Let $(V, 0)$ be an isolated hypersurface singularity defined by the holomorphic function $f: (\mathbb{C}^n, 0)\rightarrow (\mathbb{C}, 0)$. In our previous work, we introduced a series of novel Lie algebras associated to $(V,0)$, i.e., $k$-th Yau algebra $L^k(V), k\geq 0$. It was defined to be the Lie algebra of derivations of the $k$-th moduli algebras $A^k(V)= \mathcal{O}_n/(f, m^k J(f)), k\geq 0$, where $m$ is the maximal ideal of $\mathcal{O}_n$. I.e., $L^k(V):=\text{Der}(A^k(V), A^k(V))$. The dimension of $L^k(V)$ was denoted by $\lambda^k(V)$. The number $\lambda^k(V)$, which was called $k$-th Yau number, is a subtle numerical analytic invariant of $(V, 0)$. Furthermore, we formulated two conjectures for these $k$-th Yau number invariants: a sharp upper estimate conjecture of $\lambda^k(V)$ for weighted homogeneous isolated hypersurface singularities (see Conjecture \ref{conj2}) and an inequality conjecture $\lambda^{(k+1)}(V) > \lambda^k(V), k\geq0$ (see Conjecture \ref{conj1}). In this article, we verify these two conjectures when $k$ is small for large class of singularities.

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and q-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the σ-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.

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Let Y be the zero loci of a regular section of a convex vector bundle Y over Y . We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and Y which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of Y .

We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

As a continuation of\lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the``large volume limit''in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include:(1) proofs of various conjectural formulas of physicists\vk\lkm involving mirror maps and modular functions;(2) new identities involving multi-variable hypergeometric series and modular functions--generalizing\lianyaufour. In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in\P^ 4 [6, 2, 2, 1, 1], in one limit the series of the couplings are expressed in terms of the j function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of``instanton

1. INDEX THEORY, ELLIPTIC CURVES AND LOOP GROUPS One can look at elliptic genus from several different points of view; from index theory, from representation theory of Kac-Moody affine Lie algebras or from the theory of elliptic functions and modular forms. Each of them shows us some quite different interesting features of ellitic genus. On the other hand we can also combine the forces of these three different mathematical fields to derive many interesting results in topology such as rigidity, divisibility and vanishing of topological invariants..

Spin polynomial invariants, as denned by Pidstrigatch and Tyurin, are used to define differentiable invariants for Dolgachev surfaces. In order to define and compute these invariants, some machinery is developed to study the wall structure for Spin polynomials, to deal with the fact that the moduli spaces have the wrong dimension and to handle nonreduced structures. The invariants are then computed.

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and HirzebruchRiemannRoch, and some modern development of uniformization theorems, KhlerEinstein metric and the theory of DonaldsonUhlenbeckYau on Hermitian YangMills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.