Georgia Institute of Technology College of Sciences

A linear cocycle over a diffeomorphism f of a manifold M is an automorphism of a vector bundle over M that projects to f. An important example is given by the differential Df or its restriction to an invariant sub-bundle. We consider a Holder continuous linear cocycle over a hyperbolic system and explore what conclusions can be made based on its properties at the periodic points of f. In particular, we obtain criteria for a cocycle to be isometric or conformal and discuss applications and further developments.

I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.

The viruses that infect bacteria have a hallowed position in the development of modern biology, and once inspired Max Delbruck refer to them as "the atom of biology". Recently, these viruses have become the subject of intensive physical investigation. Using single-molecule techniques, it is actually possible to watch these viruses in the act of packing and ejecting their DNA. This talk will begin with a general introduction to viruses and their life cycles and will then focus on simple physical arguments about the forces that attend viral DNA packaging and ejection, predictions about the ejection process and single-molecule measurements of ejection itself.

Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some (here finite) set X; the threshold question for such an F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F? For example: about how many random edges from the complete graph K_n are typically required to produce a Hamiltonian cycle?) We'll discuss recent progress and lack thereof on a few threshold-type questions, and try to say something about a ludicrously general conjecture of G. Kalai and the speaker to the effect that there is always a pretty good naive explanation for a threshold being what it is.

The stability of a liquid drop of prescribed volume hanging from a circular cylindrical tube in a gravity field has been a problem of continuing interest. This problem was treated variationally in the late '70s by Henry Wente who showed there was a continuous family indexed by increasing volume which terminated in a final unstable equilibrium due to one or the other of two specific geometric mechanisms. I will describe a similar problem arising in mathematical biology for drops at the bottom of a rectangular tube and explain, among other things, how the associated instability occurs through exactly three physical mechanisms.

Hyperbolic actions of Z^k and R^k arise naturally in algebraic and geometric context. Algebraic examples include actions by commuting automorphisms of tori or nilmanifolds and, more generally, affine and homogeneous actions on cosets of Lie groups. In contrast to hyperbolic actions of Z and R, i.e. Anosov diffeomorphisms and flows, higher rank actions exhibit remarkable rigidity properties, such as scarcity of invariant measures and smooth conjugacy to a small perturbation. I will give an overview of results in this area and discuss recent progress.

The fields of statistical physics, discrete probability, combinatorics, and theoretical computer science have converged around efforts to understand random structures and algorithms. Recent activity in the interface of these fields has enabled tremendous breakthroughs in each domain and has supplied a new set of techniques for researchers approaching related problems. This thesis makes progress on several problems in this interface whose solutions all build on insights from multiple disciplinary perspectives. First, we consider a dynamic growth process arising in the context of DNA-based self-assembly. The assembly process can be modeled as a simple Markov chain. We prove that the chain is rapidly mixing for large enough bias in regions of Z^d. The proof uses a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile, which arises in the nanotechnology application. Moreover, we use intuition from statistical physics to construct a choice of the biases for which the Markov chain M_{mon} requires exponential time to converge. Second, we consider a related problem regarding the convergence rate of biased permutations that arises in the context of self-organizing lists. The Markov chain M_{nn} in this case is a nearest-neighbor chain that allows adjacent transpositions, and the rate of these exchanges is governed by various input parameters. It was conjectured that the chain is always rapidly mixing when the inversion probabilities are positively biased, i.e., we put nearest neighbor pair x < y in order with bias 1/2 <= p_{xy} <= 1 and out of order with bias 1-p_{xy}. The Markov chain M_{mon} was known to have connections to a simplified version of this biased card-shuffling. We provide new connections between M_{nn} and M_{mon} by using simple combinatorial bijections, and we prove that M_{nn} is always rapidly mixing for two general classes of positively biased {p_{xy}}. More significantly, we also prove that the general conjecture is false by exhibiting values for the p_{xy}, with 1/2 <= p_{xy} <= 1 for all x < y, but for which the transposition chain will require exponential time to converge. Finally, we consider a model of colloids, which are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. This clustering has proved elusive to verify, since all local sampling algorithms are known to be inefficient at high density, and in fact a new nonlocal algorithm was recently shown to require exponential time in some cases. We characterize the high and low density phases for a general family of discrete interfering binary mixtures by showing that they exhibit a "clustering property" at high density and not at low density. The clustering property states that there will be a region that has very high area, very small perimeter, and high density of one type of molecule. Special cases of interfering binary mixtures include the Ising model at fixed magnetization and independent sets.

In the talk some problems related with the famous Chernoff square root of n - lemma in the theory of approximation of some semi-groups of operators will be discussed. We present some optimal bounds in these approximations (one of them is Euler approximation) and two new classes of operators, generalizing sectorial and quasi-sectorial operators will be introduced. The talk is based on two papers [V. Bentkus and V. Paulauskas, Letters in Math. Physics, 68, (2004), 131-138] and [V. Paulauskas, J. Functional Anal., 262, (2012), 2074-2099]

Erd\H{o}s and Rothschild asked to estimate the maximum number, denotedby $h(n,c)$, such that every $n$-vertex graph with at least $cn^2$edges, each of which is contained in at least one triangle, mustcontain an edge that is in at least $h(n,c)$ triangles. In particular,Erd\H{o}s asked in 1987 to determine whether for every $c>0$ there is$\epsilon>0$ such that $h(n,c)>n^{\epsilon}$ for all sufficientlylarge $n$. We prove that $h(n,c)=n^{O(1/\log \log n)}$ for every fixed$c<1/4$. This gives a negative answer to the question of Erd\H{o}s,and is best possible in terms of the range for $c$, as it is knownthat every $n$-vertex graph with more than $n^2/4$ edges contains anedge that is in at least $n/6$ triangles.Joint work with Jacob Fox.

In the talk we demonstrate the usefulness of the so-called Beveridge-Nelson decomposition in asymptotic analysis of sums of values of linear processes and fields. We consider several generalizations of this decomposition and discuss advantages and shortcomings of this approach which can be considered as one of possible methods to deal with sums of dependent random variables. This decomposition is derived for linear processes and fields with the continuous time (space) argument. The talk is based on several papers, among them [V. Paulauskas, J. Multivar. Anal. 101, (2010), 621-639] and [Yu. Davydov and V. Paulauskas, Teor. Verojat. Primenen., (2012), to appear]

I present a new class of vertex cover and set cover games, with the price of anarchy bounds matching the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs. In particular, the price of anarchy is 2 for vertex cover. The basic intuition is that the members of the vertex cover form a Mafia that has to "protect" the graph, and may ask ransoms from their neighbors in exchange for the protection. These ransoms turn out to capture a good dual solution to the linear programming relaxation. For linear costs we also exhibit linear time best response dynamics that converge that mimic the classical greedy approximation algorithm of Bar-Yehuda and Even. This is a joint work with Georgios Piliouras and Tomas Valla.

The correlation of gaps in dimer systems was introduced in 1963 by Fisher and Stephenson, who looked at the interaction of two monomers generated by the rigid exclusion of dimers on the closely packed square lattice. In previous work we considered the analogous problem on the hexagonal lattice, and we extended the set-up to include the correlation of any finite number of monomer clusters. For fairly general classes of monomer clusters we proved that the asymptotics of their correlation is given, for large separations between the clusters, by a multiplicative version of Coulomb's law for 2D electrostatics. However, our previous results required that the monomer clusters consist (with possibly one exception) of an even number of monomers. In this talk we determine the asymptotics of general defect clusters along a lattice diagonal in the square lattice (involving an arbitrary, even or odd number of monomers), and find that it is given by the same Coulomb law. Of special interest is that one obtains a conceptual interpretation for the multiplicative constant, as the product of the correlations of the individual clusters. In addition, we present several applications of the explicit correlation formulas that we obtain.