Georgia Institute of Technology College of Sciences

We formulate a plasma model in which negative ions tend to a fixed, spatially-homogeneous background of positive charge. Instead of solutions with compact spatial support, we must consider those that tend to the background as x tends to infinity. As opposed to the traditional Vlasov-Poisson system, the total charge and energy are thus infinite, and energy conservation (which is an essential component of global existence for the traditional problem) cannot provide bounds for a priori estimates. Instead, a conserved quantity related to the energy is used to bound particle velocities and prove the existence of a unique, global-in-time, classical solution. The proof combines these energy estimates with a crucial argument which establishes spatial decay of the charge density and electric field.

Biological aggregations such as insect swarms, bird flocks, and fish schools are arguably some of the most common and least understood patterns in nature. In this talk, I will discuss recent work on swarming models, focusing on the connection between inter-organism social interactions and properties of macroscopic swarm patterns. The first model is a conservation-type partial integrodifferential equation (PIDE). Social interactions of incompressible form lead to vortex-like swarms. The second model is a high-dimensional ODE description of locust groups. The statistical-mechanical properties of the attractive-repulsive social interaction potential control whether or not individuals form a rolling migratory swarm pattern similar to those observed in nature. For the third model, we again return to a conservation-type PIDE and, via long- and short-wave analysis, determine general conditions that social interactions must satisfy for the population to asymptotically spread, contract, or reach steady state.

In this talk, we will introduce the classical Cramer's Theorem. The pattern of proof is one of the two most powerful tools in the theory of large deviations. Namely, the upper bound comes from optimizing over a family of Chebychef inequalities; while the lower bound comes from introducing a Radon-Dikodym factor in order to make what was originally "deviant" behavior look like typical behavior. If time permits, we will extend the Cramer's Theorem to a more general setting and discuss the Sanov Theorem. This talk is based on Deuschel and Stroock's .

Since the seminal work of Erdos and Renyi the phase transition of the largest components in random graphs became one of the central topics in random graph theory and discrete probability theory. Of particular interest in recent years are random graphs with constraints (e.g. degree distribution, forbidden substructures) including random planar graphs. Let G(n,M) be a uniform random graph, a graph picked uniformly at random among all graphs on vertex set [n]={1,...,n} with M edges. Let P(n,M) be a uniform random planar graph, a graph picked uniformly at random among all graphs on vertex set [n] with M edges that are embeddable in the plane. Erodos-Renyi, Bollobas, and Janson-Knuth-Luczak-Pittel amongst others studied the critical behaviour of the largest components in G(n,M) when M= n/2+o(n) with scaling window of size n^{2/3}. For example, when M=n/2+s with s=o(n) and s \gg n^{2/3}, a.a.s. (i.e. with probability tending to 1 as n approaches \infty) G(n,M) contains a unique largest component (the giant component) of size (4+o(1))s. In contract to G(n,M) one can observe two critical behaviour in P(n,M), when M=n/2+o(n) with scaling window of size n^{2/3}, and when M=n+o(n) with scaling window of size n^{3/5}. For example, when M=n/2+s with s = o(n) and s \gg n^{2/3}, a.a.s. the largest component in P(n,M) is of size (2+o(1))s, roughly half the size of the largest component in G(n,M), whereas when M=n+t with t = o(n) and t \gg n^{3/5}, a.a.s. the number of vertices outside the giant component is \Theta(n^{3/2}t^{-3/2}). (Joint work with Tomasz Luczak)