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Invited lecture given at meeting on ‘Espaces de Lacets’,
, 1994
"... A theory of integration for anticommuting paths is described. This is combined with standard Itô calculus to give a geometric theory of Brownian paths on curved supermanifolds. This lecture concerns a generalisation of Brownian motion and Itô calculus to include paths in spaces of anticommuting vari ..."
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A theory of integration for anticommuting paths is described. This is combined with standard Itô calculus to give a geometric theory of Brownian paths on curved supermanifolds. This lecture concerns a generalisation of Brownian motion and Itô calculus to include paths in spaces of anticommuting variables. The motivation for this work comes originally from physics, where anticommuting variables were first introduced by Martin [7] in order to extend Feynman’s path integral methods to Fermionic systems. Subsequently various geometric applications Research supported by a Royal Society University Research Fellowship 1 to this approach have emerged, and additionally anticommutiong variables have been found to play a significant rôle in the quantizaton of systems with gauge symmetry. 1 Functions of anticommuting variables Suppose that θ 1,..., θ n are n anticommuting variables, so that θ i θ j = −θ j θ i. (1) Then, since (θ i) 2 is zero, the natural space of functions to consider is the 2 ndimensional space of functions of the form f(θ 1,..., θ n) = ∑ µ∈Mn fµθ µ where µ denotes a multiindex µ1...µµ, with 1 ≤ µ1 <... < µµ  ≤ n, Mn denotes the set of all such multiindices (including the empty multiindex ∅) and θ µ = 1θ µ1... θ µ µ. The detailed nature of this function space is determined by the choice of space in which the coefficient functions fµ lie. This may be the real numbers, the complex numbers or some space of commuting and anticommuting variables. The importance of such functions for Fermionic physics is that they allow realisations of the canonical anticommutation relations (2) ψ i ψ j + ψ j ψ i = 2δ ij
The Power of the Works of Papakyriakopoulos in 3Manifolds
, 2001
"... Christos Papakyriakopoulos was in Princeton when I arrived. I was a graduate student, from Arkansas; I believe that I had been admitted to graduate school at Princeton University in 1956 because Ralph Fox had been on my side. Fox communicated with Papakyriakopoulos frequently, and I learned that Pa ..."
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that Papa (as we called him) had just done some amazing work in the theory of 3manifolds. Princeton was a center of topology in those days. Solomon Lefschetz
Study of the Rolling Manifolds Model and of its Controllability
, 2012
"... soutenue le 27/11/2012 par Petri KOKKONEN Étude du modèle des variétés roulantes et de sa commandabilité ..."
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soutenue le 27/11/2012 par Petri KOKKONEN Étude du modèle des variétés roulantes et de sa commandabilité
SEMIINFINITE COHOMOLOGY AND SUPERCONFORMAL ALGEBRAS
, 2000
"... Abstract We describe representations of certain superconformal algebras in the semiinfinite Weil complex related to the loop algebra of a complex finitedimensional Lie algebra and in the semiinfinite cohomology. We show that in the case where the Lie algebra is endowed with a nondegenerate invar ..."
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Cited by 2 (2 self)
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certaines algèbres superconformes dans le complexe de Weil semiinfini de l’algèbre des lacets d’une algèbre de Lie complexe de dimension finie et dans la cohomologie semiinfinie. Nous démontrons que dans le cas où l’algèbre de Lie est munie d’une forme symétrique non dégénérée invariante, la cohomologie
Systitmes dynamiqueslDynamica / Systems The principal loopbundle and dynamical systems
"... Abstract. The purpose of this Note is to announce the proof of the nonexistence of expansive homeomorphisms on simply connected closed manifolds, provided one assumes locally connectedness of local stable and unstable sets. We also show the nonexistence of homoclinic points in the universal coveri ..."
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Abstract. The purpose of this Note is to announce the proof of the nonexistence of expansive homeomorphisms on simply connected closed manifolds, provided one assumes locally connectedness of local stable and unstable sets. We also show the nonexistence of homoclinic points in the universal
The Geometry of Syzygies  A second course in Commutative Algebra and Algebraic Geometry
, 2001
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DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON SURFACES
"... A. – We prove a universal inequality between the diastole, defined using a minimax process on the onecycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closedRiemannian surface can be swept out by a family of multiloops whose lengths are bounded in terms of ..."
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l’espace des 1cycles, et l’aire d’une surface riemannienne fermée. De manière informelle, nous prouvons que toute surface riemannienne fermée peut être balayée par une famille de multilacets dont les longueurs sont contrôlées par l’aire de la surface. Cette inégalité diastolique, qui repose sur
Principe variationnel et groupes Kleiniens
"... Summary Let Γ be a nonelementary Kleinian group acting on a CartanHadamard manifold ˜X; denote by Λ(Γ) the nonwandering set of the geodesic flow (φt) acting on the unit tangent bundle T 1 ( ˜ X/Γ). When Γ is convex cocompact (i.e. Λ(Γ) is compact), the restriction of (φt) to Λ(Γ) is an Axiom A ..."
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Cited by 14 (1 self)
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Summary Let Γ be a nonelementary Kleinian group acting on a CartanHadamard manifold ˜X; denote by Λ(Γ) the nonwandering set of the geodesic flow (φt) acting on the unit tangent bundle T 1 ( ˜ X/Γ). When Γ is convex cocompact (i.e. Λ(Γ) is compact), the restriction of (φt) to Λ(Γ) is an Axiom A
VARIÉTÉS FAIBLEMENT SPÉCIALES À COURBES ENTIÈRES DÉGÉNÉRÉES
, 2008
"... * Summary: A complex projective manifold is said to be weaklyspecial if none of its finite étale covers has a dominat rational map to a positivedimensional manifold of general type. Weaklyspecial manifolds are conjectured in [HT 00] to be potentially dense if defined over a number field. In [Ca ..."
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Cited by 4 (2 self)
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* Summary: A complex projective manifold is said to be weaklyspecial if none of its finite étale covers has a dominat rational map to a positivedimensional manifold of general type. Weaklyspecial manifolds are conjectured in [HT 00] to be potentially dense if defined over a number field. In [Ca
RESEARCH STATEMENT
"... Abstract. We pursue two related areas of research. One is the combinatorial study of generic immersed curves in closed surfaces. The other is the combinatorial extension of knots and links to threemanifolds. 1. Summary 1.1. Immersed Curves in Surfaces. We classify generic immersions of curves in cl ..."
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Abstract. We pursue two related areas of research. One is the combinatorial study of generic immersed curves in closed surfaces. The other is the combinatorial extension of knots and links to threemanifolds. 1. Summary 1.1. Immersed Curves in Surfaces. We classify generic immersions of curves
Results 1  10
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107