## Seminars and Colloquia by Series

Tuesday, April 19, 2011 - 16:00 , Location: Skiles 006 , Craig A. Sloane , School of Mathematics, Georgia Tech , Organizer: Michael Loss
Classical Hardy, Sobolev, and Hardy-Sobolev-Maz'ya inequalities are well known results that have been studied for awhile. In recent years, these results have been been generalized to fractional integrals. This Dissertation proves a new Hardy inequality on general domains, an improved Hardy inequality on bounded convex domains, and that the sharp constant for any convex domain is the same as that known for the upper halfspace. We also prove, using a new type of rearrangement on the upper halfspace, based in part on Carlen and Loss' concept of competing symmetries, the existence of the fractional Hardy-Sobolev-Maz'ya inequality in the case p = 2, as well as proving the existence of minimizers, at least in limited cases.
Tuesday, March 8, 2011 - 09:00 , Location: Skiles 006 , Benjamin Webb , School of Mathematics, Georgia Tech , Organizer:
Real world networks typically consist of a large number of dynamical units with a complicated structure of interactions. Until recently such networks were most often studied independently as either graphs or as coupled dynamical systems. To integrate these two approaches we introduce the concept of an isospectral graph transformation which allows one to modify the network at the level of a graph while maintaining the eigenvalues of its adjacency matrix. This theory can then be used to rewire dynamical networks, considered as dynamical systems, in order to gain improved estimates for whether the network has a unique global attractor. Moreover, this theory leads to improved eigenvalue estimates of Gershgorin-type. Lastly, we will discuss the use of Schwarzian derivatives in the theory of 1-d dynamical systems.
Tuesday, March 1, 2011 - 14:00 , Location: Skiles 005 , Maria del Carmen Reguera Rodriquez , School of Mathematics, Georgia Tech , Organizer:
Thursday, February 3, 2011 - 15:00 , Location: Skiles 006 , Sergio Angel Almada , School of Mathematics, Georgia Tech , Organizer:
A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (nonstandard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks.
Tuesday, October 5, 2010 - 15:05 , Location: Skiles 114 , Tobias Hurth , School of Mathematics, Georgia Tech , Organizer:
We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute the unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero or infinity and derive analogues of classical probability theory results such as central limit theorem and large deviation principle.
Thursday, August 19, 2010 - 10:00 , Location: Skiles 114 , Carl Yerger , School of Mathematics, Georgia Tech , Organizer: Robin Thomas
A graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In this thesis, we study the structure of critical graphs on higher surfaces. One major result in this area is Carsten Thomassen's proof that there are finitely many 6-critical graphs on a fixed surface. This proof involves a structural theorem about a precolored cycle C of length q. In general terms, he proves that a coloring \phi of C can be extended inside the cycle, or there exists a subgraph H with at most 5^{q^3} vertices such that \phi cannot be extended to a 5-coloring of H. In Chapter 2, we provide an alternative proof that reduces the number of vertices in H to be cubic in q. In Chapter 3, we find the nine 6-critical graphs among all graphs embeddable on the Klein bottle. Finally, in Chapter 4, we prove a result concerning critical graphs related to an analogue of Steinberg's conjecture for higher surfaces. We show that if G is a 4-critical graph embedded on surface \Sigma, with Euler genus g and has no cycles of length four through ten, then |V(G)| \leq 2442g + 37.
Thursday, July 1, 2010 - 10:30 , Location: Skiles 154 , Ramazan Tinaztepe , School of Mathematics, Georgia Tech , Organizer: Christopher Heil
Monday, May 3, 2010 - 11:00 , Location: Skiles 255 , Ian Palmer , Georgia Tech , , Organizer:
A construction is given for which the Hausdorﬀ measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorﬀ measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorﬀ and box dimensions diﬀer—in particular, it does not depend on any self-similarity or regularity conditions on the space or an embedding in an ambient space. The only restriction on the space is that it have positive s-dimensional Hausdorﬀ measure, where s is the Hausdorﬀ dimension of the space, assumed to be ﬁnite.
Tuesday, April 27, 2010 - 13:00 , Location: Skiles 269 , Ulfar Stefansson , School of Mathematics, Georgia Tech , Organizer: Doron Lubinsky
Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior follows, as well as estimates for the smallest and largest zeros. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval and the asymptotic properties of the associated Christoffel functions.
Monday, April 26, 2010 - 15:00 , Location: Skiles 255 , Armen Vagharshakyan , School of Mathematics, Georgia Tech , Organizer: Michael Lacey
We improve the lower bound for the L_\infty norm of the discrepancy function. This result makes a partial step towards resolving the Discrepancy Conjecture. Being a theorem in the theory of irregularities of distributions, it also relates to corresponding results in approximation theory (namely, the Kolmogorov entropy of spaces of functions with bounded mixed derivatives) and in probability theory (namely, Small Ball Inequality - small deviation inequality for the Brownian sheet). We also provide sharp bounds for the exponential Orlicz norm and the BMO norm of the discrepancy function in two dimensions. In the second part of the thesis we prove that any sufficiently smooth one-dimensional Calderon-Zygmund convolution operator can be recovered through averaging of Haar shift operators. This allows to generalize the estimates, which had been previously known for Haar shift operators, to Calderon-Zygmund operators. As a result, the A_2 conjecture is settled for this particular type of Calederon-Zygmund operators.