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It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_{\infty}$ -operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $𝒫$ be an operad, $M$ a $𝒫$ -module and $U$ a $𝒫$ -algebra. An $M$ -trace over $U$ consists of a space $V$ and a module homomorphism $T : M \to {End}_{U, V}$ over the operad homomorphism $𝒫 \to {End}_{U}$ given by the algebra structure on $U$ . Let $𝒞_{1}$ be the little 1-cubes...

From astrology to topology via Feynman diagrams and Lie algebras

Bar-Natan, Dror (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

Summary of the three lectures. These notes are available electronically at http://www.ma.huji.ac.il/~drorbn/Talks/Srni-9901/notes.html.

From the Fermi-Walker to the Cartan connection

Lafuente, Javier, Salvador, Beatriz (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

Let $M$ be a $C^{\infty}$ -manifold with a Riemannian conformal structure $C$ . Given a regular curve $γ$ on $M$ , the authors define a linear operator on the space of (differentiable) vector fields along $γ$ , only depending on $C$ , called the Fermi-Walker connection along $γ$ . Then, the authors introduce the concept of Fermi-Walker parallel vector field along $γ$ , proving that such vector fields set up a linear space isomorphic to the tangent space at a point of $γ$ . This allows to consider the Fermi-Walker horizontal lift of...

Further remarks on extended umbral calculus

Kwaśniewski, A. K., Grądzka, E. (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

The paper deals with extensions of the finite operator calculus of G.-C. Rota called $ψ$ -extensions which give a framework for corresponding quantum group investigations. This also covers the instance of the well-known $q$ -analogue of umbral calculus. The article also contains glossaries of the most important terms and notations used by Ward, Viskov, Markowsky and Roman on one side and the Rota-oriented notations on the other side.

Gauge-natural field theories and Noether theorems: canonical covariant conserved currents

Palese, Marcella, Winterroth, Ekkehart (2006)

Proceedings of the 25th Winter School "Geometry and Physics"

Summary: We specialize in a new way the second Noether theorem for gauge-natural field theories by relating it to the Jacobi morphism and show that it plays a fundamental role in the derivation of canonical covariant conserved quantities. In particular we show that Bergmann-Bianchi identities for such theories hold true covariantly and canonically only along solutions of generalized gauge-natural Jacobi equations. Vice versa, all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms...

G-connections as twisted formal solutions of systems of PDE's related to geometric structures

Ortaçgil, Ercüment (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

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