Seminars and Colloquia by Series

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.

The structure of graphs excluding a fixed immersion

Series
Graph Theory Seminar
Time
Thursday, April 19, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Paul Wollan – ISyE, GT and The Sapienza University of Rome
A graph $G$ contains a graph $H$ as an immersion if there exist distinct vertices $\pi(v) \in V(G)$ for every vertex $v \in V(H)$ and paths $P(e)$ in $G$ for every $e \in E(H)$ such that the path $P(uv)$ connects the vertices $\pi(u)$ and $\pi(v)$ in $G$ and furthermore the paths $\{P(e):e \in E(H)\}$ are pairwise edge disjoint. Thus, graph immersion can be thought of as a generalization of subdivision containment where the paths linking the pairs of branch vertices are required to be pairwise edge disjoint instead of pairwise internally vertex disjoint. We will present a simple structure theorem for graphs excluding a fixed $K_t$ as an immersion. The structure theorem gives rise to a model of tree-decompositions based on edge cuts instead of vertex cuts. We call these decompositions tree-cut decompositions, and give an appropriate definition for the width of such a decomposition. We will present a ``grid" theorem for graph immersions with respect to the tree-cut width. This is joint work with Paul Seymour.

Fluids, vortex sheets, and the skew mean curvature flow.

Series
School of Mathematics Colloquium
Time
Thursday, April 19, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Boris Khesin – IAS/University of Toronto
We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.

Agler Decompositions on the Bidisk

Series
Analysis Seminar
Time
Wednesday, April 18, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kelly Bickel – Washington University - St. Louis
It is well-known that every Schur function on the bidisk can be written as a sum involving two positive semidefinite kernels. Such decompositions, called Agler decompositions, have been used to answer interpolation questions on the bidisk as well as to derive the transfer function realization of Schur functions used in systems theory. The original arguments for the existence of such Agler decompositions were nonconstructive and the structure of these decompositions has remained quite mysterious. In this talk, we will discuss an elementary proof of the existence of Agler decompositions on the bidisk, which is constructive for inner functions. We will use this proof as a springboard to examine the structure of such decompositions and properties of their associated reproducing kernel Hilbert spaces.

Computation of limit cycles and their isochrons: Applications to biology.

Series
Mathematical Biology Seminar
Time
Wednesday, April 18, 2012 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gemma Huguet – NYU
 In this talk we will present a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not reached yet the asymptotic state (a limit cycle). That is we extend the computation of the phase resetting curves (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRS). These are very useful tools for the study of synchronization of coupled oscillators. To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and the Lie symmetries approach), which allow to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we design a numerical scheme to compute both the isochrons and the PRSs of a given oscillator. Finally, we will show some examples of the computations we have carried out for some well-known biological models. This is joint work with Toni Guillamon and R. de la Llave

Sparse and low rank estimation problems

Series
Research Horizons Seminar
Time
Wednesday, April 18, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vladimir Koltchinskii – Georgia Tech
Recently, there has been a lot of interest in estimation of sparse vectors in high-dimensional spaces and large low rank matrices based on a finite number of measurements of randomly picked linear functionals of these vectors/matrices. Such problems are very basic in several areas (high-dimensional statistics, compressed sensing, quantum state tomography, etc). The existing methods are based on fitting the vectors (or the matrices) to the data using least squares with carefully designed complexity penalties based on the $\ell_1$-norm in the case of vectors and on the nuclear norm in the case of matrices. Proving error bounds for such methods that hold with a guaranteed probability is based on several tools from high-dimensional probability that will be also discussed.

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