Seminars and Colloquia by Series

Tight small Seifert fibered manifolds

Series
Geometry Topology Seminar
Time
Monday, October 28, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bulent Tosun – University of Virginia
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.

Essential spunnormal surfaces via tropical geometry

Series
Geometry Topology Seminar
Time
Monday, October 7, 2013 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew Brasile – University of Illinois at Chicago –
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.

Relative Symplectic Caps, 4-Genus and Fibered Knots

Series
Geometry Topology Seminar
Time
Monday, September 30, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dheeraj Kulkarni – Georgia Tech
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

Congruence subgroups of braid groups

Series
Geometry Topology Seminar
Time
Monday, September 23, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tara Brendle – U Glasgow
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.

The decategorification of bordered Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, September 16, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lawrence Roberts – University of Alabama
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.

Complete nonnegatively curved planes

Series
Geometry Topology Seminar
Time
Monday, September 9, 2013 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor Belegradek – Georgia Tech
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.

The arc complex and contact geometry: non-destabilizable planar open book decompositions of the tight contact 3-sphere

Series
Geometry Topology Seminar
Time
Wednesday, September 4, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Youlin Li – Georgia Tech
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.

Sutured manifolds, limits and knot Heegaard Floer homology

Series
Geometry Topology Seminar
Time
Monday, August 26, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech
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.

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