Mathematics

Mathematical theory of wind-driven water waves

Samuel Walsh (Missouri-Columbia)

Abstract: It is easy to see that wind blowing over a body of water can create waves. But this simple observation leads to a more fundamental question: Under what conditions on the velocity profile of the wind will persistent surface water waves be generated? This problem has been studied intensively in the applied fluid dynamics community since the first efforts of Kelvin in 1871. In this talk, we will present a mathematical treatment of the predominant model for wind-wave generation, the so-called quasi-laminar model of J. Miles. Essentially, this entails determining the (linear) stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We give a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, a complete proof of the celebrated instability criterion of Miles. In particular, our analysis incorporates both the effects of surface tension and a vortex sheet on the air--sea interface. We are thus able to give a unified equation connecting the Kelvin--Helmholtz and quasi-laminar models of wave generation.

Equivalence relations and the Borel reducibility hierarchy

Anush Tserunyan (UIUC)

Abstract: For the past twenty years, a major focus of descriptive set theory has been the study of equivalence relations on Polish spaces that are definable (Borel, analytic, etc.) when viewed as sets of pairs; e.g. orbit equivalence relations induced by continuous actions of Polish groups are analytic. This study provides appropriate framework and tools for understanding the nature of classification of mathematical objects (measure-preserving transformations, unitary operators, Riemann surfaces, etc.) up to some notion of equivalence (isomorphism, conjugacy, conformal equivalence, etc.), and measuring the complexity of such classification problems. Due to its broad scope, it has natural interactions with other areas of mathematics, such as ergodic theory and topological dynamics, functional analysis and operator algebras. In this talk, I will give an introduction to this fascinating subject.

Stochastic Navier-Stokes Equation with Levy noise

Sivaguru S. Sritharan   [email] (DRCSI, Naval Postgraduate School)

Abstract: The subject of stochastic Navier-Stokes equation has grown in to a rich area in the past couple of decades with topics of research ranging from martingale solutions, invariant measures, large deviations, Malliavin calculus and control theory. There is also a natural connections to PDEs in infinite dimensions. In this talk we will give an introduction to this subject, touch upon some of the research areas, and indicate some open problems.

Subdivisions of a large clique in C_6-free graphs

Maryam Sharifzadeh (UIUC Math)

Abstract: Mader conjectured that every $C_{4}$-free graph has a subdivision of a clique of order linear in its average degree. We show that every $C_{6}$-free graph has such a subdivision of a large clique. We also prove the dense case of Mader's conjecture in a stronger sense, i.e., for every $c$, there is a $c'$ such that every $C_{4}$-free graph with average degree $cn^{1/2}$ has a subdivision of a clique $K_{\ell}$ with $\ell=\lfloor c'n^{1/2}\rfloor$ where every edge is subdivided exactly $3$ times. Joint work with József Balogh and Hong Liu.