Seminars and Colloquia by Series

Overconvergent Lattices and Berkovich Spaces

Series
Algebra Seminar
Time
Tuesday, April 24, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew Dudzik – UC Berkeley
The construction of the Berkovich space associated to a rigid analytic variety can be understood in a general topological framework as a type of local compactification or uniform completion, and more generally in terms of filters on a lattice. I will discuss this viewpoint, as well as connections to Huber's theory of adic spaces, and draw parallels with the usual metric completion of $\mathbb{Q}$.

Universality of the global fluctuations for the eigenvectors of Wigner random matrices

Series
Stochastics Seminar
Time
Tuesday, April 24, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
F. Benaych-Georges – Universite Pierre et Marie Curie
Many of the asymptotic spectral characteristics of a symmetric random matrix with i.i.d. entries (such a matrix is called a "Wigner matrix") are said to be "universal": they depend on the exact distribution of the entries only via its first moments (in the same way that the CLT gives the asymptotic fluctuations of the empirical mean of i.i.d. variables as a function of their second moment only). For example, the empirical spectral law of the eigenvalues of a Wigner matrix converges to the semi-circle law if the entries have variance 1, and the extreme eigenvalues converge to -2 and 2 if the entries have a finite fourth moment. This talk will be devoted to a "universality result" for the eigenvectors of such a matrix. We shall prove that the asymptotic global fluctuations of these eigenvectors depend essentially on the moments with orders 1, 2 and 4 of the entries of the Wigner matrix, the third moment having surprisingly no influence.

Perturbation Theory and its Application to Complex Biological Networks --A quantification of systematic features of biological networks

Series
Dissertation Defense
Time
Tuesday, April 24, 2012 - 11:00 for 2 hours
Location
Skiles 006
Speaker
Yao Li – School of Mathematics, Georgia Tech
The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency between modules by injecting external noise. To accomplish this, a deep study of stochastic dynamical systems would be essential. The first part is about the stochastic dynamical system theory. The classical estimation of invariant measures of Fokker-Planck equations is improved by the level set method. Further, we develop a discrete Fokker-Planck-type equation to study the discrete stochastic dynamical systems. In the second part, we quantify systematic measures including degeneracy, complexity and robustness. We also provide a series of results on their properties and the connection between them. Then we apply our theory to the JAK-STAT signaling pathway network.

Optimization of two-link and three-link snake-like locomotion

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 23, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Fangxu Jing – GT Math
We analyze two-link (or three-link) 2D snake like locomotions and discuss the optimization of the motion. The snake is modeled as two (or three) identical links connected via hinge joints and the relative angles between the links are prescribed as periodic actuation functions. An essential feature of the locomotion is the anisotropy of friction coefficients. The dynamics of the snake is analyzed numerically, as well as analytically for small amplitude actuations of the relative angles. Cost of locomotion is defined as the ratio between distance traveled by the snake and the energy expenditure within one period. Optimal conditions of the highest efficiency in terms of the friction coefficients and the actuations are discussed for the model.

On a weak form of Arnold diffusion in arbitrary degrees of freedom

Series
CDSNS Colloquium
Time
Monday, April 23, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vadim Kaloshin – Univ. of Maryland
Consider a generic perturbation of a nearly integrable system of {\it arbitrary degrees of freedom $n\ge 2$ system}\[H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T,\]with strictly convex $H_0$. Jointly with P.Bernard and K.Zhang we prove existence of orbits $(\th,p)(t)$ exhibiting Arnold diffusion \[\|p(t)-p(0) \| >l(H_1)>0 \quad \textup{independently of }\eps.\]Action increment is independent of size of perturbation$\eps$, but does depend on a perturbation $\eps H_1$.This establishes a weak form of Arnold diffusion. The main difficulty in getting rid of $l(H_1)$ is presence of strong double resonances. In this case for $n=2$we prove existence of normally hyperbolic invariant manifolds passing through these double resonances. (joint with P. Bernard and K. Zhang)

Discrete Mathematical Biology Working Seminar

Series
Other Talks
Time
Monday, April 23, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Will Perkins – Georgia Tech
A discussion of the paper "RNA folding with soft constraints: reconciliation of probing data and thermodynamic secondary structure prediction" by Washietl et al (NAR, 2012).

Matchings in hypergraphs

Series
Combinatorics Seminar
Time
Friday, April 20, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tomasz Luczak – Emory University and Adam Mickiewicz University, Poznan –
Let H_k(n,s) be a k-uniform hypergraphs on n vertices in which the largest matching has s edges. In 1965 Erdos conjectured that the maximum number of edges in H_k(n,s) is attained either when H_k(n,s) is a clique of size ks+k-1, or when the set of edges of H_k(n,s) consists of all k-element sets which intersect some given set S of s elements. In the talk we prove this conjecture for k = 3 and n large enough. This is a joint work with Katarzyna Mieczkowska.

Stability of ODE with colored noise forcing.

Series
CDSNS Colloquium
Time
Friday, April 20, 2012 - 11:10 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Timothy Blass – Carnegie Melon University
I will discuss recent work on the stability of linear equations under parametric forcing by colored noise. The noises considered are built from Ornstein-Uhlenbeck vector processes. Stability of the solutions is determined by the boundedness of their second moments. Our approach uses the Fokker-Planck equation and the associated PDE for the marginal moments to determine the growth rate of the moments. This leads to an eigenvalue problem, which is solved using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes into "ladder operators." The results are given in terms of a perturbation expansion in the size of the noise. We have found very good agreement between our results and numerical simulations. This is joint work with L.A. Romero.

The one dimensional free Poincare inequality

Series
Stochastics Seminar
Time
Thursday, April 19, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Ionel Popescu – Georgia Institute of Technology, School of Mathematics
This is obtained as a limit from the classical Poincar\'e on large random matrices. In the classical case Poincare is obtained in a rather easy way from other functional inequalities as for instance Log-Sobolev and transportation. In the free case, the same story becomes more intricate. This is joint work with Michel Ledoux.

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