Related papers: The Witten equation and its virtual fundamental cy…
We consider complementary dynamical systems related to stationary Korteweg-de Vries hierarchy of equations. A general approach for finding elliptic solutions is given. The solutions are expressed in terms of Novikov polynomials in general…
In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to…
We propose quantum algorithms for complex-valued nonlinear partial differential equations in the strongly nonlinear regime, where the dynamics is governed by vortex cores, phase singularities, and nonlinear vortex interactions. Examples…
In this paper, using geometric properties of the field rotation parameters, we present a solution of Smale's Thirteenth Problem on the maximum number of limit cycles for Li\'{e}nard's polynomial system. We also generalize the obtained…
We introduce a general theory of homological Milnor-Witt cycle modules over an excellent base scheme equipped with a dimension function, extending both Rost's cycle modules and Feld's theory over fields. To any such module we associate a…
We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a…
Path integrals developed by Richard Feynman have been an important tool in Physics in studying quantum field theory. In mathematics, it has also been widely used in providing formal proofs in the study of Index theorem and asymptotic…
The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space $H$. If $(i, H, B)$ is a Wiener triplet associated to $H$, the quantum state space will be the space of $L^2$ functions on $B$ with…
We consider semilinear parabolic stochastic PDEs driven by additive noise. The question addressed in this note is that of the regularity of transition probabilities. If the equation satisfies a Hormander 'bracket condition', then any…
We study the Cauchy problem for the Whitham modulation equations for monotone increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is…
This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying…
We present a simple new way - called Schrodingerisation - to simulate general linear partial differential equations via quantum simulation. Using a simple new transform, referred to as the warped phase transformation, any linear partial…
Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one…
This paper is a sequel to our previous work, where we proved the ``modularity theorem'' for algebraic Witt vectors over imaginary quadratic fields. This theorem states that, in the case of imaginary quadratic fields $K$, the algebraic Witt…
Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the…
We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to…
We revisit the classic Wigner semi-circle from two different angles. One consists in studying the Stieltjes transform directly on the real axis, which does not converge to a fixed value but follows a Cauchy distribution that depends on the…
We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the…
We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend…
We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with $m\geq 2$ vertices. Their global Poisson structure is characterised by quasi-Hamiltonian algebras related to these quivers, which were studied…