For the past several years, studies on affine processes have been worked out by many researchers about moment explosions, implied volatilities, and long-term behaviors. Recently, Glasserman and Kim, and Keller-Ressel investigated the moment explosions of the canonical affine models of Dai and Singleton, and general two-factor affine stochastic volatility models, respectively, and presented their long-term behaviors. On the other hand, Benaim and Friz, and Lee showed that implied volatilities at extreme strikes are linked to the moment explosions of stock prices at given option maturities. In this work, we characterize the regions in which moment explosions happen for some time or at a given time, and relate them to the long-term behavior of stock prices and to implied volatilities, extending previous works on moment explosions for affine processes.

Contact geometry can be regarded as an odd-dimensional counterpart of
symplectic geometry. In 3-dimensional case we have a dichotomy of
contact structures: tight vs. overtwisted. Overtwisted contact
structures are not very interesting from a topological point of view,
but tight structures have intimate and subtle relationship with
topology. In this talk, I introduce contact structures of odd
dimensional manifolds and some interesting results about contact
topology including the result of Colin, Giroux, and Honda that states a
3-manifold admits infinitely many tight contact structures if and only
if it has an incompressible torus.

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. The study of this notion is motivated by frequency assignment problems in wireless networks. We introduce and study the list-coloring (or choice) version of this notion. Joint work with Panagiotis Cheilaris.

We'll discuss joint work with J. Colliander, G. Staffilani, H. Takaoka, and T.Tao. The particular result is motivated by the much harder goal of understanding how energy can be exchanged between the different modes (or `frequencies') of nonlinear partial differential equations (PDE). The energy of solutions to such PDE is often conserved, but one can ask whether it's possible for smoother norms to grow in time. Such growth provides at least some quantitative measure of how energy might cascade from lower modes to arbitrarily high modes of the solution - a phenomenon sometimes described as "turbulence".

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.

Given a Hamiltonian Lie group action on a symplectic manifold, I will explain the construction of a quantum deformation of the Kirwan map. It is defined by counting solutions of the symplectic vortex equations, which are a gauge theoretical deformation of the Cauchy-Riemann equations. Based on this map, I will formulate a quantum abelianization conjecture (joint work with Chris Woodward).

Recent work of the authors and colleagues on the turbulent mixing of compressible fluids is developed
and extended with an emphasis on the multiscale aspects of this work.

Hilbert proposed his 12-th problem at the Paris ICM in 1900 that

"it may be possible to find, for an arbitrary number field $K$, a transcendental function whose values generate the abelian extensions of $K$."

 In 1920's Takagi established the existence of Class Fields. Given a number field $K$ and a generalized ideal class group $G$ of $K$, his theory asserts that there exists a unique abelian extension of $K$ with $G$ its Galois group. Such an abelian extension is called the

class field of $K$ corresponding to $G$. Unfortunately, his argument could not provide us any explicit algorithm to construct class fields.

Therefore, we shall talk about Complex Multiplication Theorem which enables us to construct class fields over imaginary quadratic fields explicitly, due to Kronecker, Hecke, Hasse, Deuring and Shimura.

There is no general theory for solving nonlinear partial differential equations. The calculus of variations and maximum principle are useful methods to solve such nonlinear elliptic partial differential equations.
In this talk, I will briefly review the basic concepts of the calculus of variations and maximum principle and prove the nonexistence of solutions for some elliptic system using the maximum principle.

In this talk, we discuss different approaches to the
well-posedness problem of the one-dimensional cubic nonlinear
Schrodinger equation (NLS.)  First, we review how one can obtain local
well-posedness in H^s, s>1/2, by energy method and in L^2 by
Bourgain's periodic Strichartz estimate.  Then, we show that NLS is
"ill-posed" below L^2.

In order to construct solutions below L^2, we renormalize the
nonlinearity (called Wick ordering.)   Then, we consider the
renomarlized NLS with randomized initial data and show that it is
locally well-posed almost surely in H^s, s> -1/3.  Finally, we discuss
how to extend such solutions to global ones by combining probabilistic
local well-posedness theory with Bourgain's high-low argument (= an
argument for showing (deterministic)  global well-posedness, developed
by Bourgain, and it is a precursor to the I-method.)  This part is a
joint work with James Colliander (University of Toronto.)

There is a summary of these results in a survey paper with Catherine
Sulem (University of Toronto.)

We confirm that Entropy Conjecture holds
for every continuous self-map
of a compact K( π, 1) manifold
with the fund. gr. 

π virtually nilpotent,
e.g. for every continuous map of an infra-nilmanifold.
In fact, a lower estimate of the topological entropy
by the logarithm of the spectral radius of exterior power
of an assoc. ”linearization matrix” with integer entries.
This & estimates of Mahler measure of polynom.,
give some absolute lower bounds for the entropy.

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.

In this talk, we consider the relativistic Vlasov equations coupled with some nonlinear electromagnetic fields. The nonlinear electromagnetic fields considered here can be regarded as some special cases of the nonlinear electromagnetic field of the Born-Infeld theory. We will show that classical solutions exist globally in time for some
restricted cases.

In this seminar, I will introduce to wave equation. Wave equation is an important partial differential equation that describes the propagation of a variety of wave, such as sound waves, light waves and water waves.

Usually, I proved the existence of solutions of wave equation by using Faedo-Galerkin method. I shall explain to Faedo-Galerkin method. Next, I will apply to wave equation by using Faedo-Galerkin method.

The Boltzmann-BGK model is widely used in the kinetic theory of gases as a qualitatively correct model for the Boltzmann equation. Recently, a semi-Lagrangian scheme for the BGK model was suggested and tested successfully for various flow problems arsing in gase dynamics.

In this talk, I will breifly review the basic aspects of collisional kinetic equations and show that the approximate distribution function of the scheme converges to the smooth solution of the BGK model.

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'.

In a non-relativistic case, the dynamics of N-particle system is governed by the Schrödinger equation. The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this talk, I will introduce estimates on the rate of convergence of the quantum mechanical evolution towards the Hartree dynamics. Some basic concepts and tools of mathematical physics will also be covered.

It is an important conjecture that the Chow group of a smooth projective variety over a number field is a finitely generated Z-module, and no one knows how to solve it so far. On the other hand people considered two modified questions which are more accessible. One is whether the Chow group modulo n is finite or not, and the other is whether the torsion part is finite or not.

In this talk I survey recent progress about these two questions made by Shuji Saito, Kanetomo Sato and the speaker

The narrow escape problem consists of deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibit the nonlinear interaction of many small absorbing targets. 

I will talk about problems on a Siegel type domain in the complex Euclidean space. Here the Siegel type domain means the Siegel domain of first and second type in the sense of Pyatetskii-Shapiro or a domain defined by a weighted homogeneous polynomial. It has been known that most of model objects in Complex Geometry of negatively curved Kaehler manifolds or Complex Analysis of bounded domains are biholomorphic to Siegel type domain. Although Siegel type domains are unbounded in C^n, they have many affine automorphisms, so the Siegel type realization of models has been employed in wide area of Several Complex Variables. In this talk I will discuss a geometric problem on the realization of a Siegel type domain as a bounded domain.

 It is clear that the physically meaningful quantity is not the potential itself, but its difference or gradient. However, the theory is based on the potential itself. In this talk the speaker will introduce new concept of relative potential.

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a random graph. They have been a subject of intensive study during the last two decades and have seen numerous applications both in Combinatorics and Computer Science.

Starting with the work of Thomason and Chung, Graham, and Wilson, there have been many works which established the equivalence of various properties of graphs to quasi-randomness. In this talk, I will give a survey on this topic, and provide a new condition which guarantees quasi-randomness. This result answers an open question raised independently by Janson, and Shapira and Yuster.

Joint work with Hao Huang (UCLA).

During last few years, there has been extensive study on 'Optimal mass transportation' problem. In this talk, we will describe some theories on this area and show how these can be applied to (evolutionary) Partial Differential Equations.

In this talk we introduce the theory of graph skein modules associted with the Yamada polynomial. We compute the graph algebra in some simple cases. Then, we apply graph skein techniques to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime.