dynamical ones where the elements (cells, genes, agents, etc) are interacting dynamical systems. Recently a general approach to the studies of dynamical networks with arbitrary topology was developed. This approach is based on a symbolic dynamics and is in a sense similar to the one introduced by Sinai and the speaker for Lattice Dynamical Systems, where the graph of interactions is a lattice. The new approach allows to analyse a combined effect of all three features which characterize a dynamical network (topology, dynamics of elements of the network and interactions between these elements) on its evolution. The networks are of the most general type, e.g. the local systems and interactions need not to be homogeneous, nor restrictions are imposed on a structure of the graph of interactions. Sufficient conditions on stability of dynamical networks are obtained. It is demonstrated that some subnetworks can evolve regularly while the others evolve chaotically. This approach is a very natural one and thus gives a hope that in many other problems (some will be discussed) on dynamical networks a progress could be expected.