Seminars and Colloquia by Series

Spinal Open Books and Symplectic Fillings

Series
Geometry Topology Seminar
Time
Monday, February 21, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeremy Van Horn-Morris – AIM
A theorem of Chris Wendl allows you to completely characterize symplectic fillings of certain open book decompositions by factorizations of their monodromy into Dehn twists. Olga Plamenevskaya and I use this to generalize results of Eliashberg, McDuff and Lisca to classify the fillings of certain Lens spaces. I'll discuss this and a newer version of Wendl's theorem, joint with Wendl and Sam Lisi, this time for spinal open books, and discuss a few more applications.

Legendrian and transverse knots in cabled knot types

Series
Geometry Topology Seminar
Time
Monday, February 14, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bulent Tosun – Ga Tech
In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain

Hyperbolic polyhedra and the Jones polynomial

Series
Geometry Topology Seminar
Time
Monday, February 7, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roland van der Veen – UCBerkeley –
For knots the hyperbolic geometry of the complement is known to be relatedto itsJones polynomial in various ways. We propose to study this relationship morecloselyby extending the Jones polynomial to graphs. For a planar graph we will showhow itsJones polynomial then gives rise to the hyperbolic volume of the polyhedronwhose1-skeleton is the graph. Joint with Francois Gueritaud and FrancoisCostantino.

Braid groups and symplectic groups

Series
Geometry Topology Seminar
Time
Monday, February 7, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Margalit – Georgia Tech
The braid group embeds in the mapping class group, and so the symplectic representation of the mapping class group gives rise to a symplectic represenation of the braid group. The basic question Tara Brendle and I are trying to answer is: how can we describe the kernel? Hain and Morifuji have conjectured that the kernel is generated by Dehn twists. I will present some progress/evidence towards this conjecture.

Gromov's knot distortion

Series
Geometry Topology Seminar
Time
Friday, January 28, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John Pardon – Princeton University –
Gromov defined the distortion of an embedding of S^1 into R^3 and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of S^1 into R^3 with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will show how to give a nontrivial lower bound on the distortion of torus knots, which is sharp in the case of (p,p+1) torus knots. I will also mention some natural conjectures about the distortion, for example that the distortion of the (2,p)-torus knots is unbounded.

Caratheodory's conjecture on umbilical points of convex surfaces

Series
Geometry Topology Seminar
Time
Monday, January 24, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mohammad Ghomi – Ga Tech
Caratheodory's famous conjecture, dating back to 1920's, states that every closed convex surface has at least two umbilics, i.e., points where the principal curvatures are equal, or, equivalently, the surface has contact of order 2 with a sphere. In this talk I report on recent work with Ralph howard where we apply the divergence theorem to obtain integral equalities which establish some weak forms of the conjecture.

Quantum Curves in Chern-Simons Theory

Series
Geometry Topology Seminar
Time
Wednesday, January 19, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tudor Timofte – IAS, Princeton –
I will discuss a new general framework for cutting and gluing manifolds in topological quantum field theory (TQFT). Applying this method to Chern-Simons theory with gauge group SL(2,C) on a knot complement M leads to a systematic quantization of the SL(2,C) character variety of M. In particular, the classical A-polynomial of M becomes an operator "A-hat", the same operator that appears in the recursion relations of Garoufalidis et al. for colored Jones polynomials.

The geometry of right-angled Artin subgroups of mapping class groups

Series
Geometry Topology Seminar
Time
Monday, January 10, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matt Clay – Allegheny College
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin subgroup quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. This is joint work with Chris Leininger and Johanna Mangahas.

Hyperbolicity of hyperplane complements

Series
Geometry Topology Seminar
Time
Monday, December 6, 2010 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor Belegradek – Georgia Tech
We will discuss properties of manifolds obtained by deleting a totally geodesic ``divisor'' from hyperbolic manifold. Fundamental groups of these manifolds do not generally fit into any class of groups studied by the geometric group theory, yet the groups turn out to be relatively hyperbolic when the divisor is ``sparse'' and has ``normal crossings''.

A homomorphic universal finite type invariant of knotted trivalent graphs

Series
Geometry Topology Seminar
Time
Monday, November 29, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Zsuzsanna Dancso – University of Toronto
Knotted trivalent graphs (KTGs) along with standard operations defined on them form a finitely presented algebraic structure which includes knots, and in which many topological knot properties are defineable using simple formulas. Thus, a homomorphic invariant of KTGs places knot theory in an algebraic context. In this talk we construct such an invariant: the starting point is extending the Kontsevich integral of knots to KTGs. This was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki in the late 90's using the theory of associators. We present an elementary construction building on Kontsevich's original definition, and discuss the homomorphic properties of the invariant, which, as it turns out, intertwines all the standard KTG operations except for one, called the edge unzip. We prove that in fact no universal finite type invariant of KTGs can intertwine all the standard operations at once, and present an alternative construction of the space of KTGs on which a homomorphic universal finite type invariant exists. This space retains all the good properties of the original KTGs: it is finitely presented, includes knots, and is closely related to Drinfel'd associators. (Partly joint work with Dror Bar-Natan.)

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