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Series: Geometry Topology Seminar

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers. This is joint work with Asa Levi.

Series: Geometry Topology Seminar

We show that after stabilizations of opposite parity and braid isotopy, any twobraids in the same topological link type cobound embedded annuli. We use this to prove thegeneralized Jones conjecture relating the braid index and algebraic length of closed braidswithin a link type, following a reformulation of the problem by Kawamuro. This is joint workwith Doug Lafountain.

Series: Geometry Topology Seminar

For a fixed integer n, consider the nerve L_n of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space L_n has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points. Note that p-toral groups are a Lie analogue of finite p-groups, thus if we are interested in the U(n)-space L_n at a fixed prime p, only the p-toral subgroups of U(n) play a significant role. The space L_n is strongly related to the K-theory analogues of the symmetric powers of spheres and the Weiss tower for the functor that assigns to a vector space V the classifying space BU(V). Our results are a step toward a K-theory analogue of the Whitehead conjecture as part of the program of Arone-Dwyer-Lesh. This is joint work with J.Bergner, R.Joachimi, K.Lesh, K.Wickelgren.

Series: Geometry Topology Seminar

Frohman and Gelca showed that the Kauffman bracket skein module of the
thickened torus is the Z_2 invariant subalgebra A'_q of the quantum torus
A_q. This shows that the Kauffman bracket skein module of a knot complement
is a module over A'_q. We discuss a conjecture that this module is
naturally a module over the double affine Hecke algebra H, which is a
3-parameter family of algebras which specializes to A'_q. We give some
evidence for this conjecture and then discuss some corollaries. If time
permits we will also discuss a related topological construction of a
2-parameter family of H-modules associated to a knot in S^3. (All results
in this talk are joint with Yuri Berest.)

Series: Geometry Topology Seminar

Let g be a positive integer and let Gamma_g be the mapping
class group of the genus g closed orientable surface. We show that
every finite group is involved in Gamma_g. (Here a group G is said to
be involved in a group Gamma if G is isomorphic to a quotient of a
subgroup of Gamma of finite index.) This answers a question asked by
U. Hamenstadt. The proof uses quantum representations of mapping class
groups. (Joint work with A. Reid.)

Series: Geometry Topology Seminar

The notion of distance for a Heegaard splitting of athree-dimensional manifold $M$, introduced by John Hempel, has provedto be a very powerful tool for understanding the geometry and topologyof $M$. I will describe how distance, and a slight generalizationknown as subsurface projection distance, can be used to explore theconnection between geometry and topology at the center of the moderntheory hyperbolic three-manifolds.In particular, Schalremann-Tomova showed that if a Heegaard splittingfor $M$ has high distance then it will be the only irreducibleHeegaard splitting of $M$ with genus less than a certain bound. I willexplain this result in terms of both a geometric proof and atopological proof. Then, using the notion of subsurface distance, Iwill describe a construction of a manifold with multiple distinctlow-distance Heegaard splittings of the same (small) genus, and amanifold with both a high distance, low-genus Heegaard splitting and adistinct, irreducible high-genus, low-distance Heegaard splitting.

Series: Geometry Topology Seminar

Contact geometry in three dimensions is a land of two disjoint classes ofcontact structures; overtwisted vs. tight. The former ones are flexible,means their geometry is determined by algebraic topology of underlying twoplane fields. In particular their existence and classification areunderstood completely. Tight contact structure, on the other hand, arerigid. The existence problem of a tight contact structure on a fixed threemanifold is hard and still widely open. The classification problem is evenharder. In this talk, we will focus on the classification of tight contactstructures on Seifert fibered manifolds on which the existence problem oftight contact structures was settled recently by Lisca and Stipsicz.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

In a paper published in 2012, Nathan Dunfield and StavrosGaroufalidis gave simple, sufficient conditions for a spunnormal surface tobe essential in a compact, orientable 3-manifold with torus boundary. Thistalk will discuss a generalization of this result which utilizes a theoremfrom tropical geometry.

Series: Geometry Topology Seminar

The $4$-genus of a knot is an important measure of complexity, related tothe unknotting number. A fundamental result used to study the $4$-genusand related invariants of homology classes is the Thom conjecture,proved by Kronheimer-Mrowka, and its symplectic extension due toOzsvath-Szabo, which say that closed symplectic surfacesminimize genus.Suppose (X, \omega) is a symplectic 4-manifold with contact type bounday and Sigma is a symplectic surface in X such that its boundary is a transverse knot in the boundary of X. In this talk we show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, \Sigma) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in the 3-sphere.We will also discuss a relative version of Giroux's criterion of Stein fillability. This is joint work with Siddhartha Gadgil