Monday, February 20, 2012 - 14:05 , Location: Skiles 005 , Stavros Garoufalidis , Georgia Tech , firstname.lastname@example.org , Organizer: Stavros Garoufalidis
The Neumann-Zagier equations are well-understood objects of classical hyperbolic geometry. Our discovery is that they have a nontrivial quantum content, (that for instance captures the perturbation theory of the Kashaev invariant to all orders) expressed via universal combinatorial formulas. Joint work with Tudor Dimofte.
Friday, February 17, 2012 - 14:00 , Location: Skiles 006 , Alexandra Pettet , University of British Columbia , Organizer: Dan Margalit
The outer automorphism group Out(F) of a non-abelian free group F of finite rank shares many properties with linear groups and the mapping class group Mod(S) of a surface, although the techniques for studying Out(F) are often quite different from the latter two. Motivated by analogy, I will present some results about Out(F) previously well-known for the mapping class group, and highlight some of the features in the proofs which distinguish it from Mod(S).
Monday, January 23, 2012 - 15:05 , Location: Skiles 006 , Alexander Getmanenko , IPMU Japan , email@example.com , Organizer: Stavros Garoufalidis
In a joint work with D.Tamarkin we study analytic continuability of solutions of theLaplace-transformed Schroedinger equation by methods of Kashiwara-Schapira style microlocal theoryof sheaves.
Monday, January 23, 2012 - 14:05 , Location: Skiles 006 , Jenny Wilson , University of Chicago , Organizer: Dan Margalit
In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the H^k(P\Sigma_n, \Q) tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n, that is, in some precise sense, the description of the decomposition of the cohomology group into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.
Monday, December 5, 2011 - 14:00 , Location: Skiles 005 , Emmy Murphy , Stanford University , Organizer: John Etnyre
In the talk, I plan to give a definition of loose Legendrian knots inside contact manifolds of dimension 5 or greater. The definition is significantly different from the 3 dimensional case, in particular loose knots exist in local charts. I'll discuss an h-principle for such knots. This implies their classification, a bijective correspondence with their formal (algebraic topology) invariants. I'll also discuss applications of this result, comparisons with 3D contact toplogy, and some open questions.
Monday, November 28, 2011 - 14:00 , Location: Skiles 005 , Doug LaFountain , Aarhus Universitet , Organizer: John Etnyre
For a genus g surface with s > 0 punctures and 2g+s > 2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner.
Monday, November 14, 2011 - 14:00 , Location: Skiles 005 , Jen Hom , Columbia University , Organizer: John Etnyre
We will use a new concordance invariant, epsilon, associated to the knot Floer complex, to define a smooth concordance homomorphism. Applications include a new infinite family of smoothly independent topologically slice knots, bounds on the concordance genus, and information about tau of satellites. We will also discuss various algebraic properties of this construction.
Monday, November 7, 2011 - 14:00 , Location: Skiles 005 , Clay Shonkwiler , UGA , Organizer: John Etnyre
In 1997 Hausmann and Knutson discovered a remarkable correspondence between complex Grassmannians and closed polygons which yields a natural symmetric Riemannian metric on the space of polygons. In this talk I will describe how these symmetries can be exploited to make interesting calculations in the probability theory of the space of polygons, including simple and explicit formulae for the expected values of chord lengths. I will also give a simple and fast algorithm for sampling random polygons--which serve as a statistical model for polymers--directly from this probability distribution.