arXiv research

Survey recent constructions of cyclic cocycles for Lie groups.

Paper calculates indices for group actions using cocycles.

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal functions relative to the boundary. Closure under convolution and other module …

Study bi-graded Lie algebras and their applications.

The study proves a geometric result related to Harish-Chandra's theorem.

The paper extends algebraic constructions to $\mathbb Z$-graded manifolds and Lie algebroids.

Harish-Chandra's volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing…

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary fa…

Extends Kostant's results to symmetric pairs in Clifford algebras.

We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these glo…

Paper defines and computes a new weight system for gl_N Lie algebra.

The paper proves estimates for Hodge Laplacians on Lie groups.

We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initi…

The article discusses extensions of Harish-Chandra's admissibility theorem.

It is known that there exists a natural functor $Φ$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to di…

We set up foundations of representation theory over $S$, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat $S$-Lie algebras and their representations, characters, $gl_n(S)$-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differentia…

This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case …

Motivated by the study of Hörmander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized function…

Schwartz functions smoothly extend to real projective spaces.

We explain how the Harish-Chandra Plancherel Theorem and results in relative Lie algebra cohomology can be used in order to compute in a uniform way the $L^2$-Betti numbers, the Novikov-Shubin invariants, and the $L^2$-torsion of compact locally symmetric spaces thus completing results previously obtained by Borel, Lot…

The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.

We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\mathbb C$ which acts on $V…

The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula…

Two sweeps of the Brennan-Schwartz algorithm solve American options under negative rates.

New metric defined for bounded symmetric domains.

The Schwartz-Smith model parameters are estimated using Kalman Filter with additional constraints.

Let $λ: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,ω)$ of the symplectic vector space $(V,ω)$ by the metaplectic group $\tilde{G}=Mp(V,ω).$ In this case, $λ$ is also a representation of $\tilde{G}$ on the vector space $V$ and thus, it gives rise to the representation of $\tilde{G…

The paper constructs Ricci-flat Kähler manifolds with specific decay properties.

A new two-step LSMC method improves game option pricing accuracy.

The paper explores global index formulas for one-dimensional holomorphic foliations.

Resolving Schwartz's quadratic meander number conjecture

Describes representations of modular group into SL(3,R)/SO(3).

Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called space-time foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been presented, among them, …

We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized s…

In the paper "Pappus's theorem and the modular group", R. Schwartz constructed a 2-dimensional family of faithful representations $ρ_Θ$ of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ into the group $\mathscr{G}$ of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subg…

We use the theory of dual of Fréchet-Schwartz (DFS) spaces to establish a sufficient condition for top-degree solvability for the differential complex associated to a hypocomplex locally integrable structure. As an application, we show that the top-degree cohomology of left-invariant hypocomplex structures on a compact…

Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.

We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as gr…

We propose a new method for studying $n$- and $Γ$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $Γ$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $Γ$-cohomology of…

Paper optimizes neural networks for Bermudan option pricing with faster convergence and risk management tools.

PDSim simulates and estimates commodity futures prices using polynomial diffusion models.

A beta function for double layers is defined and analyzed.

We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly supported semiregular distribution on $M\times M$ is the Schwartz kernel of a classical …

We deduce a recent theorem by R. Schwartz on the structure of the so-called Poncelet grid from complete integrability of the billiard in an ellipse

Wave equation map reveals manifold's structure.

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^kłf(x+tξ),ξ^m\r\,dt. $ We give the range characterization for the operator $f\mapsto(I^0\!f,I^1\!f,\dots, I^m\!f)$ on the Schwartz space of rank $m$ smo…