Related papers: Elliptic Schubert Classes and the Poincare Duality
Following Petersson, we study the parabolic, hyperbolic and elliptic expansions of holomorphic cusp forms and the associated Poincar\'e series. We show how these ideas extend to the space of second-order cusp forms.
We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the ordinary locus of the modular curve in terms of certain $p$-adic analytic moment functions associated to Katz' two-variable…
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many…
The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the…
Consider a pair of symplectic varieties dual with respect to 3D-mirror symmetry. The K-theoretic limit of the elliptic duality interface is an equivariant K-theory class of the product. We show that this class provides correspondences in…
The classical Poincar\'e formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several…
A version of the twisted Poincar\'{e} duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure…
We show that Koszul duality between differential graded categories and pointed curved coalgebras interchanges smooth and proper Calabi-Yau structures. This result is a generalization and conceptual explanation of the following two…
Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over $\Q$ and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer…
In this paper we construct an equivariant Poincar\'e duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group $C^*$-algebras of extended…
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of…
In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincar\'e series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which…
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…
The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of…
The purpose of this paper was to give an algebraic analog of Poincare duality. But there is a mistake in the proof of the main theorem. It will be corrected as soon as possible.
We give a short proof of the duality theorem for the reduced $L_p$-cohomology of a complete oriented Riemannian manifold.
The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood from the multiplication in the space of…
We introduce a new class of duality symmetries amongst quantum field theories. The new class is based upon global spacetime symmetries, such as Poincare invariance and supersymmetry, in the same way as the existing duality transformations…
In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincar\'e duality for \'etale cohomology with mod $p$-coefficients : the varieties satisfying "primitive…
The aim of these notes, originally intended as an appendix to a book on the foundations of equivariant cohomology, is to set up the formalism of the $G$-equivariant Poincar\'e duality for oriented $G$-manifolds, for any connected compact…