The Topology / Geometry seminar meets at 3:00 p.m. on Tuesdays in 210 Deady, except as noted. See also the University of Oregon Mathematics Department webpage.
## Fall 2017
## Winter 2018
## Spring 2018
## Abstracts## Fall 2017September 26, 2017. Nathan Perlmutter (Stanford University), "Parametrized Morse theory, cobordism categories, and positive scalar curvature".Abstract: Click here (PDF).
Abstract: T-duality arose as an agreement between the predictions of different versions of string theory. There is an underlying topological agreement as well, which can be expressed as an isomorphism between the twisted K-theory of certain circle bundles. I will carefully explain this through examples, and then I will discuss my work with Westerland and Sati on T-duality for more general sorts of fiber bundles. In particular, I will describe the universal T-duality theory for sphere bundles of a fixed rank and its relationship with algebraic K-theory. Time permitting, I will discuss current work on a T-duality isomorphism in chromatic homotopy theory.
October 10, 2017. Nikolai Saveliev (University of Miami), "Instanton knot homology and equivariant gauge theory".Abstract: Singular instanton knot homology is a Floer theory defined by Kronheimer and Mrowka using via gauge theory on orbifolds; they used it to prove that Khovanov homology is an unknot detector. We show how replacing gauge theory on an orbifold with an equivariant gauge theory on its double branched cover simplifies the matters and allows for a number of explicit calculations. This is a joint work with Prayat Poudel.
October 17, 2017. Jeremy Van Horn-Morris (University of Arkansas), "Contact and symplectic topology and the topology of 3- and 4-manifolds".
Abstract: Contact and symplectic geometry give a lens to the study of manifolds that sits somewhat at the intersection or midpoint between complex and Riemannian geometry. The tools here are more flexible than in either complex or Riemannian geometry while still retaining some of the strong constraints of the first and the some of the ubiquity of the second. Particularly in 3- and 4-dimensions, contact and symplectic structures have strong connections to the underlying topology of the manifold. We'll survey the motivating open questions in the field with a particular emphasis on 3- and 4-dimensions and along the way we'll introduce some of the many Floer theories that have become the dominant toolkit in the modern development of the field.
October 24, 2017. Eric Hogle (University of Oregon) "On the RO(C2)-graded cohomology of certain equivariant Grassmannians".
Abstract: The Grassmannian manifold of k-planes in R^n has a group action if R^n is taken to be a real representation of the group. When the group is C2 (the cyclic group on two elemnts), the Schubert cell construction of the Grassmannian generalizes to an equivariant representation-cell structure. However, this cell structure depends an identificiation of the representation with R^n via some equivariant flag. A different choice of flag can give a very different cell structure.
I will explain a program to compute the RO(C2)-graded Bredon cohomology of these important spaces. A theorem of Kronholm dictates that these must be free modules over the cohomology of a point, but the degrees of the generators is, in general, unknown. The ambiguity introduced by the choice mentioned above turns out to be an asset for this task. I use a computation by Dan Dugger of the cohomology of an infinite equivariant Grassmannian, and prove some theorems about equivariant flag manifolds to find the cohomologies of several infinite families of finite-dimensional equivariant Grassmannians. I will present the main ideas behind how this is done. October 31, 2017. Clover May (University of Oregon) "A structure theorem for RO(C_2)-graded cohomology".Abstract: Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C_2 the cyclic group of order two. In this talk I will present a structure theorem for RO(C_2)-graded cohomology with coefficients in the constant Z/2 Mackey functor that substantially simplifies computations. The structure theorem says the cohomology of any finite C_2-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will sketch the proof, which depends on a Toda bracket calculation, and give some examples. November 7, 2017. Keegan Boyle (University of Oregon) "The virtual cosmetic surgery conjecture".Abstract: The cosmetic surgery conjecture, which has been around since the early 1990s, asks when surgery along two different framing curves for the same knot produce the same 3-manifold. The virtual cosmetic surgery conjecture is a generalization of this question to coverings between surgeries. I will explain the conjecture, discuss how far you can get with elementary techniques, and present an application of hyperbolic geometry to the conjecture. Time permitting, I will also discuss the relevance of equivariant Floer homology theories. November 14, 2017. Beibei Liu (UC Davis) "Heegaard Floer homology of L-space links with two components".Abstract: We compute different version of link Floer homology for any L-space link with two components and prove that they are determined by the Alexander polynomials of every sublink of the L-space link. As an application, we compute the Thurston polytope and Thurston norm of the link and give some explicit examples. November 21, 2017. Mauricio Gomez Lopez (University of Oregon) "The homotopy type of the PL cobordism category".Abstract: Over the past 20 years, diffeomorphism groups and their classifying spaces have been the subject of intense research in both algebraic and geometric topology. The interest in these kinds of spaces is largely due to the Madsen-Weiss Theorem, which is a result that identifies the homology of the stable mapping class group with the homology of the infinite loop space of a certain Thom spectrum. The significance of this result, and of the previous work of Tillman that led up to it, lies in the fact that it introduced novel ways of using methods from homotopy theory in the study of automorphism groups of manifolds.
A key tool in several of the existing proofs of the Madsen-Weiss Theorem is the smooth cobordism category. By performing a systematic study of this category, Galatius and Randal-Williams produced a substantially elementary proof of the Madsen-Weiss Theorem which bypasses a lot of the heavy machinery used in the earlier proofs of this result. Moreover, Galatius and Randal-Williams later refined the methods of this proof to obtain a version of the Madsen-Weiss Theorem for higher dimensional manifolds. In this talk, besides providing a general introduction to the Madsen-Weiss Theorem, I will report on my progress in translating this body of results to the category of piecewise linear manifolds. More specifically, in this talk I will introduce a PL cobordism category, which can be viewed as the PL version of the cobordism category used in the context of diffeomorphism groups, and explain the main result that I have proven in this project. Concretely, my main result shows that the classifying space of the PL cobordism category is weak homotopy equivalent to the infinite loop space of a spectrum built out of spaces of PL manifolds, thus providing a PL analogue of a result of Galatius and Randal-Williams.
December 5, 2017 in McKenzie 349. Ben Knudsen (Harvard) "Subdivisional spaces and graph braid groups".Abstract: We develop an approach to the study of the configuration spaces of a cell complex X that is both flexible and suitable for computation. We proceed by viewing X, together with its subdivisions, as a "subdivisional space," a kind of diagram object, which has associated to it certain diagrammatic versions of configuration spaces. These objects, which model the correct homotopy types, mix the discrete and the continuous, and they may be attacked by combining techniques drawn from discrete Morse theory and factorization homology. We apply our theory in the 1-dimensional example of a graph, obtaining an enhanced version of a family of chain models for graph braid groups originally studied by Swiatkowski. These complexes come equipped with a robust computational toolkit, which we exploit in numerous calculations, old and new. This is joint work with Byung Hee An and Gabriel Drummond-Cole. |