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Series: Geometry Topology Seminar

I will describe new
techniques for computing the homology of braid groups with coefficients
in certain exponential coefficient systems. An unexpected side of this
story (at least to me) is a connection with the cohomology of certain
braided Hopf
algebras — quantum shuffle algebras and Nichols algebras — which are
central to the classification of pointed Hopf algebras and quantum
groups. We can apply these tools to get a bound on the growth of the
cohomology of Hurwitz moduli spaces of branched covers
of the plane in certain cases. This yields a weak form of Malle’s
conjecture on the distribution of fields with prescribed Galois group in
the function field setting. This is joint work with Jordan Ellenberg
and TriThang Tran.

Series: Geometry Topology Seminar

We consider two knot diagrams to be equivalent if they are isotopic without Reidemeister moves, and prove a method for determining if the equivalence class of a knot diagram contains a representative that is the Lagrangian projection of a Legendrian knot. This work gives us a new tool for determining if a Legendrian knot can be de-stabilized.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including RPn and CPn, matching a mirror prediction due to Huybrechts and Thomas. The idea of the proof can be interpreted as a "mirror" of the construction in algebraic geometry, realized by a new surgery and cobordism construction. This is a joint work with Cheuk-Yu Mak.

Series: Geometry Topology Seminar

Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X? For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers. In joint work with Ghaswala, we completely answer the question for cyclic branched covers. When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X. In a family where the answer is no, we find a presentation for the group of isotopy classes of homeomorphisms of the sphere that do lift, which is a finite index subgroup of the mapping class group of the sphere. Our family introduces new examples of orbifold Picard groups of subloci of Teichmuller space that are finitely generated but not cyclic.

Series: Geometry Topology Seminar

Existence of tight contact structures is a fundamental question of contact topology. Etnyre and Honda first gave the example which doesn't admit any tight structure. The existence of fillable tight structures is also a subtle question. Here we give some new examples of hyperbolic 3-manifolds which do not admit any fillable structures.

Series: Geometry Topology Seminar

We discuss the higher Teichmuller space A_{G,S} defined by Fockand Goncharov. This space is defined for a punctured surface S withnegative Euler characteristic, and a semisimple, simply connected Lie groupG. There is a birational atlas on A_{G,S} with a chart for each idealtriangulation of S. Fock and Goncharov showed that the transition functionsare positive, i.e. subtraction-free rational functions. We will show thatwhen G has rank 2, the transition functions are given by explicit quivermutations.

Series: Geometry Topology Seminar

Given a plane field $dz-xdy$ in $\mathbb{R}^3$. A Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in $\mathbb{R}^3$, $J^1(S^1)$, and $\#^k(S^1\times S^2)$ as well as a few Legendrian knot invariants. We will also look at the relationships between a few of these knot invariants. No knowledge of Legendrian knots will be assumed though some knowledge of basic knot theory would be useful.

Series: Geometry Topology Seminar

Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using an obstruction coming from Heegaard Floer homology, we will provide infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition. This is joint work with Lidman.

Series: Geometry Topology Seminar

In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1,2,4, 8. This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace. I have been studying Hurewicz maps for infinite loopspaces by showing how a filtration of the homotopy groups coming from stable homotopy theory (the Adams filtration) interacts with a filtration of the homology groups coming from infinite loopspace theory. There are some clean and tidy consequences that lead to a new proof of Milnor's theorem, and other applications.