Abstracts of Talks at the Cornell Topology Festival,
May 6-9, 2005
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.
SOME INVARIANTS OF TRANSVERSALLY ORIENTED FOLIATIONS
- Abstract (pdf file).
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.
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.).
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 S2n-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.
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.
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
- 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.
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.
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.
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
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.
Back to Topology Festival home page.
Last modified: Mon Mar 14 16:23:28 Eastern Standard Time 2005