Monday, February 4, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nicolai Haydn – USC
The theorem of Shannon-McMillan-Breiman states that for every
generating partition on an ergodic system,
the exponential decay rate of the measure of cylinder sets
equals the metric entropy almost everywhere (provided the entropy is finite).
We show that the measure of cylinder sets are lognormally
distributed for strongly mixing systems and infinite partitions and show that the rate of convergence
is polynomial provided the fourth moment of the information function is finite.
We also show that it satisfies the almost sure invariance principle.
Unlike previous results by Ibragimov and others which only apply to finite partitions,
here we do not require any regularity of the conditional entropy function.
Monday, February 4, 2013 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Lipton – LSU
Metamaterials are a new form of structured materials used to control
electromagnetic waves through localized resonances. In this talk we
introduce a rigorous mathematical framework for controlling localized
resonances and predicting exotic behavior inside optical metamaterials.
The theory is multiscale in nature and provides a rational basis for
designing microstructure using multiphase nonmagnetic materials to create
backward wave behavior across prescribed frequency ranges.
Monday, February 4, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Russell Avdek – USC
We introduce a new surgery operation for contact manifolds called the Liouville connect sum. This operation -- which includes Weinstein handle attachment as a special case -- is designed to study the relationship between contact topology and symplectomorphism groups established by work of Giroux and Thurston-Winkelnkemper. The Liouville connect sum is used to generalize results of Baker-Etnyre-Van Horn-Morris and Baldwin on the existence of "monodromy multiplication cobordisms" as well as results of Seidel regarding squares of symplectic Dehn twists.
I'll discuss two methods for finding bounds on sums of graph eigenvalues (variously for the Laplacian, the renormalized Laplacian, or the adjacency matrix). One of these relies on a Chebyshev-type estimate of the statistics of a subsample of an ordered sequence, and the other is an adaptation of a variational argument used by P. Kröger for Neumann Laplacians. Some of the inequalities are sharp in suitable senses. This is ongoing work with J. Stubbe of EPFL
Friday, February 1, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Huseyin Acan – Ohio State University
A permutation of the set {1,2,...,n} is connected if there is no k < n such
that the set of the first k numbers is invariant as a set under the
permutation. For each permutation, there is a corresponding graph whose
vertices are the letters of the permutation and whose edges correspond to
the inversions in the permutation. In this way, connected permutations
correspond to connected permutation graphs.
We find a growth process of a random permutation in which we start with the
identity permutation on a fixed set of letters and increase the number of
inversions one at a time. After the m-th step of the process, we obtain a
random permutation s(n,m) that is uniformly distributed over all
permutations of {1,2,...,n} with m inversions. We will discuss the evolution
process, the connectedness threshold for the number of inversions of s(n,m),
and the sizes of the components when m is near the threshold value. This
study fits into the wider framework of random graphs since it is analogous
to studying phase transitions in random graphs. It is a joint work with my
adviser Boris Pittel.
In this talk, we are going to introduce Linearized Proximal Alternating Minimization Algorithm and its variants for total variation based variational model. Since the proposed method does not require any special inner solver (e.g. FFT or DCT), which is quite often required in augmented Lagrangian based approach (ADMM), it shows better performance for large scale problems. In addition, we briefly introduce new regularization method (nonconvex higher order total variation).
Friday, February 1, 2013 - 11:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John Etnyre – Ga Tech
In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.
Thursday, January 31, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geordie Richards – IMA
The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering. In this talk we present invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized. As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.
Wednesday, January 30, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
James Scurry – Georgia Tech
We will study one and two weight inequalities for several different
operators from harmonic analysis, with an emphasis on vector-valued
operators. A large portion of current research in the area of one weight
inequalities is devoted to estimating a given operators' norm in terms
of a weight's A_p characteristic; we consider some related problems
and the extension of several results to the vector-valued setting. In
the two weight setting we consider some of the difficulties of
characterizing a two weight inequality through Sawyer-type testing
conditions.