scholarly journals Relations in the cohomology ring of the moduli space of flat SO (2 n  + 1)-connections on a Riemann surface

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170427
Author(s):  
Elisheva Adina Gamse ◽  
Jonathan Weitsman
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Integrable Systems ◽  
Marked Point ◽  
Cohomology Ring ◽  
Line Bundles ◽  
Chern Classes ◽  
Theme Issue ◽  
Gauge Transformations ◽  
Finite Dimensional

We consider the moduli space of flat SO (2 n  + 1)-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalizing a conjecture of Newstead. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Asymptotic velocity for four celestial bodies

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170426
Author(s):  
Andreas Knauf
Keyword(s):  
Celestial Mechanics ◽  
Integrable Systems ◽  
Energy Surface ◽  
Initial Conditions ◽  
Theme Issue ◽  
Asymptotic Velocity ◽  
Pair Potentials ◽  
Finite Dimensional ◽  
Celestial Bodies ◽  
Almost All

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree − α for α ∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Open problems, questions and challenges in finite- dimensional integrable systems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170430 ◽  
Author(s):  
Alexey Bolsinov ◽  
Vladimir S. Matveev ◽  
Eva Miranda ◽  
Serge Tabachnikov
Keyword(s):  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Theme Issue ◽  
Open Problems ◽  
Finite Dimensional

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference ‘Finite-dimensional Integrable Systems, FDIS 2017’ held at CRM, Barcelona in July 2017. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Convexity of the moment map image for torus actions on b m -symplectic manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170420 ◽  
Author(s):  
Victor W. Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Manifolds ◽  
Moment Map ◽  
Torus Action ◽  
Convexity Theorem ◽  
Theme Issue ◽  
Torus Actions ◽  
Finite Dimensional ◽  
The Moment

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a b m -symplectic manifold. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals The Mod Two Cohomology of the Moduli Space of Rank Two Stable Bundles on a Surface and Skew Schur Polynomials

2019 ◽  
Vol 71 (03) ◽  
pp. 683-715 ◽  
Author(s):  
Christopher W. Scaduto ◽  
Matthew Stoffregen
Keyword(s):  
Riemann Surface ◽  
Group Action ◽  
Moduli Space ◽  
Class Group ◽  
Cohomology Ring ◽  
Mapping Class ◽  
Schur Polynomials ◽  
Stable Bundles ◽  
Cohomology Rings ◽  
Cup Product

AbstractWe compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

scholarly journals CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

2010 ◽  
Vol 21 (04) ◽  
pp. 497-522 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAINAK PODDAR
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Prime Number ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Stable Vector Bundles

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

scholarly journals A survey on polynomial in momenta integrals for billiard problems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170418 ◽  
Author(s):  
Misha Bialy ◽  
Andrey E. Mironov
Keyword(s):  
Constant Curvature ◽  
Integrable Systems ◽  
Theme Issue ◽  
Algebraic Version ◽  
Finite Dimensional ◽  
Short Survey ◽  
Magnetic Billiards ◽  
Algebraic Technique

In this paper, we give a short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique, we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Symplectic invariants for parabolic orbits and cusp singularities of integrable systems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170424 ◽  
Author(s):  
Alexey Bolsinov ◽  
Lorenzo Guglielmi ◽  
Elena Kudryavtseva
Keyword(s):  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Normal Forms ◽  
Integrable Hamiltonian Systems ◽  
Theme Issue ◽  
New Techniques ◽  
Finite Dimensional ◽  
Parabolic Orbits ◽  
Symplectic Invariants ◽  
Cusp Singularities

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals On the integrability of Birkhoff billiards

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170419 ◽  
Author(s):  
Vadim Kaloshin ◽  
Alfonso Sorrentino
Keyword(s):  
Integrable Systems ◽  
Theme Issue ◽  
Finite Dimensional

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

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