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Series: Geometry Topology Seminar

The so-called integral Burau representation gives a
symplectic representation of the braid group. In this talk we will
discuss the resulting congruence subgroups of braid groups, that is,
preimages of the principal congruence subgroups of the symplectic
group. In particular, we will show that the level 4 congruence braid
group is equal to the group generated by squares of Dehn twists. One
key tool is a "squared lantern relation" amongst Dehn twists. Joint work with Dan Margalit.

Series: Geometry Topology Seminar

Khovanov homology is an invariant of a link in S^3 which refines the Jones polynomial of the link. Recently I defined a version of Khovanov homology for tangles with interesting locality and gluing properties, currently called bordered Khovanov homology, which follows the algebraic pattern of bordered Floer homology. After reviewing the ideas behind bordered Khovanov homology, I will describe what appears to be the Jones polynomial-like structure which bordered Khovanov homology refines.

Series: Geometry Topology Seminar

It is known that any complete nonnegatively curved metric on the plane is conformally equivalent to the Euclidean metric. In the first half of the talk I shall explain that the conformal factors that show up correspond precisely to smooth subharmonic functions of minimal growth. The proof is function-theoretic. This characterization of conformal factors can be used to study connectedness properties of the space of complete nonnegatively curved metrics on the plane. A typical result is that the space of metrics cannot be separated by a finite dimensional subspace. The proofs use infinite-dimensional topology and dimension theory. This is a joint work with Jing Hu.

Series: Geometry Topology Seminar

We introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this result to show that there are planer open books of the standard contact structure on the 3-sphere with 5 (or any number larger than 5) boundary components that do not destabilize. We also show that any planar open book with 4 or fewer boundary components does destabilize. This is joint work with John Etnyre.

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.