Related papers: Tropical Theta Functions and Log Calabi-Yau Surfac…

We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new…

We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's…

In the first part of the paper, we prove a mirror symmetry isomorphism between integral tropical homology groups of a pair of mirror tropical Calabi-Yau hypersurfaces. We then apply this isomorphism to prove that a primitive patchworking of…

We introduce a two-parameter refinement of the Jucys-Murphy theory, that we call the CJT-refinement, unifying Schur, zonal, and, conjecturally, Jack actions of the ring of symmetric functions on the Fock space. Applications of this…

The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial. The fibers of the tangent bundle to this moduli space carry algebra structures, which are…

In 2006, Kenyon and Okounkov computed the moduli space of Harnack curves of degree $d$ in $\mathbb{C}\mathbb{P}^2$. We generalize to any projective toric surface some of the techniques used there. More precisely, we show that the moduli…

We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix…

In this thesis we probe various interactions between toric geometry and string theory. First, the notion of a top was introduced by Candelas and Font as a useful tool to investigate string dualities. These objects torically encode the local…

In the recent works \cite{Cannarsa-Chen-Cheng} and \cite{Cannarsa-Cheng3}, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution…

The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way.…

Compactifications of heterotic string theory on Generalized Calabi-Yau manifolds have been expected to give the same type of flexibility that type IIB compactifications on Calabi-Yau orientifolds have. In this note we generalize the work…

This manuscript from August 1995 (revised February 1996) studies the Kaehler cone of Calabi-Yau threefolds via symplectic methods. For instance, it is shown that if two Calabi-Yau threefolds are general in complex moduli and are symplectic…

We show that, once translated to the dual setting of convex triangulations of lattice polytopes, results and methods from previous tropical works by Arnal-Renaudineau-Shaw, Renaudineau-Shaw, Renaudineau-Rau-Shaw, and Jell-Rau-Shaw extend to…

For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regular functions on the $\mathrm{SL}_3(\mathbb{C})$-character variety,…

We study type one generalized complex and generalized Calabi--Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the…

We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our…

We show that generalised Calabi-Yau dg (co)algebras are Koszul dual to generalised symmetric dg (co)algebras, without needing to assume any smoothness or properness hypotheses. Similarly, we show that Gorenstein and Frobenius are Koszul…

We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta…

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We…

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi-Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated…