Georgia Institute of Technology College of Sciences

Collective movement is one of the most prevailing observations in nature. Yet, despite considerable progress, many of the theoretical principles underlying the emergence of large scale synchronization among moving individuals are still poorly understood. For example, a key question in the study of animal motion is how the details of locomotion, interaction between individuals and the environment contribute to the macroscopic dynamics of the hoard, flock or swarm. The talk will present some of the prevailing models for swarming and collective motion with emphasis on stochastic descriptions. The goal is to identify some generic characteristics regarding the build-up and maintenance of collective order in swarms. In particular, whether order and disorder correspond to different phases, requiring external environmental changes to induce a transition, or rather meta-stable states of the dynamics, suggesting that the emergence of order is kinetic. Different aspects of the phenomenon will be presented, from experiments with locusts to our own attempts towards a statistical physics of collective motion.

We prove results concerning the equidistribution of some "sparse" subsets of orbits of horocycle flows on $SL(2, R)$ mod lattice. As a consequence of our analysis, we recover the best known rate of growth of Fourier coefficients of cusp forms for arbitrary noncompact lattices of $SL(2, R)$, up to a logarithmic factor. This talk addresses joint work with Livio Flaminio, Giovanni Forni and Pankaj Vishe.

I will describe several old and new applications of topological and algebraic methods in the derivation of combinatorial results. In all of them the proofs provide no efficient solutions for the corresponding algorithmic problems.

We find a good characterization for the following problem: Given a rational row vector c and a lattice L in R^n which contains the integer lattice Z^n, do all lattice points of L in the half-open unit cube [0,1)^n lie on the hyperplane {x in R^n : cx = 0}? This work generalizes a theorem due to G. K. White, which provides sufficient and necessary conditions for a tetrahedron in R^3 with integral vertices to have no other integral points. Our approach is based on a novel proof of White's result using number-theoretic techniques due to Morrison and Stevens. In this talk, we illustrate some of the ideas and describe some applications of this problem.