We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Engli\v s and A. Loi.

We prove that for every non-negative integer g, there exists a bound on the number of ends of a complete, embedded minimal surface M in R3 of genus g and finite topology. This bound on the finite number of ends when M has at least two ends implies that M has finite stability index which is bounded by a constant that only depends on its genus.

In this companion piece to [9], some variations on the main results there are sketched. In particular, • the recursions in [9], which we interpreted as the quantum Lefschetz, is re- formulated in terms of Givental’s quantization formalism, or equivalently, a summation of finitely many graphs; • varietiesofmodificationoftheauxilliaryspaces(masterspaces)forthefixed point localization are given, leading to different (looking) recursions; • applications of this circle of ideas to derive (apparently) new relations of Gromov–Witten invariants.

Based on large N Chern-Simons/topological string duality, in a series of papers [36, 21, 19], Labastida, Marino, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

No abstract uploaded!