Related papers: The Singularity of Kontsevich's Solution for $QH^{…
We describe interdependencies among the quantum cohomology associativity relations. We strengthen the first reconstruction theorem of Kontsevich and Manin by identifying a subcollection of the associativity relations which implies the full…
We construct some explicit quasihomogeneous algebraic solutions to the associativity (WDVV) equations by using analytical methods of the finite gap integration theory. These solutions are expanded in the uniform way to non-semisimple…
We apply a recently developed method to exactly solve the $\Phi^3$ matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a…
The associativity of the multiplication on a Frobenius manifold is equivalent to the WDVV equation of a symmetric cubic form in flat coordinates. Frobenius manifold could be regarded a very special type of statistical manifold. There is a…
We consider the Swift-Hohenberg equation on manifolds with conical singularities and show existence, uniqueness and maximal regularity of the short time solution in terms of Mellin-Sobolev spaces. Moreover, we give a necessary and…
A well known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper we present a generalization of this construction for the…
For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K],…
Let G denote a complex, semisimple, simply-connected group. We identify the equivariant quantum differential equation for the cotangent bundle to the flag variety of G with the affine Knizhnik-Zamolodchikov connection of Cherednik and…
These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial…
We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted H\"older and Sobolev spaces with discrete…
We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten-Dijkgraaf-Verlinde-Verlinde…
In this paper we give an explicit representation of the solutions of a characteristic Cauchy problem for a class of PDEs with singular coefficients. We give the explicit solutions in terms of the Gauss hypergeometric functions, which enable…
Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more…
We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the…
In this paper we study uniqueness properties of solutions of the k-generalized Korteweg-de Vries equation. Our goal is to obtain sufficient conditions on the behavior of the difference $u_1-u_2$ of two solutions $u_1, u_2$ of the equation…
In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical…
In this paper, we study coupled complex modified Korteweg-de Vries (ccmKdV) equation by combining the Hirota's method and the Kadomtsev-Petviashvili (KP) reduction method. First, we show that the bilinear form of the ccmKdV equation under…
In this paper, we construct a pair of solutions to the open WDVV equations associated with the infinite-dimensional Frobenius manifolds that underlie the genus-zero universal Whitham hierarchy, and for the resulting flat F-manifolds, we…
We compute the C*-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C^2/G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated…
In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlev\'e equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that…