Georgia Institute of Technology College of Sciences

Gas-liquid transition is one of the most basic problem to study in equilibrium phase transitions. In the pressure-temperature phase diagram, the gas-liquid coexistence curve terminates at a critical point C, also called the Andrews critical point. It is, however, still an open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. To answer this basic question, using the Landau's mean field theory and the Le Chatelier principle, a dynamic model for the gas-liquid phase transitions is established. With this dynamic model, we are able to derive a theory on the Andrews critical point C: 1) the critical point is a switching point where the phase transition changes from the first order with latent heat to the third order, and 2) the liquid-gas phase transition going beyond Andrews point is of the third order. This clearly explains why it is hard to observe the liquid-gas phase transition going beyond the Andrews point. In addition, the study suggest an asymmetry principle of fluctuations, which appears also in phase transitions in ferromagnetic systems. The analysis is based on the dynamic transition theory we have developed recently with the philosophy to search the complete set of transition states. The theory has been applied to a wide range of nonlinear problems. A brief introduction for this theory will be presented as well. This is joint with Tian Ma.

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this talk, which is based on a survey with Benny Sudakov, we discuss some of these applications.

We study the spectral properties of matrices of long-range percolation model. These are N*N random real symmetric matrices H whose elements are independent random variables taking zero value with probability 1-\psi((i-j)/b), b\in \R^{+}, where \psi is an even positive function with \psi(t)<1 and vanishing at infinity. We show that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the ensemble we study in the limit N,b\to\infty. In the second part, we study the leading term of the correlation function of the resolvent G(z)=(H-z)^{-1} with large enough |Imz| in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3<\alpha<1. We show that this leading term, when considered in the local spectral scale leads to an expression found earlier by other authors for band random matrix ensembles. This shows that the ensemble we study and that of band random matrices belong to the same class of spectral universality.

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^h) copies of H can be made H-free by removing o(n^2) edges. We give a new proof which avoids Szemeredi’s regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.