Topology Seminar
Johns Hopkins University

Unless otherwise noted, this seminar meets every Monday, from 4:30pm to 5:30pm Room 308 Krieger Hall. If you would like to fill one of the blank spots in the schedule below, or if you know a potential speaker, please contact one of the seminar co-organizers, John Lind or José Manuel Gómez.

Upcoming talk

Monday May 7. Frank Quinn. Virginia Tech.
Title: The missing global theory of smooth manifolds
Abstract: High-dimensional manifold theory developed largely between 1940 and 1980, with surgery, s-cobordism theorem, etc. Looking back after the dust settled and the material has been digested, we see that there are two main families. The topological family includes PL, top, and ANR homology manifolds, which differ slightly but are very similar. There is a slick and effective global theory describing these. The smooth family includes differentiable, real analytic, and probably real algebraic. The first two are essentially the same (I'm not clear about the third). The current global theory is far less effective than the topological case. I'll explain this, and describe "design criteria" for a better theory. Ingredients include duality with modern ring spectra, the image of J, and clues from algebraic geometry.

Complete list of talks

Monday Jan 9. Andrew Linshaw. Brandeis University.
Title: Jet schemes and invariant theory
Abstract: Given a complex, reductive algebraic group G and a G-module V, the mth jet scheme G_m acts on the mth jet scheme V_m, for all m>0. I will discuss the ring of G_m invariant functions on V_m, and its relationship to the ring of functions on (V//G)_m, where V//G is the categorical quotient. Time permitting, I will discuss some applications of our results to vertex algebras. This is a joint work with Gerald Schwarz (Brandeis) and Bailin Song (University of Science and Technology of China).

Monday Feb 6. Jack Morava. Johns Hopkins University.
Title: Cobordism of quasitoric manifolds
Abstract: A quasitoric manifold M^{2n} is (roughly) a compact complex-oriented manifold with an almost-free action of an n-torus. They are very easy to work with, because M can be reconstructed from the quotient M/T^n, which is a complex polytope. Any complex-oriented manifold is cobordant to a quasitoric manifold, but cobordism of qtoric manifolds is a finer equivalence relation; it is very rich, and its study is just beginning, cf http://arxiv.org/abs/1201.3127

Monday Feb 20. David Ayala. Harvard University.
Title: Locally standard geometries and algebraic structures
Abstract: For M a suitable category of topological spaces which are locally prescribed, I will characterize homology theories on M in terms of algebraic data. Examples of such a category includes that of n-manifolds for some fixed n, and that of 3-manifolds with an embedded 1-manifold; examples of such algebraic data is that of an E_n algebra, or an associative algebra with a suitable action of an E_3 algebra. For such a collection M, I will also discuss Poincare' duality with coefficients in an algebra in the sense of this algebraic data. The techniques involve (higher) operads and category theory. This work is joint with John Francis and Hiro Tanaka.

Monday Feb 27. Fred Cohen. University of Rochester.
Title: Packings of pennies in the plane and their stable decompositions
Abstract: Topological spaces given by either (1) complements of coordinate planes in Euclidean space or (2) spaces of non-overlapping hard-disks in a fixed disk ("packings of pennies in the plane") have several features in common. The main results, in joint work with L. Taylor, and S. Gitler, give decompositions for the so-called "stable structure" of these spaces as well as consequences of these decompositions.
This talk will present definitions, basic properties together with some recent results.

Monday March 5. Kirsten Wickelgren. Harvard University.
Title: 2-nilpotent real section conjecture
Abstract: Sullivan's conjecture, proven by Haynes Miller and Gunnar Carlsson, relates the fixed points to the homotopy fixed points of p-group actions on finite complexes. Applying this result to algebraic curves defined over R with the action of complex conjugation gives the real analogue of Grothendieck's section conjecture predicting that the rational points on curves over finitely generated fields are determined by maps between etale fundamental groups. By examining the symmetric powers of curves, we show a 2-nilpotent section conjecture over R: for a curve X over R such that each component of its normalization has real points, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the topological fundamental group or etale fundamental group with its Z/2 action. This implies that the set of real points equipped with a real tangent direction of a smooth compact curve X is determined by the maximal 2-nilpotent quotient of the absolute Galois group of the function field, showing a 2-nilpotent birational real section conjecture.

Monday March 12. No Seminar (Kempf Lectures).
Title: TBA
Abstract: TBA

Monday March 19. No Seminar (Spring Break).
Title: TBA
Abstract: TBA

Monday March 26. Inna Zakharevich. MIT.
Title: A K-theoretic perspective on scissors congruence problems
Abstract: Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. We will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.

Monday April 2. Nitu Kitchloo. Johns Hopkins University.
Title: The Stable Symplectic Category
Abstract: I will describe a stabilization procedure on the Symplectic category (which was introduced by A. Weinstein as an attempt in formalizing the geometric quantization functor). Our stable symplectic category has several appealing properties including the fact that it is indeed a stable category (in terms of stable homotopy theory). I'll provide as much motivation as I can. No special background is needed, except some familiarity with the concept of a symplectic manifold. If you are a champion at geometric quantization, please come.

Monday April 9. Mike Hoffman. U.S. Naval Academy.
Title: Multiple zeta values: Old conjectures and new results
Abstract: In the 1990's Zagier made a conjecture about the dimension of the rational vector space spanned by multiple zeta values (MZVs), and I refined the conjecture by suggesting an actual basis. It's very hard to prove most particular MZVs irrational (e.g., even zeta(5) isn't known to be irrational), so proving the linear independence of the purported basis seems out of reach. But recently Francis Brown did the next best thing by showing the purported basis spans the MZVs. I will talk about this, concentrating on a key ingredient supplied by Zagier.

Monday April 16. Igor Kriz. University of Michigan.
Title: Field theories, infinite loop spaces and Khovanov homology
Abstract: In this talk (reporting on joint work with Po Hu and Daniel Kriz), I will discuss 1+1 topological quantum field theories valued in symmetric bimonoidal categories, and a construction of resulting field theories valued in module spectra. As an application, I will discuss the results of Lipshitz and Sarkar and ours on refinements of Khovanov homology to K-theory and stable homotopy theory

Monday April 23. Hiro Lee Tanaka. Northwestern University.
Title: Factorization homology and link invariants
Abstract: Factorization homology is a way of constructing invariants of smooth n-manifolds out of an E_n algebra A. In this talk I will discuss joint work with David Ayala and John Francis, in which we generalize factorization homology to create invariants of stratified n-manifolds, where now the input data is an E_n algebra A together with some modules. I will discuss some elementary properties of factorization homology, including a manifold version of the excision axiom familiar from usual homology theories. One consequence of our work is that one can define topological field theories for cobordism categories whose objects and morphisms are stratified manifolds, as in Witten's Chern-Simons TFT. If time allows I will show that factorization homology of the free (E_3,E_1)-algebra can recover the homotopy type of link complements, so defines in particular a link invariant.

Monday April 30. Angelica Osorno. University of Chicago.
Title: Stable homotopy 1-types and Picard groups
Abstract: It is a classical result that groupoids model homotopy 1-types, in the sense that there is an equivalence between the homotopy categories, via the classifying space and fundamental groupoid functors. We extend this result to stable homotopy 1-types and Picard groupoids, that is, symmetric monoidal groupoids in which every object has a weak inverse. Using an algebraic description of Picard groupoids, we identify the Postnikov data associated to a stable 1-type; the abelian goups $\pi_0$ and $\pi_1$, and the unique $k$-invariant. We relate this data to the exact sequences of Picard groupoids developed by Vitale. Joint with Niles Johnson

Monday May 7. Frank Quinn. Virginia Tech.
Title: The missing global theory of smooth manifolds
Abstract: High-dimensional manifold theory developed largely between 1940 and 1980, with surgery, s-cobordism theorem, etc. Looking back after the dust settled and the material has been digested, we see that there are two main families. The topological family includes PL, top, and ANR homology manifolds, which differ slightly but are very similar. There is a slick and effective global theory describing these. The smooth family includes differentiable, real analytic, and probably real algebraic. The first two are essentially the same (I'm not clear about the third). The current global theory is far less effective than the topological case. I'll explain this, and describe "design criteria" for a better theory. Ingredients include duality with modern ring spectra, the image of J, and clues from algebraic geometry.