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Series: School of Mathematics Colloquium

In this talk I will outline recent results of G-Q Chen, Dehua Wang, and me on the problem of isometric embedding a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Remarkably there is very pretty duality between this problem and the equations of steady 2-D gas dynamics. Compensated compactness (L.Tartar and F.Murat) yields proof of existence of solutions to an initial value problem when the prescribed metric is the one associated with the catenoid.

Series: School of Mathematics Colloquium

A new estimate on weak solutions of the Navier-Stokes equations in three dimensions gives some information about the partial regularity of solutions. In particular, if energy concentration takes place, the dimension of the microlocal singular set cannot be too small. This estimate has a dynamical systems proof. These results are joint work with M. Arnold and A. Biryuk.

Series: School of Mathematics Colloquium

This is joint work with Andrei Okounkov. The ``honeycomb dimer model'' is a natural model of discrete random surfaces in R^3. It is possible to write down a ``Law of Large Numbers" for such surfaces which describes the typical shape of a random surface when the mesh size tends to zero. Surprisingly, one can parameterize these limit shapes in a very simple way using analytic functions, somewhat reminiscent of the Weierstrass parameterization of minimal surfaces. This is even more surprising since the limit shapes tend to be facetted, that is, only piecewise analytic. There is a large family of boundary conditions for which we can obtain exact solutions to the limit shape problem using algebraic geometry techniques. This family includes the (well-known) solution to the limit shape of a ``boxed plane partition'' and has many generalizations.

Series: School of Mathematics Colloquium

The study of partition identities has a long history going back to Euler, with applications ranging from Analysis to Number Theory, from Enumerative Combina- torics to Probability. Partition bijections is a combinatorial approach which often gives the shortest and the most elegant proofs of these identities. These bijections are then often used to generalize the identities, find "hidden symmetries", etc. In the talk I will present a modern approach to partition bijections based on the geometry of random partitions and complexity ideas.

Series: School of Mathematics Colloquium

Molecular topology is an application of graph theory to fields like chemistry, biology and pharmacology, in which the molecular structure matters. Its scope is the topological characterization of molecules by means of numerical invariants, called topological indices, which are the main ingredient of the molecular topological models. These models have been instrumental in the discovery of new applications of naturally occurring molecules, as well as in the design of synthetic molecules with specific chemical, biological or pharmacological properties. The talk will focus on pharmacological applications.

Series: School of Mathematics Colloquium

Consider the 2-d ideal incompressible fluid moving inside a bounded domain (say 2-d torus). It is described by 2-d Euler equations which have unique global solution; thus, we have a dynamical system in the space of sufficiently regular incompressible vector fields. The global properties of this system are poorly studied, and, as much as we know, paradoxical. It turns out that there exists a global attractor (in the energy norm), i.e. a set in the phase space attracting all trajectories (in spite the fact that the system is conservative). This apparent contradiction leads to some deep questions of non-equilibrium statistical mechanics.

Series: School of Mathematics Colloquium

In this talk we will review some of the global asymptotic results obtained during the last two decades in the theory of the classical Painleve equations with the help of the Isomonodromy - Riemann-Hilbert method. The results include the explicit derivation of the asymptotic connection formulae, the explicit description of linear and nonlinear Stokes phenomenon and the explicit evaluation of the distribution of poles. We will also discuss some of the most recent results emerging due to the appearance of Painleve equations in random matrix theory. The Riemann-Hilbert method will be outlined as well.

Series: School of Mathematics Colloquium

Issai Schur (1918) considered a class of polynomials with integer coefficients and simple zeros in the closed unit disk. He studied the limit behavior of the arithmetic means s_n for zeros of such polynomials as the degree n tends to infinity. Under the assumption that the leading coefficients are bounded, Schur proved that \limsup_{n\to\infty} |s_n| \le 1-\sqrt{e}/2. We show that \lim_{n\to\infty} s_n = 0 as a consequence of the asymptotic equidistribution of zeros near the unit circle. Furthermore, we estimate the rate of convergence of s_n to 0. These results follow from our generalization of the Erdos-Turan theorem on discrepancy in angular equidistribution of zeros. We give a range of applications to polynomials with integer coefficients. In particular, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding \liminf_{n\to\infty} s_n was developed further by Siegel and many others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line.

Series: School of Mathematics Colloquium

In this talk, we discuss 1.) the nonlinear instability and unstable manifolds of steady solutions of the Euler equation with fixed domains and 2.) the evolution of free (inviscid) fluid surfaces, which may involve vorticity, gravity, surface tension, or magnetic fields. These problems can be formulated in a Lagrangian formulation on infinite dimensional manifolds of volume preserving diffeomorphisms with an invariant Lie group action. In this setting, the physical pressure turns out to come from the combination of the gravity, surface tension, and the Lagrangian multiplier. The vorticity is naturally related to an invariant group action. In the absence of surface tension, the well-known Rayleigh-Taylor and Kelvin-Helmholtz instabilities appear naturally related to the signs of the curvatures of those infinite dimensional manifolds. Based on these considerations, we obtain 1.) the existence of unstable manifolds and L^2 nonlinear instability in the cases of the fixed domains and 2.) in the free boundary cases, the local well-posedness with surface tension in a rather uniform energy method. In particular, for the cases without surface tension which do not involve hydrodynamical instabilities, we obtain the local existence of solutions by taking the vanishing surface tension limit.

Series: School of Mathematics Colloquium

We prove that a smooth compact submanifold of codimension $2$ immersed in $R^n$, $n>2$, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman. These results consist of joint works with Stephanie Alexander and Jeremy Wong, and Robert Greene.