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Series: Geometry Topology Seminar

Legendrian contact homology is an invariant in contact geometry that assigns to each Legendrian submanifold a dg-algebra. While well-defined, it depends upon counts of holomorphic curves that can be hard to calculate in practice. In this talk, we introduce a class of Legendrian tori constructed as the product of collections of Legendrian knots. For this class, we discuss how to explicitly compute the dg-algebra invariant of the tori in terms of diagram projections of the constituent Legendrian knots.

Series: Geometry Topology Seminar

Note different time and day.

A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a stabilization-invariant property of open books which corresponds to tightness of the corresponding contact structure. We will mention applications to the classification of contact 3-folds, and also to the question of whether tightness is preserved under Legendrian surgery.

Series: Geometry Topology Seminar

It is a natural question to ask whether one can deduce topological
properties of a finite--volume three--manifold from its Riemannian
invariants such as volume and systole. In all generality this is
impossible, for example a given manifold has sequences of finite covers
with either linear or sub-linear growth. However under a geometric
assumption, which is satisfied for example by some naturally defined
sequences of arithmetic manifolds, one can prove results on the
asymptotics of the first integral homology. I will try to explain these
results in the compact case (this is part of a joint work with M. Abert,
N. Bergeron, I. Biringer, T. Gelander, N. Nikolov and I. Samet) and time
permitting I will discuss their extension to manifolds with cusps such
as hyperbolic knot complements.

Series: Geometry Topology Seminar

A group is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. This class encompasses many examples of interest: hyperbolic and relatively hyperbolic groups, Out(F_n) for n>1, all but finitely many mapping class groups, most fundamental groups of 3-manifolds, groups acting properly on proper CAT(0) spaces and containing rank 1 elements, 1-relator groups with at least 3 generators, etc. On the other hand, many results known for these particular classes can be naturally generalized in the context of acylindrically hyperbolic groups. In my talk I will survey some recent progress in this direction. The talk is partially based on my joint papers with F. Dahmani, V. Guirardel, M.Hull, and A. Minasyan.

Series: Geometry Topology Seminar

A few years ago I proved that any rectangular diagram of the
unknot admits monotonic simplification by elementary moves. More recently
M.Prasolov and I addressed the question: when a rectangular diagram of a
link admits at least one step of simplification? It turned out that an
answer can be given naturally in terms of Legendrian links. On this way,
we resolved positively a conjecture by V.Jones on the invariance of the
algebraic crossing number of a minimal braid, and a few similar questions.

Series: Geometry Topology Seminar

Thurston's gluing equations are polynomial equations invented byThurston to explicitly compute hyperbolic structures or, more generally, representations in PGL(2,C). This is done via so called shape coordinates.We generalize the shape coordinates to obtain a parametrization ofrepresentations in PGL(n,C). We give applications to quantum topology, anddiscuss an intriguing duality between the shape coordinates and thePtolemy coordinates of Garoufalidis-Thurston-Zickert. The shapecoordinates and Ptolemy coordinates can be viewed as 3-dimensional analogues of the X- and A-coordinates on higher Teichmuller spaces due toFock and Goncharov.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Note: this is a 40 minute talk.

We will explore the notion of surgery on transverse knots in contact 3-manifolds. We will see situations when this operation does or does not preserves properties of the original contact structure, and avenues for further research.

Series: Geometry Topology Seminar

We introduce a new surgery operation for contact manifolds called the Liouville connect sum. This operation -- which includes Weinstein handle attachment as a special case -- is designed to study the relationship between contact topology and symplectomorphism groups established by work of Giroux and Thurston-Winkelnkemper. The Liouville connect sum is used to generalize results of Baker-Etnyre-Van Horn-Morris and Baldwin on the existence of "monodromy multiplication cobordisms" as well as results of Seidel regarding squares of symplectic Dehn twists.

Series: Geometry Topology Seminar

Given any smooth manifold, there is a canonical symplectic structure on its cotangent bundle. A long standing idea of Arnol'd suggests that the symplectic topology of the cotangent bundle should contain a great deal of information about the smooth topology of its base. As a contrast, I show that when X is an open 4-manifold, this symplectic structure on T^*X does not depend on the choice of smooth structure on X. I will also discuss the particular cases of smooth structures on R^4 and once-punctured compact 4-manifolds.