The fundamental gap is the difference of the first two eigenvalues of the Laplace operator, which is important both in mathematics and physics and has been extensively studied. For the Dirichlet boundary condition, the log-concavity estimate of the first eigenfunction plays a crucial role, which was established for convex domains in the Euclidean space and the round sphere. Joint with G. Khan, H. Nguyen, and G. Wei, we obtain log-concavity estimates of the first eigenfunction for convex domains in surfaces of positive curvature and consequently establish fundamental gap estimates. In a subsequent work, together with G. Khan and G. Wei, we improve the log-concavity estimates and obtain stronger gap estimates which recover known results on the round sphere.

Abstract:   Motived by the notion of the algebraic hyperbolicity introduced by X. Chen, we introduce the notion of Nevanlinna hyperbolicity for a pair of (X, D), where X is a projective variety and D is an effective Catier divisor on X. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Paun and Sibony. This is a joint work with Yan He.

 In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can show that the classical PDE theory holds on triple junction surfaces. As a consequence, we can prove a type of Generalized Bernstein Theorem and talk about the Morse index on minimal triple junction surfaces.

The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in the mid-70s, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local  boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced  operators adapted to the boundary taking centre stage in formulating and understanding non-local boundary conditions.

That being said, much of this analysis has been confined to the situation when adapted boundary operators can be chosen self-adjoint. Dirac-type operators are the quintessential example. Nevertheless, natural geometric operators such as the Rarita-Schwinger operator on 3/2-spiniors, arising from physics in the study of the so-called Delta baryon, falls outside of this class. Analytically, this requires analysis beyond self-adjoint operators. In recent work with Bär,  the compact boundary case is handled for general first-order elliptic operators, using spectral theory to choose adapted boundary operators to be invertible bi-sectorial. The Fourier circle methods present in the self-adjoint analysis are replaced by the  bounded holomorphic functional calculus, coupled with pseudo-differential operator theory and semi-group techniques. This allows for a full understanding of the maximal domain of the interior operator as a bounded surjection to a space on the boundary of mixed Sobolev regularity, constructed from spectral projectors associated to the adapted boundary operator. Regularity and Fredholm extensions are also studied.

For the noncompact case, a preliminary trace theorem as well as regularity theory are handed by resorting to the case with compact boundary. This necessitates  deforming the coefficients of the interior operator in a compact neighbourhood. Therefore, even for Dirac-type operators, allowing for fully general symbols in the compact boundary case is paramount. Under slightly stronger geometric assumptions near the noncompact boundary (automatic  for the compact case) and  when the interior operator admits a self-adjoint adapted boundary operator, an upgraded trace theorem mirroring the compact setting is obtained. Importantly, there is no spectral assumptions other than self-adjointness on the adapted boundary operator. This, in particular, means that the spectrum of this operator can be the entire real line. Again, the primarily tool that is used in the analysis is the bounded holomorphic functional calculus.

I will introduce G-algebroids — structures related to the symmetries and low-energy effective descriptions of membranes, strings, and particles. I will describe some basic properties of these algebroids and show how they relate to the more standard Lie and Courant algebroids. I will finish by discussing the main classification results and some (spherical) examples. This is a joint work with M. Bugden, O. Hulik, and D. Waldram.