Related papers: Period Integrals and the Riemann-Hilbert Correspon…

The proposal made 50 years ago by Schulman (1968), Laidlaw & Morette-DeWitt (1971) and Dowker (1972) to decompose the propagator according to the homotopy classes of paths was a major breakthrough: it showed how Feynman functional integrals…

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable…

We consider the homology theory of \'etale groupoids introduced by Crainic and Moerdijk, with particular interest to groupoids arising from topological dynamical systems. We prove a K\"unneth formula for products of groupoids and a…

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg…

The existence of periodic solutions in $\Gamma$-symmetric Newtonian systems $\ddot{x}=-\nabla f(x)$ can be effectively studied by means of the $(\Gamma\times O(2))$-equivariant gradient degree with values in the Euler ring $U(\Gamma\times…

In this article, we study mirror symmetry for pairs of singular Calabi--Yau manifolds which are double covers of toric manifolds. Their period integrals can be seen as certain `fractional' analogues of those of ordinary complete…

We introduce tautological system defined by prehomogenous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly…

In this paper we study those polynomials orthogonal with respect to a particular weight over the union of disjoint intervals first introduced by N.I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This…

We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure…

This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A…

We develop a new method to study intersection theory of the main component of the Hilbert scheme of points on complex manifolds. The main result is an iterated residue formula for tautological integrals. We formulate a Chern-Segre-type…

Let G be a compact connected Lie group with a maximal torus T\subsetG. In the context of Schubert calculus we obtain a canonical presentation for the integral cohomology ring H^{\ast}(G/T) of the complete flag manifold G/T. The result have…

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a…

Deligne's celebrated "Riemann--Hilbert correspondence" relates representations of the fundamental group of a smooth complex algebraic variety and regular-singular integrable connections. In this work, we show how to arrive at a similar…

In this paper, we study the sub-Riemannian problem associated with contact structures on connected, simply connected, solvable, non-nilpotent, regular three-dimensional Lie groups. For these groups, the vertical component of the Hamiltonian…

Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, there are still very few examples of racks whose homology has…

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector…

We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher loop homology of a manifold in order to compute the homology of the spaces of continuous and holomorphic maps of the Riemann sphere into a complex projective space.…

In 1989 H. Tverberg proposed a quite general conjecture in Discrete geometry, which could be considered as the common basis for many results in Combinatorial geometry and at the same time as a discrete analogue of the common transversal…

Realizing the possibility suggested by Hardouin [6], we show that her own Picard-Vessiot Theory for iterative $q$-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [2], of artinian…