TBA

Let P be a homogeneous polynomial in N+1 complex variables of degree d. The logarithmic Mahler Measure of P (denoted by m(P) ) is the integral of log|P| over the sphere in C^{N+1} with respect to the usual Hermitian metric and measure on the sphere.  Now let X be a smooth variety embedded in CP^N by a high power of an ample line bundle and let $\Delta$ denote a generalized discriminant of X wrt the given embedding , then $\Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables.  In this talk I will discuss work in progress whose aim is to find an asymptotic expansion of m(\Delta) in terms of elementary functions of the degree of the embedding.  

We will discuss the isotopy problem of symplectic forms in a fixed symplectic class on compact four manifolds.

We will introduce a nonlinear Hodge flow for a general approach. We will also discuss the hypersymplectic flow, introduced by Fine-Yao to study hypersymplectic four manifolds.

Abstract: We present a uniform description of  SU(3) structures in dimension 6 as well as  G_2 structures in dimension 7 in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach sheds new light on connections with torsion and their invariants.

It is well known that the semi-linear elliptic Allen-Cahn equation arising in phase transition theory is closely related to the theory of minimal surfaces. Earlier works of Modica and Sternberg et. al in the 1970’s studied minimizing solutions in the framework of De Giorgi’s Gamma-convergence theory. The more profound regularity theory for stationary and stable solutions were obtained by the deep work of Tonegawa and Wickramasekera, building upon the celebrated Schoen-Simon regularity theory for stable minimal hypersurfaces. This is recently used by Guaraco to develop a new approach to min-max constructions of minimal hypersurfaces via the Allen-Cahn equation. In this talk, we will discuss about the boundary behaviour for limit interfaces arising in the Allen-Cahn equation on bounded domains (or, more generally, on compact manifolds with boundary). In particular, we show that, under uniform energy bounds, any such limit interface is a free boundary minimal hypersurface in the generalised sense of varifolds. Moreover, we establish the up-to-the-boundary integer rectifiability of the limit varifold. If time permits, we will also discuss what we expect in the case of stable solutions. This is on-going joint work with Davide Parise (UCSD) and Lorenzo Sarnataro (Princeton). This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China.