Seminars and Colloquia by Series

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 MargalitGeorgia 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 PardonPrinceton 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 GhomiGa 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 TimofteIAS, 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 ClayAllegheny 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 BelegradekGeorgia 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 DancsoUniversity 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.)

Curve operators and Toeplitz operators in TQFT.

Series
Geometry Topology Seminar
Time
Friday, November 19, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Julien Marche Paris VII & Ecole Polytechnique
Topological quantum field theory associates to a surface a sequence of vector spaces and to curves on the surface, sequence of operators on that spaces. It is expected that these operators are Toeplitz although there is no general proof. I will state it in some particular cases and give applications to the asymptotics of quantum invariants like quantum 6-j symbols or quantum invariants of Dehn fillings of the figure eight knot. This is work in progress with (independently) L. Charles and T. Paul.

Dilatation vs self-intersection number for point-pushing pseudo-Anosovs

Series
Geometry Topology Seminar
Time
Monday, November 15, 2010 - 17:00 for 1 hour (actually 50 minutes)
Location
Room 326, Boyd Graduate Studies (UGA)
Speaker
Spencer DowdallUniversity of Chicago
This talk is about the dilatations of pseudo-Anosov mapping classes obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. I'll also give bounds on the least dilatation of any pseudo-Anosov in the point-pushing subgroup and describe the asymptotic dependence on self-intersection number. All of the upper bounds involve analyzing explicit examples using train tracks, and the lower bound is obtained by lifting to the universal cover and studying the images of simple closed curves.

Semi-infinite cycles in Floer Theory

Series
Geometry Topology Seminar
Time
Monday, November 15, 2010 - 15:45 for 1 hour (actually 50 minutes)
Location
Room 326, Boyd Graduate Studies (UGA)
Speaker
Max LipyanskiyColumbia University

Please Note: This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will begin at 5:00. (NOTE: These talks are on the UGA campus.)

I will survey the program of realizing various versions of Floer homology as a theory of geometric cycles. This involves the description of infinite dimensional manifolds mapping to the relevant configuration spaces. This approach, which goes back to Atiyah's address at the Herman Weyl symposium, is in some ways technically simpler than the traditional construction based on Floer's version of Morse theory. In addition, it opens up the possibility of defining more refined invariants such as bordism andK-theory.

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