Since introduced by Wigner, random matrix theory has become a powerful tool in mathematical phyics. The subject now also plays important roles in various fileds such as combinatorics, probability theory, statistical physics, number theory, nuclear physics, game theory, wireless communication, etc. In this talk, I will explain some important results on random matrices such as Wigner semi-circle law and Dyson sine kernel. Some applications will also be introduced.

We will survey how to improve computational efficiency in various pairing computations.

The zeta-function of a variety defined over the integers encoded the number of solutions with coefficients in all finite fields. Surprisingly, the value of this zeta function is related to other interesting invariants of the variety. A good example is the analytic class number formula, which related the value of the Dedekind zeta-function of a number field to the class number and other invariants. We will discuss generalizations of this to varieties over finite fields.

Given a hypergraph H = (V,E), a coloring of its vertices is said to be conflict-free if for every hyperedge S ∈ E there is at least one vertex whose color is distinct from the colors of all other vertices in S. When all hyperedges in H have cardinality two (i.e., when H is a simple graph), this coloring coincides with the classical graph coloring. The study of this coloring is motivated by frequency assignment problems in wireless networks and several other applications. We will survey this notion and introduce some fascinating open problems.

Manin showed how the Brauer group of schemes together with class field theory(reciprocity laws) accounts for many counterexamples to the Hasse principle(local-global principle) and to weak approximation for rational points of projective varieties. Only recently did one start to investigate  an analogous approach for integral points of affine varieties. Here the relevant approximation property is strong approximation,a generalization of the chinese remainder theorem. I shall recall what is known or conjectured for rational points.
I will then go on to describe what has been achieved for integral points :
unconditional results for many classes of homogeneous spaces of linear algebraic groups,
with a connexion to the classical study of integral quadratic forms(F. Xu and the speaker, D. Harari, M. Borovoi, C. Demarche) ; computations and conjectures for curves (D. Harari and F. Voloch) ; computations for certain  cubic surfaces (O. Wittenberg and the speaker).
I will in particular discuss the classical problem of representation of an integer as sum of three cubes of integers.

Solitons are nonlinear solutions that maintain its shape and travel at a constant speed. They widely arise in focusing nonlinear dispersive equations. They are occurred as a balance of nonlinear reinforcing effects and dispersive effects in the medium. In this talk, I will briefly review long-standing history of solitons and discuss past and current issues around soliton solutions. It will include a brief explanation to the inverse scattering method, the orbital (asymptotic) stability, and then an introduction to the 'soliton resolution conjecture'.