A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
New definition of Born geometry connects to known geometries.
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
Joachimsthal integrals characterize conics in various geometries.
Derives path-integrals for superstrings on curved backgrounds using string geometry theory.
Cauchy used infinitesimals in differential geometry and integral geometry.
New integrable systems derived from Nijenhuis geometry.
Geometric approach links hydrodynamic integrability to compatible nets.
Lecture notes introduce contact complete integrability for odd-dimensional manifolds.
Survey of open problems linking integrable systems and Nijenhuis geometry.
We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler manifold is so related to an integrable system. The cotangent bundle of a special …
Derives path integrals for perturbative strings on various backgrounds.
We introduce a notion of the noncommutative integrability within a framework of contact geometry.
Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.
We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the…
This paper demostrates a method for analysing almost CR geometries , by uniquley defining a partially integrable structure from the same data. Thus two almost CR geometries and are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries …
Extends integrability to cosymplectic manifolds.
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the following aspects of integrable non-Hamiltonian systems: local and semi-local normal …
In 5D, integrability is linked to curvature constraints of subconformal structures.
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i…
New framework for logarithmically divergent integrals on manifolds with corners.
A general method for analytic inversion in integral geometry is proposed. All classical and some new reconstruction formulas of Radon-John type are obtained by this method. No harmonic analysis and PDE is used.
Study reveals new geometric structures for magnetic field Hamiltonian systems.
Research covers geometry, analysis, and integration on infinite-dimensional spaces.
Study path geometries with constant torsion and cone structures.
The paper evaluates integrals of planes and their relation to convex set angles.
This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solutio…
Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R^n equipped with the metric derived from the p-norm. This has, in effect, been investigated intensively for 1<p<\infty, but not for p=1. We show that integral …
In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measu…
In this article, we treat G_2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G_2-structure; in particular, we discuss existence and make a number of identifications of the spa…
Study of geometric structures on manifolds, focusing on integrability conditions.
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…
Paper extends curvature estimates to new tensor types.
We study the types of non-integrable -structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a $\Spin(7)$-structure play a special role. Any geometry of that type admits a unique c…
We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surfaces of constant negative Gauss curvature and isothermic surfaces. These are purely expository notes written to accompany some lectures I gave in Fukuoka in May 2015.
Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrov…
Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.
Calculates spinor heat flow using Gaussian-Grassmann integrals.
For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of t…
A general method for analytic inversion of geometric integral transforms is proposed
In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plücker coordinates. Particular emphasis is put on the restriction to Lie circle g…
Unified approach to constructing integrable systems using Stäckel lifts.
The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd symplectic superspace and for more clear interpretation of the Batalin-Vilkovisky formalism geometry.
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.