Georgia Institute of Technology College of Sciences

A discussion of the paper "Evaluation of the information content of RNA structure mapping data for secondary structure prediction" by Quarrier et al (RNA, 2010).

An important question in circle dynamics is regarding the absolute continuity of an invariant measure. We will consider orientation preserving circle homeomorphisms with break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is well known that the invariant measures of sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesgue measure. But in the case of homeomorphisms with break points the results are quite dierent. We will discuss conjugacies between two circle homeomorphisms with break points. Consider the class of circle homeomorphisms with one break point b and satisfying the Katznelson-Ornsteins smoothness condition i.e. Df is absolutely continuous on [b; b + 1] and D2f 2 Lp(S1; dl); p > 1: We will formulate some results concerning the renormaliza- tion behavior of such circle maps.

The phenomenon of wave run-up has the capital importance for the beach erosion, coastal protection and flood hazard estimation. In the present talk we will discuss two particular aspects of the wave run-up problem. In this talk we focus on the wave run-up phenomena on a sloping beach. In the first part of the talk we present a simple stochastic model of the bottom roughness. Then, we quantify the roughness effect onto the maximal run-up height using Monte-Carlo simulations. A critical comparison with more conventional approaches is also performed.In the second part of the talk we study the run-up of simple wave groups on beaches of various geometries. Some resonant amplification phenomena are unveiled. The maximal run-up height in resonant cases can be 20 times higher than in regular situations. Thus, this work can provide a possible mechanism of extreme tsunami run-up conventionally ascribed to "local site effects".References:Dutykh, D., Labart, C., & Mitsotakis, D. (2011). Long wave run-up on random beaches. Phys. Rev. Lett, 107, 184504.Stefanakis, T., Dias, F., & Dutykh, D. (2011). Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett., 107, 124502.

Radar imaging is a technology that has been developed, verysuccessfully, within the engineering community during the last 50years. Radar systems on satellites now make beautiful images ofregions of our earth and of other planets such as Venus. One of thekey components of this impressive technology is mathematics, and manyof the open problems are mathematical ones.This lecture will explain, from first principles, some of the basicsof radar and the mathematics involved in producing high-resolutionradar images.

Recently, Sarkar showed that a smooth marked cobordism between two knots K_1 , K_2 induces a map between the knot Floer homology groups of the two knots HFK(K_1 ), HFK(K_2 ). It has been conjectured that this map is well defined (with respect to smooth marked cobordisms). After outlining what needs to be shown to prove this conjecture, I will present my current progress towards showing this result for the combinatorial version of HFK. Specifically, I will present a generalization of Carter and Saito's movie move theorem to grid diagrams, give a very brief introduction to combinatorial knot Floer homology and then present a couple of the required chain homotopies needed for the proof of the conjecture.

The critical group of a graph G is an abelian group K(G) whose order is the number of spanning forests of G. As shown by Bacher, de la Harpe and Nagnibeda, the group K(G) has several equivalent presentations in terms of the lattices of integer cuts and flows on G. The motivation for this talk is to generalize this theory from graphs to CW-complexes, building on our earlier work on cellular spanning forests. A feature of the higher-dimensional case is the breaking of symmetry between cuts and flows. Accordingly, we introduce and study two invariants of X: the critical group K(X) and the cocritical group K^*(X), As in the graph case, these are defined in terms of combinatorial Laplacian operators, but they are no longer isomorphic; rather, the relationship between them is expressed in terms of short exact sequences involving torsion homology. In the special case that X is a graph, torsion vanishes and all group invariants are isomorphic, recovering the theorem of Bacher, de la Harpe and Nagnibeda. This is joint work with Art Duval (University of Texas, El Paso) and Caroline Klivans (Brown University).

Modeling stochasticity in gene regulation is an important and complex problem in molecular systems biology. This talk will introduce a stochastic modeling framework for gene regulatory networks. This framework incorporates propensity parameters for activation and degradation and is able to capture the cell-to-cell variability. It will be presented in the context of finite dynamical systems, where each gene can take on a finite number of states and where time is a discrete variable. One of the new features of this framework is that it allows a finer analysis of discrete models and the possibility to simulate cell populations. A background to stochastic modeling will be given and applications will use two of the best known stochastic regulatory networks, the outcome of lambda phage infection of bacteria and the p53-mdm2 complex.

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.

 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

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.

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.

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.

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.

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.

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.