This is joint work with Alan Noell and Sivaguru Ravisankar. We prove a Hartogs-Bochner type theorem for a bounded domain $U$ with smooth boundary in ${\mathbb C}^n \times {\mathbb R}$ with nodegenerate, flat, and elliptic CR singularities (all natural conditions for the problem). That is, a CR function on the boundary $\partial U$ extends to a CR function on $U$, smooth up to the boundary.

Self-avoiding walk of length n on the integer lattice $Z^d$ is the uniform measure on nearest-neighbour walks in $Z^d$ that begin at the origin and are of length $n$. If such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. The result of doing so is a self-avoiding polygon. We investigate the numbers of self-avoiding walks, polygons, and in particular the "closing" probability that a length n self-avoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo Duminil-Copin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on self-avoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most $n^{-1/2 + o(1)}$ in any dimension at least two. Developing a quite different method of polygon joining employed by Madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. We further make use of the snake method and polygon joining technique at once to prove upper bounds on the closing probability below $n^{-1/2}$ in the two-dimensional setting.

Fundamental biological molecular processes, such as protein folding, molecular recognition, and molecular assemblies, are mediated by surrounding aqueous solvent (water or salted water). Continuum description of solvent is an efficient approach to understanding such processes. In this work, we develop a solvent fluid model and computational methods for solvent dynamics and solute-solvent interface motion. The key components in our model include the Stokes equation for the incompressible solvent fluid which governs the motion of the solute-solvent interface, the ideal-gas law for solutes, and the balance on the interface of viscous force, surface tension, van der Waals type dispersive force, and electrostatic force. We use the ghost fluid method to discretize the flow equations that are reformulated into a set of Poisson equations, and design special numerical boundary conditions to solve such equations to allow the change of solute volume. We move the interface with the level-set method. To stabilize our schemes, we use the Schur complement and least-squres techniques. Numerical tests in both two and three-dimensional spaces will be shown to demonstrate the convergence of our method, and to demonstrate that this new approach can capture dry and wet hydration states as observed in experiment and molecular dynamics simulations.

Two of the most important examples in $II_1$ factors are the amenable $II_1$ factor and the free group factors. The first one is well-understood, thanks to Connes' fundamental work, while the structure of free group factors is still under intense study. Regarding amenable subalgebras inside free group factors, Jesse Peterson conjectured that any diffuse amenable subalgebra of a free group factor has a unique maximal amenable extension. In this talk I will show that any diffuse subalgebra of the radial masa in a free group factor with finitely many generators, has a unique maximal amenable extension.

We will describe the problem of detecting linear dependence of points in Mordell-Weil groups A(F) of abelian varieties. This is done via reduction maps. We determine the sufficient conditions for the reduction maps to detect linear dependence in A(F).
We also show that our conditons are very close to be or perhaps are the best possible. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite number of reductions. The methods combine applications of transcedental, l-adic and mod v techniques. This is joint work with G. Banaszak.

Let $G/K$ be a Riemannian symmetric space. Its complexification $G^C / K^C$ is a Stein manifold, and left-translations by $G$ are holomorphic transformations of $G^C / K^C$. In this setting, invariant domains and their envelopes of holomorphy are natural objects of study. If $G/K$ is compact, then every invariant domain $D$ in $G^C / K^C$ intersects a complex torus orbit in a lower dimensional Reinhardt domain $\Omega_D$. In this case, complex analytic properties of $D$ can be expressed in terms of those of $\Omega_D$. If $G/K$ is a non-compact, then the situation is fully understood only in the rank-one case. In this talk we present some univalence results for the envelope of holomorphy of a $G$-invariant domain in $G^C / K^C$, when the space $G/K$ is a non-compact Hermitian symmetric space (joint work with A. Iannuzzi).