Abstracts of Talks at the Cornell Topology Festival, May 6-9, 2005

DENIS AUROUX

SYMPLECTIC  4-MANIFOLDS,  MAPPING  CLASS  GROUPS,  AND  FIBER  SUMS

This talk will be about Lefschetz fibrations (i.e., fibrations over the 2-sphere with at most nodal fibers), their relation to symplectic 4-manifolds, and their characterization in terms of quasipositive factorizations in mapping class groups. We will then discuss the classification of Lefschetz fibrations, and in particular the manner in which it simplifies if fibrations are "stabilized" by suitable fiber sum operations.

AUGUSTIN BANYAGA

SOME  INVARIANTS  OF  TRANSVERSALLY  ORIENTED  FOLIATIONS

Abstract (pdf file).

PAUL BIRAN

ALGEBRAIC  FAMILIES  AND  LAGRANGIAN  CYCLES

This talk will be concerned with the study of families of algebraic varieties by means of symplectic topology. We shall explain how symplectic invariants give rise to new restrictions on algebraic families, which apparently cannot be detected on a purely algebraic level. We shall than present some applications to various problems from classical algebraic geometry on projective embeddings and deformations of singularities.

THOMAS DELZANT

FUNDAMENTAL  GROUPS  OF  KAEHLER  MANIFOLDS

This is joint work with Misha Gromov. A group is Kaehler if it can be realized as the fundamental group of a compact Kaehler manifold (e.g., complex projective). A classical problem is to identify which groups are Kaehler. I will describe some new constraints coming from combinatorial group theory (small cancelation theory, existence of a splitting or a cubing, etc.).

YAKOV ELIASHBERG

GEOMETRY  OF  CONTACT  TRANSFORMATIONS:  ORDERABILITY  VS.  SQUEEZING

I will duscuss in the talk two tightly linked problems: analogues of symplectic non-squeezing in contact geometry and existence of a non-trivial partial order on the group of contact transformations. It turns out that the non-squeezing results in contact geometry exhibit a quantum character: they hold only in a large scale. This is related to a quite amazing fact. The group of contactomorphisms of S^2n-1 for n≥1 admits a contractible loop generated by a positive Hamiltonian. On the other hand, such loops do not exist for large classes of contact manifolds, e.g. spaces of contact elements. This is joint work with S.-S. Kim and L. Polterovich.

ÉTIENNE GHYS

MINIMAL  SETS  OF  HOLOMORPHIC  FOLIATIONS  ON  THE  COMPLEX  PROJECTIVE  PLANE:  A  SURVEY

One of the most basic theorems in dynamics is due to Poincaré and Bendixson: the closure of any trajectory of a vector field on the 2-sphere contains a periodic orbit or a singular point. Is there a similar theorem for polynomial differential equations in the complex projective plane? This simple question is still open today! In this talk, I plan to survey some partial results going in this direction. In particular, I would like to describe a recent example by Deroin which concerns a question which is a symplectic analogue of the original question. Hopefully, this will fit with the general theme of the Topology Festival: Symplectic geometry and topology.

YAEL KARSHON

TORI  IN  SYMPLECTOMORPHISM  GROUPS

Maximal tori play a crucial role in the theory of compact Lie groups. If G is a compact Lie group, it contains a torus of positive dimension, and any two maximal tori in G are conjugate. Neither of these properties holds in general if G = Sympl(M,ω) is the automorphism group of a symplectic manifold (M,ω). However, if M is four dimensional and simply connected, every two dimensional torus in G = Sympl(M,ω) is maximal, and G contains only finitely many two dimensional tori, up to conjugation.

The proof combines the theory of symplectic toric manifolds with Gromov's theory of pseudo holomorphic curves. The strategy is due to Martin Pinsonnault; the result is joint with Liat Kessler and Martin Pinsonnault.

JOHN MORGAN

RICCI  FLOW  AND  TOPOLOGY  OF  3-MANIFOLDS

An overview of the theory of Ricci flow and Ricci flow with surgery in dimension three as developed by Hamilton and extended by Perelman. Application to the Poincaré Conjecture and geometrization of 3-manifolds.

SHAHAR MOZES

LATTICES  IN  PRODUCTS  OF  TREES

Motivated by the analogy between groups of automorphisms of a (regular) tree and rank 1 algebraic groups one is led to study the structure of several classes of closed subgroups of an automorphism group of a tree and in particular those which act locally primitively. Their structure and linear representation theory is then applied to study cocompact lattices in the automorphism group of a product of trees and the associated quotient spaces - finite square complexes with certain local conditions. We obtain various rigidity results and apply these to construct interesting groups. Based on joint works with Marc Burger and Bob Zimmer.

YANN OLLIVIER

A  PANORAMA  OF  RANDOM  GROUPS

We shall give a description of the basic ideas behind random groups (sometimes called 'the sociological approach to group theory'), try to review the currently known properties and discuss some applications.

BRENDAN OWENS

UNKNOTTING  INFORMATION  FROM  HEEGAARD  FLOER  THEORY

A classical theorem of Montesinos says that if a knot K has unknotting number u, then the double-branched cover of K is given by half-integral surgery on a u-component link L in S³. This has been used by Lickorish to give an obstruction to u=1; more recently this obstruction was considerably strengthened by Ozsváth and Szabó using Heegaard Floer theory.

I will describe an enhancement of Montesinos' theorem, the proof of which uses Kirby calculus and a result of Cochran and Lickorish. This leads to a generalisation of the Ozsváth-Szabó obstruction to the u>1 case, and to the completion of the table of unknotting numbers for prime knots with 9 crossings or less.

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