Georgia Institute of Technology College of Sciences

Groups, rings, modules, and compact Hausdorff spaces have underlying sets ("forgetting" structure) and admit "free" constructions. Moreover, each type of object is completely characterized by the shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding provides formulas for direct and inverse limits. After we describe a typical encounter with adjunctions, monads, and their algebras, we introduce a new "homotopy coherent" version of this adjoint duality together with a graphical calculus that is used to define a homotopy coherent algebra in quite general contexts, such as appear in abstract homotopy theory or derived algebraic geometry.

In the talk, an asymptotic behaviour of first order properties of the Erdos-Renyi random graph G(n,p) will be considered. The random graph obeys the zero-one law if for each first-order property L either its probability tends to 0 or tends to 1. The random graph obeys the zero-one k-law if for each property L which can be expressed by first-order formula with quantifier depth at most k either its probability tends to 0 or tends to 1. Zero-one laws were proved for different classes of functions p=p(n). The class n^{-a} is at the top of interest. In 1988 S. Shelah and J.H. Spencer proved that the random graph G(n,n^{-a}) obeys zero-one law if a is positive and irrational. If a is rational from the interval (0,1], then G(n,n^{-a}) does not obey the zero-one law. I obtain zero-one k-laws for some rational a from (0,1]. For any first-order property L let us consider the set S(L) of a from (0,1) such that a probability of G(n,n^{-a}) to satisfy L does not converges or its limit is not zero or one. Spencer proved that there exists L such that S(L) is infinite. Recently in the joint work with Spencer we obtain new results on a distribution of elements of S(L) and its limit points.

Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. This talk will be focused on one of the famous mean-field spin glasses, the Sherrington-Kirkpatrick model. We will present results on the conjectured properties of the Parisi measure including its uniqueness and quantitative behaviors. This is based on joint works with A. Auffinger.

The existence of large BV (total variation) solution for compressible Euler equations in one space dimension is a major open problem in the hyperbolic conservation laws, where the small BV existence was first established by James Glimm in his celebrated paper in 1964. In this talk, I will discuss the recent progress toward this longstanding open problem joint with my collaborators. The singularity (shock) formation and behaviors of large data solutions will also be discussed.

It is known that certain medium, for example electromagnetic field and Bose Einstein condensate, has positive speed of sound. It is observed that if the medium is in its equilibrium state, then an invading subsonic particle will slow down due to friction; and the speed of a supersonic particle will slow down to the speed of sound and the medium will radiate. This is called Cherenkov radiation. It has been widely discussed in physical literature, and applied in experiments. In this talk I will present some rigorous mathematical results. Joint works with Juerg Froehlich, Israel Michael Sigal, Avy Soffer, Daniel Egli, Arick Shao.