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Series: Stochastics Seminar

The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.

Series: School of Mathematics Colloquium

In this talk, we discuss 1.) the nonlinear instability and unstable manifolds of steady solutions of the Euler equation with fixed domains and 2.) the evolution of free (inviscid) fluid surfaces, which may involve vorticity, gravity, surface tension, or magnetic fields. These problems can be formulated in a Lagrangian formulation on infinite dimensional manifolds of volume preserving diffeomorphisms with an invariant Lie group action. In this setting, the physical pressure turns out to come from the combination of the gravity, surface tension, and the Lagrangian multiplier. The vorticity is naturally related to an invariant group action. In the absence of surface tension, the well-known Rayleigh-Taylor and Kelvin-Helmholtz instabilities appear naturally related to the signs of the curvatures of those infinite dimensional manifolds. Based on these considerations, we obtain 1.) the existence of unstable manifolds and L^2 nonlinear instability in the cases of the fixed domains and 2.) in the free boundary cases, the local well-posedness with surface tension in a rather uniform energy method. In particular, for the cases without surface tension which do not involve hydrodynamical instabilities, we obtain the local existence of solutions by taking the vanishing surface tension limit.

Series: ACO Student Seminar

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 analyze 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. Some natural graph theoretical and dynamical questions appear in the farther developments of this approach. No preliminary knowledge of anything besides basic calculus and linear algebra is required to understand what is going on.

Wednesday, November 5, 2008 - 11:00 ,
Location: Skiles 255 ,
Melissa Kemp ,
Dept of Biomedical Engineering, Georgia Tech ,
Organizer: Christine Heitsch

Hydrogen peroxide has been long considered a harmful reactive oxygen species, but is increasingly appreciated as a cellular signaling molecule. The mechanism by which the cell buffers against intracellular H2O2 accumulation during periods of oxidative stress is not fully understood. I will introduce a detailed network model of the known redox reactions and cellular thiol modifications involved in H2O2 buffering. The model includes anti-oxidative contributions of catalase, glutathione peroxidase, peroxiredoxin, and glutaredoxin, in addition to the cytoplasmic redox buffers, thioredoxin and glutathione. Based on ordinary differential equations, the model utilizes mass action kinetics to describe changes in concentration and redox state of cytoplasmic proteins upon exposure to physiologically relevant concentrations of extracellular H2O2. Simulations match experimental observations of a rapid and transient oxidation of thioredoxin upon exposure to extracellular peroxide. The increase in the concentration of oxidized proteins predicted by the model is simultaneously accompanied by an increase in protein S-glutathionylation, possibly regulating signal transduction in cells undergoing oxidative stress. Ultimately, this network analysis will provide insight into how to target antioxidant therapies for enhanced buffering without impacting the necessary protein oxidation used by cells for signaling purposes.

Series: Research Horizons Seminar

Orthogonal polynomials play a role in myriads of problems ranging from approximation theory to random matrices and signal processing. Generalizations of orthogonal polynomials - such as biorthogonal polynomials, cardinal series, Muntz polynomials, are used for example, in number theory and numerical analysis. We discuss some of these, and some potential research projects involving them.

Series: Analysis Seminar

Pseudodifferential operators and affine pseudodifferential operators arise naturally in the study of wireless communications. We discuss the origins of these operators and give new conditions on the kernels and symbols of pseudodifferential and affine pseudodifferential operators which ensure the operators are trace class (and more generally, Schatten p-class).

Series: Dissertation Defense

The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.
In the second part we consider some classes of cellular automata called Deterministic Walks in Random Environments on \mathbb Z^1. At first we deal with the system with constant rigidity and Markovian distribution of scatterers on \mathbb Z^1. It is shown that these systems have essentially the same properties as DWRE on \mathbb Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.

Series: Combinatorics Seminar

We will discuss some extensions/generalizations of the striking and elegant fact (proved independently by Furstenberg and Sarkozy) that any subset of the integers of positive upper density necessarily contains two distinct elements whose difference is a perfect square. This is joint work with Akos Magyar.

Series: Stochastics Seminar

Consider a class of multidimensional degenerate diffusion processes of the following form

X_t = x+\int_0^t (X_s) ds+\int_0^t \sigma(X_s) dW_s,

Y_t = y+\int_0^t F(X_s)ds,

where b,\sigma, F are assumed to be smooth and b,\sigma bounded. Suppose now that \sigma\sigma^* is uniformly elliptic and that \nabla F does not degenerate. These assumptions guarantee that only one Poisson bracket is needed to span the whole space. We obtain a parametrix representation of Mc Kean-Singer type for the density of (X_t,Y_t) from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. This particular representation then allows to give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The "weak" degeneracy allows to use the local limit Theorem in Gaussian regime but also induces some difficulty to define the suitable approximating process. In particular two time scales appear. Another difficulty w.r.t. the standard literature on the topic, see e.g. Konakov and Mammen (2000), is the unboundedness of F.

Friday, October 31, 2008 - 14:00 ,
Location: Skiles 269 ,
Sinem Celik Onaran ,
School of Mathematics, Georgia Tech ,
Organizer: John Etnyre

It is still not known whether every genus g Lefschetz fibration over the 2-sphere admits a section or not. In this talk, we will give a brief background information on Lefschetz fibrations and talk about sections of genus two Lefschetz fibration. We will observe that any holomorphic genus two Lefschetz fibration without seperating singular fibers admits a section. This talk is accessible to anyone interested in topology and geometry.