Related papers: Tri-Hamiltonian Structure of the Ablowitz-Ladik Hi…
We prove that any three-point genus zero Gromov-Witten invariant on a type A Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these…
In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $\pi_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we…
We construct a new 20-dimensional family of projective 6-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are…
In [16] the fundamental relationship between stable quotient invariants and the B-model for local P2 in all genera was studied under some specialization of equivariant variables. We generalize the argument of [16] to full equivariant…
In the paper we first investigate symmetries of isospectral and non-isospectral four-potential Ablowitz-Ladik hierarchies. We express these hierarchies in the form of $u_{n,t}=L^m H^{(0)}$, where $m$ is an arbitrary integer (instead of a…
We give an explicit expression for the contact loci of hyperplane arrangements and show that their cohomology rings are combinatorial invariants. We also give an expression for the restricted contact loci in terms of Milnor fibers of…
We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the…
The twisted Alexander polynomials of a space, associated to a linear representation $\sigma$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they…
We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ($\omega, \mathscr{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra…
In the authors's previous work on symmetric differentials and their connection to the topological properties of the ambient manifold, a class of symmetric differentials was introduced: closed symmetric differentials ([BoDeO11] and…
A Frobenius manifold has tri-hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We focus on the case of dimension four and show that, under the assumption of semisimplicity, the corresponding…
It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a…
We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the…
We introduce a Hamiltonian for fermions on a lattice and prove a theorem regarding its topological properties. We identify the topological criterion as a $\mathbb{Z}_2-$ topological invariant $p(\textbf{k})$ (the Pfaffian polynomial). The…
We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We…
We theoretically introduce a quasi-1D magnetic heterostructure of alternating 2D topological and normal insulator strips. Its low-energy physics is governed by a hybrid Hamiltonian intertwining the Su-Schrieffer-Heeger and Shockley models,…
Let $M$ be a smooth closed orientable manifold and $\mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $\mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on…
We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of…
We consider a 3$\times$3 spectral problem which generates four-component CH type systems. The bi-Hamiltonian structure and infinitely many conserved quantities are constructed for the associated hierarchy. Some possible reductions are also…
We consider the deformations of Whitham systems including the "dispersion terms" and having the form of Dubrovin-Zhang deformations of Frobenius manifolds. The procedure is connected with B.A. Dubrovin problem of deformations of Frobenius…