2022-2023

Winter 2023

Talks held in-person on Tuesdays, 3-4pm Pacific, in University Hall 209. E-mail J.D. Quigley for information. 

Spring 2023

Talks held in-person on Tuesdays, 3-4pm Pacific, in University Hall 210. E-mail J.D. Quigley for information. 

Abstracts

Spring

May 23. Dennis Nguyen, Complex sections and cobordisms.
Abstract: There is a classical problem of finding linearly independent vector fields on a manifold. Bokstedt, Dupont and Svane approached this problem by looking for an obstruction to finding a cobordant manifold with linearly independent vector fields. We find the cobordism obstruction to the existence of complex sections on an almost complex manifold. We describe this obstruction in terms of the Chern classes of the manifold. Moreover, we define a notion of complex section cobordism groups analogous to the vector field cobordism groups defined in Bokstedt, Dupont and Svane. We compute these groups for low numbers of sections using the Adams spectral sequence. Finally, we show that this obstruction vanishes for certain manifolds in the complex cobordism ring.

May 16. David Wraith, Intermediate curvatures: an overview and some recent developments.
Abstract: In addition to the classical notions of scalar, Ricci and sectional curvature in Riemannian geometry, there are also natural notions of curvature which interpolate between these. Studying intermediate curvatures offers a more nuanced view of curvature in Riemannian geometry. Indeed, one might hope to arrive at an enhanced understanding of the classical curvatures in this way. In the first part of the talk, I will define intermediate curvatures, and discuss how these have arisen in the literature. In the second part of the talk, I will describe some recent developments exploring the connections between intermediate curvatures and topology, including joint work with Philipp Reiser.

May 9. Subhankar Dey, Essential surfaces in link exteriors and link Floer homology.
Abstract: Knot Floer homology has been known to be quite a strong invariant for knots and links for detection of several topological properties of them. 

In this talk we will briefly shed light on how detection of certain essential surfaces in the knot/link exteriors via Floer homology have been the central theme on results about knot/link Floer homological detection of several knots and links. This talk is based on joint work with Fraser Binns, part of which are ongoing.

May 2. Prasit Bhattacharya, Equivariant Steenrod operations.
Abstract: Classical Steenrod operations are one of the most fundamental and formidable tools in stable homotopy theory. They led to the calculation of homotopy groups of spheres, calculation of cobordism rings,  characteristic classes, and many other celebrated applications of homotopy theory to  geometry. However, equivariant Steenrod operations are not known beyond the group of order 2. In this talk, I will demonstrate a geometric construction of the classical Steenrod operations and generalize it to construct G-equivariant Steenrod operations for any finite group G. Time permitting, I will discuss applications to equivariant geometry.

April 25. Ryan Budney, The Schoenflies monoid.
Abstract: I will describe Smale's proof that Diff(S^2) has the homotopy-type of O_3.  Then describe Cerf's re-imagining of this proof.  Lastly I will describe how Cerf's proof allows us to argue that the Schoenflies monoid (isotopy classes of embedded, oriented co-dimension one spheres in S^n with the connect-sum operation) is a group for all n>=2.  Given this monoid is known to have only one element when n is different from 4, this result is perhaps interesting only in dimension 4, although the fact that the proof is general is of some interest of its own.

April 18. Eva Belmont, A normalizer decomposition for compact Lie groups.
Abstract: If G is a finite group, the normalizer decomposition is a way of expressing BG, up to p-completion, as a homotopy colimit of smaller groups. Building on a construction due to Libman for p-local finite groups, we obtain a normalizer decomposition for compact Lie groups. We recover decompositions due to Dwyer, Miller, and Wilkerson of 2-completed BSU(2) and BSO(3), and compute new decompositions of BSU(p) completed at p. The talk will also contain an introduction to fusion systems, which are related to a generalization of p-completed classifying spaces, as that is the setting for our work. This is joint work with Natalia Castellana, Jelena Grbic, Kathryn Lesh, and Michelle Strumila..

April 11. Holt Bodish, Knot Floer Homology and fibered satellite knots.
Abstract: I will discuss recent work studying a family of (1,1)-patterns and their satellite knots. We show how to use bordered Floer homology and the immersed curve interpretation to compute the knot Floer homology of these satellites. We use the fact that knot Floer homology detects both fibered knots and a particular property of the monodromy of fibered knots to show that satellite knots with these patterns have these properties. If time permits, I will discuss computations of certain invariants of the concordance type of these satellite knots as well.

April 4. Richard Wong, Endotrivial modules via Galois descent.
Abstract: Let G be a finite group, and k be a field of characteristic p, where p divides the order of the group (i.e. the setting of modular representation theory).  One object of interest is the stable module category of G, which has a homotopy-theoretic interpretation as a stable ∞-category.  This allows us to use homotopy-theoretic methods to compute invariants of representation-theoretic significance.

In this talk, we will demonstrate how to use homotopy-theoretic tools to provide novel computations of the group of endotrivial modules for both p-groups and non-p-groups, which provides new insights into the classical representation-theoretic calculations.

Winter

March 14. Søren Galatius, Equivariant cobordism categories.
Abstract: I will discuss joint work with Gergely Szucs concerning equivariant cobordism categories, arXiv:1805.12342.  The objects are closed (n-1)-manifolds equipped with a smooth action of a finite group G, and morphisms are cobordisms with smooth G-action.  Tangential structures (orientations, spin structures, etc) can also be included.  Our main result is a determination of the homotopy type of the nerve of such categories, generalizing earlier work in the non-equivariant case.

February 28. Hana Jia Kong, The modified Adams--Novikov spectral sequence and a deformation of Borel equivariant homotopy.
Abstract: The R-motivic stable homotopy category has a close connection to the C2-equivariant category via the C2-equivariant Betti realization map. 

For the bigraded homotopy groups, the C2-equivariant spectra are usually more complicated to compute than the R-motivic ones, due to the existence of the "negative cone". I will first explain this connection; then I will talk about the joint work with Gabriel Angelini-Knoll, Mark Beherens, and Eva Belmont, in which we construct the modified Adams--Novikov spectral sequence (mANSS) aiming to build an odd primary analog of this R-motivic and C2-equivariant relation. Using the mANSS filtration, one can construct a deformation of Borel equivariant homotopy, which agrees with the a-completed Artin—Tate R-motivic category for the group C2.

February 21. J.D. Quigley, The motivic Hopf invariant one problem.
Abstract: Adams showed in 1960 that the (2n-1)-st homotopy group of the n-sphere contains an element of Hopf invariant one if and only if n=2, 4, or 8. This result has interesting applications in geometry and algebra, e.g. the nonexistence of H-space structures on spheres. In this talk, I will discuss a motivic analogue of the Hopf invariant one problem, its solution over certain base fields, and consequences for representability of motivic spheres by schemes admitting unital products. No prior knowledge of motivic homotopy theory will be assumed. This is joint work with William Balderrama and Dominic Leon Culver.

February 14. Zhouli Xu, The Adams differentials on the classes hj3 .

Abstract: In Adams filtration 1, Adams computed differentials on the classes hj and solved the famous Hopf invariant problem. In Adams filtration 2, Hill–Hopkins–Ravenel solved the celebrated Kervaire invariant problem on the classes hj2, with the only exception of j = 6. The precise differentials on the classes hj2 for j ≥ 6 remain unknown. 

I will talk about joint work with Robert Burklund: In Adams filtration 3, we compute nonzero differentials on the classes hj3 for all j, which confirms a conjecture of Mahowald. Our computation uses two different deformations of stable homotopy category – C-motivic stable homotopy category and F2-synthetic homotopy category – both in an essential way. Moreover, we also show that h62 survives to the Adams E9-page.

February 7. Boris Botvinnik, Families of diffeomorphisms detected by trivalent graphs (with applications to psc-metrics).
Abstract: This is a joint work with Tadayuki Watanabe. We use earlier results by Watanabe to prove that the non-trivial elements of the homotopy groups π*BDiff∂(Dd)⊗ Q (which are detected by the Kontsevich characteristic classes valued in the algebra of trivalent graphs) are lifted to elements in π*C(Dd) of the pseudo-isotopy space. Here d > 3.

I will discuss mostly the case when dimension d is even. We also prove that those elements are lifted to corresponding moduli spaces of psc-metrics.