Seminars and Colloquia by Series

Monday, December 9, 2013 - 14:00 , Location: Skiles 006 , Kyle Ormsby , MIT , Organizer: Kirsten Wickelgren
For a group G, stable G-equivariant homotopy theory studies (the stabilizations of) topological spaces with a G-action up to G-homotopy.  For a field k, stable motivic homotopy theory studies varieties over k up to (a stable notion of) homotopy where the affine line plays the role of the unit interval.  When L/k is a finite Galois extension with Galois group G, there is a functor F from the G-equivariant stable homotopy category to the stable motivic homotopy category of k.  If k is the complex numbers (or any algebraically closed characteristic 0 field) and L=k (so G is trivial), then Marc Levine has shown that F is full and faithful.  If k is the real numbers (or any real closed field) and L=k[i], we show that F is again full and faithful, i.e., that there is a "copy" of stable C_2-equivariant homotopy theory inside of the stable motivic homotopy category of R.  We will explore computational implications of this theorem.This is a report on joint work with Jeremiah Heller.
Monday, December 2, 2013 - 14:00 , Location: Skiles 006 , Mohammad Ghomi , Georgia Tech , , Organizer: Mohammad Ghomi
The total diameter of a closed planar curve C is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when C is a circle. We also generalize these results to m dimensional submanifolds of R^n, where the "area" will be defined in terms of the mod 2 winding numbers of the submanifold about the n-m-1 dimensional affine subspaces of R^n.
Monday, November 25, 2013 - 14:00 , Location: Skiles 006 , Patricia Cahn , University of Pennsylvania , Organizer: John Etnyre
We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers. This is joint work with Asa Levi.
Friday, November 22, 2013 - 14:05 , Location: Skiles 006 , Bill Menasco , U at Buffalo , Organizer: Dan Margalit
We show that after stabilizations of opposite parity and braid isotopy, any twobraids in the same topological link type cobound embedded annuli. We use this to prove thegeneralized Jones conjecture relating the braid index and algebraic length of closed braidswithin a link type, following a reformulation of the problem by Kawamuro.  This is joint workwith Doug Lafountain.
Monday, November 18, 2013 - 14:00 , Location: Skiles 006 , Vesna Stojanoska , MIT , , Organizer: Kirsten Wickelgren
For a fixed integer n, consider the nerve L_n of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space L_n has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points. Note that p-toral groups are a Lie analogue of finite p-groups, thus if we are interested in the U(n)-space L_n at a fixed prime p, only the p-toral subgroups of U(n) play a significant role. The space L_n is strongly related to the K-theory analogues of the symmetric powers of spheres and the Weiss tower for the functor that assigns to a vector space V the classifying space BU(V). Our results are a step toward a K-theory analogue of the Whitehead conjecture as part of the program of Arone-Dwyer-Lesh. This is joint work with J.Bergner, R.Joachimi, K.Lesh, K.Wickelgren.
Monday, November 11, 2013 - 14:00 , Location: Skiles 006 , Peter Samuelson , University of Toronto , Organizer: Thang Le
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.)
Friday, November 8, 2013 - 14:05 , Location: Skiles 006 , G. Masbaum , Institut de Mathématiques de Jussieu , Organizer: Thang Le
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.)
Monday, November 4, 2013 - 14:00 , Location: Skiles 006 , Jesse Johnson , Oklahoma State University , Organizer: John Etnyre
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.
Monday, October 28, 2013 - 14:00 , Location: Skiles 006 , Bulent Tosun , University of Virginia , Organizer: John Etnyre
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.
Monday, October 14, 2013 - 14:05 , Location: Skiles 006 , None , UNC Greensboro , Organizer: Dan Margalit