Georgia Institute of Technology College of Sciences

We consider a Benney-type system modeling short wave-long wave interactions in compressible viscous fluids under the influence of a magnetic field. Accordingly, this large system now consists of the compressible MHD equations coupled with a nonlinear Schodinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R^3 when the initial data are small smooth perturbations of an equilibrium state. An important point here is that, instead of the simpler case having zero as the equilibrium state for the magnetic field, we consider an arbitrary non-zero equilibrium state B for the magnetic field. This is motivated by applications, e.g., Earth's magnetic field, and the lack of invariance of the MHD system with respect to either translations or rotations of the magnetic field. The usual time decay investigation through spectral analysis in this non-zero equilibrium case meets serious difficulties, for the eigenvalues in the frequency space are no longer spherically symmetric. Instead, we employ a recently developed technique of energy estimates involving evolution in negative Besov spaces, and combine it with the particular interplay here between Eulerian and Lagrangian coordinates. This is a joint work with Junxiong Jia and Ronghua Pan.

In recent years, small-time asymptotic methods have attracted much attention in mathematical finance. Such asymptotics are especially crucial for jump-diffusion models due to the lack of closed- form formulas and efficient valuation procedures. These methods have been widely developed and applied to diverse areas such as short-time approximations of option prices and implied volatilities, and non-parametric estimations based on high-frequency data. In this talk, I will discuss some results on the small-time asymptotic behavior of some Levy functionals with applications in finance.

In elementary calculus, we learn that (1+z/n)^n has limit exp(z) as n approaches infinity. This type of scaling limit arises in many contexts - from approximation theory to universality limits in random matrices. We discuss some examples.

Let A be a mxn matrix with singular value decomposition A=UDV', where the columns of U are left singular vectors and columns of V are right singular vectors of A. Suppose X is a mxn noise matrix whose entries are i.i.d. Gaussian random variables and consider A'=A+X. Let u_k be the k-th left singular vector of A and u'_k be its counterpart of A'. We develop sharp upper bounds for concentration of linear forms for the right singular vectors of A'.The talk is based on a joint work with Vladimir Koltchinskii.