15 research outputs found
Daniel Quillen, the father of abstract homotopy theory
In this short paper we try to describe the fundamental contribution of Quillen in the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory
The de Rham homotopy theory and differential graded category
This paper is a generalization of arXiv:0810.0808. We develop the de Rham
homotopy theory of not necessarily nilpotent spaces, using closed dg-categories
and equivariant dg-algebras. We see these two algebraic objects correspond in a
certain way. We prove an equivalence between the homotopy category of schematic
homotopy types and a homotopy category of closed dg-categories. We give a
description of homotopy invariants of spaces in terms of minimal models. The
minimal model in this context behaves much like the Sullivan's minimal model.
We also provide some examples. We prove an equivalence between fiberwise
rationalizations and closed dg-categories with subsidiary data.Comment: 47 pages. final version. The final publication is available at
http://www.springerlink.co
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
The Conley Conjecture and Beyond
This is (mainly) a survey of recent results on the problem of the existence
of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb
flows. We focus on the Conley conjecture, proved for a broad class of closed
symplectic manifolds, asserting that under some natural conditions on the
manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic
orbits. We discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb flows, the cases
where the conjecture is known to fail, the question of the generic existence of
infinitely many periodic orbits, and local geometrical conditions that force
the existence of infinitely many periodic orbits. We also show how a recently
established variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a low-energy charge
in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur