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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

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.