## Seminars and Colloquia by Series

Monday, February 9, 2015 - 14:00 , Location: Skiles 006 , Will Kazez , University of Georgia , Organizer: John Etnyre
I will discuss Eliashberg and Thurston's theorem that C^2 taut foliations can be approximated by tight contact structures.  I will try to explain the importance of their work and why it is useful to weaken their smoothness assumption.  This work is joint with Rachel Roberts.
Monday, February 2, 2015 - 14:00 , Location: Skiles 006 , Joseph Hirsh , MIT , Organizer: Kirsten Wickelgren
Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the  weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which  induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two  extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.
Monday, January 26, 2015 - 14:00 , Location: Skiles 006 , Ina Petkova , Rice University , Organizer: John Etnyre
In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss theconstruction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3 gives back a stabilized version of knot Floer homology.
Tuesday, January 20, 2015 - 14:05 , Location: Skiles 006 , , UCDavis , , Organizer: Stavros Garoufalidis
Among n-dimensional regions with fixed volume, which one hasthe least boundary?   This question is known as an isoperimetricproblem; its nature depends on what is meant by a "region".   I willdiscuss variations of an isoperimetric problem known as thegeneralized Cartan-Hadamard conjecture:  If Ω is a region in acomplete, simply connected n-manifold with curvature bounded above byκ ≤ 0, then does it have the least boundary when the curvature equalsκ and Ω is round?  This conjecture was proven when n = 2 by Weil andBol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke.  Injoint work with Benoit Kloeckner, we generalize Croke's result to mostof the case κ < 0, and we establish a theorem for κ > 0.   It was originally inspired by the problem of finding the optimal shape of aplanet to maximize gravity at a single point, such as the place wherethe Little Prince stands on his own small planet.
Monday, January 19, 2015 - 14:05 , Location: Skiles 006 , None , None , Organizer: Dan Margalit
Friday, January 9, 2015 - 14:05 , Location: Skiles 006 , , IAS, Princeton , , Organizer: Stavros Garoufalidis
Recently, a "symplectic duality" between D-modules on certainpairs of algebraic symplectic manifolds was discovered, generalizingclassic work of Beilinson-Ginzburg-Soergel in geometric representationtheory. I will discuss how such dual spaces (some known and some new) arisenaturally in supersymmetric gauge theory in three dimensions.
Monday, December 8, 2014 - 14:00 , Location: Skiles 006 , Emily Riehl , Harvard University , Organizer: Kirsten Wickelgren
Groups, rings, modules, and compact Hausdorff spaces have underlying sets ("forgetting" structure)  and admit "free" constructions. Moreover, each type of object is completely characterized by the  shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding  provides formulas for direct and inverse limits. After we describe a typical encounter with  adjunctions, monads, and their algebras, we introduce a new "homotopy coherent" version of this  adjoint duality together with a graphical calculus that is used to define a homotopy coherent  algebra in quite general contexts, such as appear in abstract homotopy theory or derived algebraic  geometry.
Monday, December 1, 2014 - 14:00 , Location: Skiles 006 , Tye Lidman , University of Texas, Austin , Organizer: John Etnyre
The Lickorish-Wallace theorem states that every closed, connected, orientable three-manifold can be expressed as surgery on a link in the three-sphere (i.e., remove a neighborhood of a disjoint union of embedded $S^1$'s from $S^3$ and re-glue).  It is natural to ask which three-manifolds can be obtained by surgery on a single knot in the three-sphere.  We discuss a new way to obstruct integer homology spheres from being surgery on a knot and give some examples.  This is joint work with Jennifer Hom and Cagri Karakurt.
Monday, November 24, 2014 - 14:05 , Location: Skiles 006 , Igor Belegradek , Georgia Tech , Organizer: Igor Belegradek
I will sketch how to detect nontrivial higher homotopy groups of the space of complete nonnegatively curved metrics on an open manifold.
Monday, November 17, 2014 - 14:00 , Location: Skiles 006 , David Gepner , Purdue University , Organizer: Kirsten Wickelgren
The algebraic K-theory of the sphere spectrum, K(S), encodes significant information in both homotopy theory and differential topology. In order to understand K(S), one can apply the techniques of chromatic homotopy theory in an attempt to approximate K(S) by certain localizations K(L_n S). The L_n S are in turn approximated by the Johnson-Wilson spectra E(n) = BP[v_n^{-1}], and it is not unreasonable to expect to be able to compute K(BP). This would lead inductively to information about K(E(n)) via the conjectural fiber sequence K(BP) --> K(BP) --> K(E(n)). In this talk, I will explain the basics of the K-theory of ring spectra, define the ring spectra of interest, and construct some actual localization sequences in their K-theory. I will then use trace methods to show that it the actual fiber of K(BP) --> K(E(n)) differs from K(BP), meaning that the situation is more complicated than was originally hoped. All this is joint work with Ben Antieau and Tobias Barthel.