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It has been recently conjectured that the exact eigenfunctions of quantum mirror curves can be obtained by combining their WKB expansion with the open topological string wavefunction. In this paper we give further evidence for this…
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links…
We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge…
In this paper we study the group K_{2n}^{(n+1)}(F) where F is the function field of a complete, smooth, geometrically irreducible curve C over a number field, assuming the Beilinson--Soul\'e conjecture on weights. In particular, we compute…
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on…
In the past few years, the unifying frameworks of 4-dimensional Chern-Simons theory and affine Gaudin models have allowed for the systematic construction of a large family of integrable $\sigma$-models. These models depend on the data of a…
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…
In this note we describe instances where values of the $K$-theoretical regulator map evaluated on topological cycles equal entropies of topological actions by a group $\Gamma$. These entropies can also be described by determinants on the…
We prove an index formula for the Dirac operator acting on two-valued spinors on a $3$-manifold $M$ which branch along a smoothly embedded graph $\Sigma \subset M$, and with respect to a boundary condition along $\Sigma$ inspired by an…
We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie…
This is the author's PhD thesis. Two main sections address various aspects of mirror symmetry for compact Calabi-Yau threefolds and the roles that classically modular varieties play in string theory compactifications. The main results…
We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries. For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by…
We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the…
We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior.…
These are notes of a series of lectures on mirror symmetry and topological string theory given at the Mathematical Sciences Center at Tsinghua University. The N=2 superconformal algebra, its deformations and its chiral ring are reviewed. A…
In this lecture we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. Our analysis is based on an approach to modular invariants using braided sector induction…
We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case.…
We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only…
We propose to extend ``invertibility'' to ``regularity'' for categories in general abstract algebraic manner. Higher regularity conditions and ``semicommutative'' diagrams are introduced. Distinction between commutative and…
We develop a new approach to the isospectrality of the orbifolds constructed by Vign\'eras. We give fine sufficient criteria for i-isospectrality in given degree i and for representation equivalence. These allow us to produce very small…