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Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

We will discuss how to define two invariants of knots using sutured Heegaard Floer homology, contact structures and limiting processes. These invariants turn out to be a reformulation of the plus and minus versions of knot Heegaard Floer homology and thus give a``sutured interpretation'' of these invariants and point to a deep connection between Heegaard Floer theory and contact geometry. If time permits we will also discuss the possibility of defining invariants of non-compact manifolds and of contact structures on such manifolds.

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.