Summer Research Conference

Lusternik-Schnirelmann Category in the New Millennium

The conference was held at Mt. Holyoke from July 29, 2001 to August 2, 2001. Photos from the conference are here.

The subject of Lusternik-Schnirelmann category is now experiencing a renaissance. Although its origins are in variational analysis and dynamical systems, Lusternik- Schnirelmann theory has proved to be an important ingredient in other subjects as well. This is a perfect time for a conference on category: old problems and conjectures have fallen and new questions have opened up a treasure chest of potential applications in dynamics, symplectic geometry and algebraic topology.

Lusternik-Schnirelmann category is an integer associated to a manifold (or, more generally, a space) which is an invariant of the homotopy type of the space and which gives a numerical measure of the complexity of the space. In particular, category is an indicator of the complexity of possible dynamics on the manifold by providing a lower bound on the number of critical points required of any function on the manifold. Lusternik and Schnirelmann themselves applied category to geometry as well as to dynamics, proving the existence of (at least) three closed geodesics on a two-dimensional sphere with any metric, a remarkable accomplishment in the early days of infinite dimensional variational analysis. The theory itself may be considered a cousin of Morse theory, and, just as Morse theory may be used to understand the homotopical structure of spaces, so too has category found a place in algebraic topology. For instance, a result of G. W. Whitehead says that the set of homotopy classes of maps of a space X into a group-like space is in fact a nilpotent group with nilpotency class bounded above by the category of X. In recent years, Lusternik- Schnirelmann category ideas have had a significant influence in areas and problems ranging from Conley index theory to rational homotopy theory to the Arnold conjecture on Hamiltonian symplectomorphisms.

In algebraic topology, one focus of study in the last several years has been the Ganea conjecture. This conjecture states that the category of a product of a space X and a sphere should be the category of X plus one. The simplicity of the statement belies the deep structural questions at its heart. In fact, it is only the recent construction of counterexamples to the conjecture which reveal a whole underlying ediface of homotopical structure built around Hopf invariants. Of course, category was invented to solve problems in dynamics, so it is not surprising that it still finds a niche in the study of dynamical systems. It is more surprising, however, that various off-shoots of category lead to a place for Hopf invariants in dynamics also. Finally, recent approaches to Arnold's conjecture that Hamiltonian symplectomorphisms on a symplectic manifold have at least as many fixed points as any function on the manifold has critical points reveal the strategic position Lusternik-Schnirelmann theory occupies in symplectic topology.

The goal of this conference is to bring together mathematicians from areas on which category has had an impact to review the state of the art, set the course for future investigations and foster cross-fertilization among areas. Lectures are expected to cover the broad range of topics mentioned above, presenting new developments in both theory and applications. In addition to specialized talks, survey talks will be given devoted to the aspects of homotopy theory, dynamical systems and symplectic topology where Lusternik- Schnirelmann category plays a vital role.

Organizers (CLOT):


Gregory Lupton Daniel Tanre
Department of Mathematics Departement de Mathematiques
Cleveland State University Universite de Lille
Cleveland OH 44115 USA Lille, France
lupton AT math.csuohio.edu tanre AT gat.univ-lille1.fr

Octav Cornea John Oprea
Departement de Mathematiques Department of Mathematics
Universite de Lille Cleveland State University
Lille, France Cleveland OH 44115 USA
cornea AT gat.univ-lille1.fr oprea AT math.csuohio.edu

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