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
AbstractsFall 2017September 26, 2017. Nathan Perlmutter (Stanford University), "Parametrized Morse theory, cobordism categories, and positive scalar curvature".
Abstract: Click here (PDF).
Abstract: Tduality 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 Ktheory of certain circle bundles. I will carefully explain this through examples, and then I will discuss my work with Westerland and Sati on Tduality for more general sorts of fiber bundles. In particular, I will describe the universal Tduality theory for sphere bundles of a fixed rank and its relationship with algebraic Ktheory. Time permitting, I will discuss current work on a Tduality 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 HornMorris (University of Arkansas), "Contact and symplectic topology and the topology of 3 and 4manifolds". 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 4dimensions, 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 4dimensions 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 kplanes 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 representationcell 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 finitedimensional 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_2CW 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 3manifold. 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 Lspace links with two components".
Abstract: We compute different version of link Floer homology for any Lspace link with two components and prove that they are determined by the Alexander polynomials of every sublink of the Lspace 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 MadsenWeiss 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 MadsenWeiss Theorem is the smooth cobordism category. By performing a systematic study of this category, Galatius and RandalWilliams produced a substantially elementary proof of the MadsenWeiss Theorem which bypasses a lot of the heavy machinery used in the earlier proofs of this result. Moreover, Galatius and RandalWilliams later refined the methods of this proof to obtain a version of the MadsenWeiss Theorem for higher dimensional manifolds. In this talk, besides providing a general introduction to the MadsenWeiss 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 RandalWilliams.
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 1dimensional 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 DrummondCole. January 16, 2018. Samantha Allen (Indiana University) "The nonorientable fourgenus of knots".
Abstract: The 4–genus of a knot K is the minimal genus of a surface in B^{4} whose boundary is K. Similarly, we can define the nonorientable 4–genus of a knot K as the minimal “nonorientable genus” of a surface in B^{4} whose boundary is K. Finding the nonorientable 4–genus of a knot can be quite intractable; existing methods exploit the relationship between nonorientable genus and normal Euler number of the nonorientable surface. In this talk, I will give an overview of the interplay between the nonorientable genus and normal Euler number of nonorientable surfaces in B^{4}. I will define both of these invariants and discuss their computation for closed surfaces and then for surfaces with boundary a knot. In particular, when fixing a knot K, we can ask what pairs of nonorientable genus and normal Euler number are realizable for a surface whose boundary is K. We will see that both classical invariants and Heegaard–Floer invariants can be used towards answering this question.
Abstract: We give homotopy invariants of maps from X to Y (sometimes called “homotopy periods”), which are conjectured to be a family of complete invariants when Y is a rational space and both X and Y are simply connected. We will start with a review of the Sullivan and Quillen approaches to rational homotopy theory, and then my work with Ben Walter which resolves the rational homotopy periods question when X is a sphere. We will then discuss MaurerCartan equations, and share progress on associating invariants to their solutions in the Lie coalgebraic setting. Concrete examples will be emphasized throughout the talk.
Abstract: Morita theory deals with the question of when two rings have equivalent categories of modules. Rickard generalized this to answer the question of when two DGAs have equivalent derived categories of dgmodules. Schwede and Shipley then adapted these ideas to the topological context, showing that any reasonable stable model category is equivalent to the model category of modules over a ring spectrum with many objects. I will review this whole story and then talk about a possible application to equivariant topology.
February 6, 2018. Benson Farb (University of Chicago) "How to make predictions in topology using number theory".
Abstract: In this talk I will explain how Melanie Wood, Jesse Wolfson and I were led to discover some surprising (to us) coincidences in topology purely by analogy with some classical analytic number theory. These coincidences are given in terms of the “homological density” of one space in another. We have no explanation as to why these topological predictions end up being true. I will also explain why the following question is not completely crazy: ``Why is the Riemann zeta function evaluated at n+1 like the 2fold loop space of projective nspace?''
Abstract: Topological spaces have open covers; in an analogous way, categories can have Grothendieck topologies. These topologies are often used to define sheaves on a category (in the usual manner), and thus allow us to talk about sheaf cohomology (as a derived functor). There is a special Grothendieck topology, called the canonical topology, which contains almost all of the topologies we can write down. In nice categories, the canonical topology has a concrete presentation. I will be talking about the homotopical version of this presentation with a brief discussion on Grothendieck topologies, the canonical topology, and homotopy colimits.
February 20, 2018. Krishanu Sankar (University of British Columbia) "Steinberg summands and symmetric powers of the equivariant sphere spectrum".
Abstract: The mod p Steenrod algebra is the (Hopf) algebra of stable operations on mod p cohomology. This algebra can be computed in several possible ways: one way is to filter the EilenbergMaclane spectrum HF_{p} using the finite symmetric powers of the sphere spectrum. The cofibers of this filtration are Steinberg summands (from the representation theory of GL_{k}(Z/p)) of the classifying spaces B(Z/p)^{k}.
Our main result is to lift this to Gequivariant stable homotopy theory, where G is any finite abelian pgroup (the main case of interest being when G is cyclic of order a power of p). We can thus compute the Gequivariant Steenrod algebra by decomposing the Gequivariant classifying space of Z/p  we'll describe this computation for G=C_{p}. When p=2 and G=C_{2}, the equivariant dual Steenrod algebra is known due to HuKriz and others, but at odd primes this is new. If there is time, we will then discuss a conjectured construction of the equivariant analogues of the Milnor operations (the indecomposables in the dual Steenrod algebra).
February 27, 2018. Kadriye Nur Saglam (UC Riverside) "New Exotic 4manifolds via Luttinger surgery on Lefschetz fibrations". Abstract: In this talk, I will present a new construction of exotic symplectic 4manifolds homeomorphic but not diffeomorphic to (2h+2k1)CP^{2}#(6h+2k+3)(CP)^{2} via Luttinger surgery for any (h,k)≠(0,1). First, I will introduce two symplectic building blocks for our construction: 1) the family of Lefschetz fibrations on Σ_{k} × S^{2}#4(h+1)(CP)^{2}, constructed by Y. Gurtas, and 2) 4manifolds obtained from Σ_{g} x T^2 via Luttinger surgery. Next, I will show how to obtain the exotic copies of (2h+2k1)CP^{2} # (6h+2k+3)(CP)^{2} by gluing these building blocks. If time permits, I will also construct new symplectic 4manifolds with the free group of finite rank and various other finitely generated groups as the fundamental group. This is a joint work with Anar Akhmedov.
March 6, 2018. Akram Alishahi (Columbia University) "Trivial tangles, compressible surfaces, and Floer homology".
Abstract: Tangles are building blocks of knots and links. In this talk, we will introduce tangles, and a notion of triviality for their components, called boundary parallelness. Then, we will sketch a way, that is checkable by computer, to detect boundary parallel components of tangles. We will also discuss the analogous question for 3manifolds boundary: does the boundary have a (homologically essential) compressing disk? This is a joint work with Robert Lipshitz.
March 13, 2018. Mauricio Gomez Lopez (University of Oregon) "Spaces of graphs and the stable homology of the automorphism groups of free groups".
Abstract: The goal of this talk is to explain the theorem of Galatius which describes the stable homology of the automorphism groups of free groups. Namely, Galatius proves that this stable homology is isomorphic to the homology of a component of the infinite loop space corresponding to the sphere spectrum, thus showing that the automorphism groups of free groups and the symmetric groups have the same stable homology.
A novel construction which Galatius introduced in his proof are certain spaces of graphs embedded in the ndimensional Euclidean space. These spaces are what provide the link between the sphere spectrum and the automorphism groups of free groups, and Galatius uses them to construct models for the classifying spaces of such groups. Besides outlining the proof of Galatius, I will also explain in this talk some of the main properties of these spaces of graphs.
Spring 2018April 24, 2018. Emmy Murphy (Northwestern University) "Arboreal singularities and loose Legendrians".
May 1, 2018. Cornelia Van Cott (University of San Francisco) "Continued fractions, nonorientable surfaces, and torus knots". May 8, 2018. Jennifer Hom (Georgia Institute of Technology) "Knot concordance in homology cobordisms".
May 15, 2018. John Etnyre (Georgia Institute of Technology) "Branched covers and contact geometry". May 15, 2018. 4:00 p.m. Victor Turchin (Kansas State), "Higher Hochschild homology and representations of Out(F_n)".
May 22, 2018. 9:00 a.m. Kristen Hendricks (Michigan State), "Connected Heegaard Floer homology and homology cobordism". May 22, 2018. 3:00 p.m. Akhil Mathew (University of Chicago), "padic Ktheory and topological cyclic homology". May 29, 2018. 11:00 a.m. Siqi He (California Institute of Technology), "A KobayashiHitchin correspondence for the extended Bogomonly Equations". June 5, 2018. Tye Lidman (North Carolina State) "Spines in fourmanifolds".
