2023-2024

Fall 2023

Talks held in-person on Tuesdays, 3-4pm Pacific, in 101 Peterson. E-mail Dan Dugger for information. 

Winter 2024

Talks held in-person on Tuesdays, 3-4pm Pacific Time, 260 Tykeson. E-mail Dan Dugger for information. 

Spring 2024

Talks held in-person on Tuesdays, 3-4pm Pacific Time, 260 Tykeson (except a different room on May 14). E-mail Dan Dugger for information. 

Abstracts

Fall

October 3. Nick Addington. On the higher K-theory of some complex varieties.

Abstract:

The algebraic K-theory of a smooth complex variety is very different from topological K-theory of the underlying manifold, but if we take Z/m coefficients they become isomorphic in high degrees.  I will argue that the point at which this happens is an interesting invariant of the variety, and relate it to more classical invariants and to birational geometry.  Time permitting, I'll use my main result to prove some new cases of the integral Hodge conjecture for complex 4-folds, and to describe the higher K-theory of some dg-categories that act like non-commutative K3 surfaces.  This is joint work with Elden Elmanto.

October 10.  Tye Lidman, Invariants of four-manifolds. 

Abstract: 

Four-manifolds are totally wacky objects. For example, there are uncountably many four-manifolds homeomorphic but not diffeomorphic to Euclidean space. I will discuss one philosophy for how to distinguish four-manifolds which are homeomorphic but not diffeomorphic, which could in principle be relevant to invariants of manifolds of other dimensions as well. This is joint work with Adam Levine and Lisa Piccirillo.

October 17. Diego Manco Berrio, K-theory, multicategories, and pseudosymmetric functors.

Abstract: Multicategories where introduced by Elmendorf and Mandell in homotopy theory as an alternative way to encode multiplicative structures in the absence of symmetric monoidal structures. In a sense, they allow us to talk about multilinear maps even when we can't talk about tensor products. This has enabled several definitions of multiplicative structure preserving K-theory as a multifunctor taking structured categories to spectra. This is a way to produce E_\infty ring spectra, modules and algebras. 

In this talk I'll introduce pseudo symmetric multifunctors, weaker versions of multifunctors introduced by Yau, including a new equivalent definition. This makes clear that they preserve symmetric multiplicative structures like E_n algebras (n=1,...,\infty). In particular, Mandell's inverse K-theory preserves E_n algebras. I'll mention some other examples and conjectures.

October 24.  Leo Yoshioka, Two graph homologies and the space of long embeddings.

Abstract:

In this talk, we introduce machinery to give geometric and non-trivial (co)cycles of the spaces of long embeddings. We construct these cocycles by integral over configuration spaces associated with 2-loop graphs. This framework is a generalization of several works for 1-loop graphs by Bott, Cattaneo, Rossi, Sakai and Watanabe. Using specific simple 2-loop graphs, we obtain higher-degree cocycles of long embeddings, even with codimension two. The key to applying our framework to larger 2-loop graphs is comparing our graphs with hairy graphs, which Arone and Turchin introduced in embedding calculus. The speaker will share the current situation of comparison results. The talk is based on the speaker’s preprint arXiv:2212.01573.

November 7.  Dev Sinha (UO)   Geometric cochains.

Abstract: This talk will be accessible with only a 600-topology background.  

de Rham forms, grounded in calculus on manifolds,  give a type of real-valued cochain used all the time to relate geometry of a manifold and its homotopy theory.  But these cochains are only sensitive to the ``non-torsion’’ parts of algebraic topology and homotopy theory.  Intersection theory has long been known as a conceptual approach to cohomology over the integers, but a full development of a coherent approach to cochains grounded in intersection theory was only recently proposed by Lipyanskiy, naming the theory as geometric cohomology.  This theory has now has been fully worked out by Friedman, Medina and myself.   It is essentially a smooth version of Chow theory, and a “baby version” of some part of Floer theory.  But it is technically difficult to set up because it relies on a robust theory of manifolds with corners, itself only recently worked out by Joyce, and moreover manifolds with corners do not have “boundary squared equal zero”!  And we have only taken a small step (in our 100+ pages) because a more robust use of this theory would require application and perhaps further development of the theory of partially defined differential graded algebras.

I will go through some of the story of setting up this theory, as well as the first application to finer understanding of the relationship between between intersection and cup product.  I will comment on how various topics in “Intermediate Algebraic Topology” are conceptually simple in this theory (though in almost no cases have details been written down to prove things in this language).  And I will share my hope/ program to use geometric cochains to faithfully model homotopy types through E-infinity structure.

November 14.  Chad Giusti (Oregon State). Tracking topological structure across neural populations.

Abstract: The stimulus space model for neural population activity describes the activity of individual neurons as points localized in a metric space, with neural activity modulated by distance to individual stimuli. Neural systems with stimulus space coding are common in the brain and amenable to study using topological tools. We will discuss our recent mathematical and computational efforts toward determining when multiple neural populations encode "the same" topologically structured information, and how biological neural systems could learn to encode such data. In particular, we will describe a method for working around the fact that persistent homology is not (in general) functorial. This is joint work with Niko Schonsheck, Iris Yoon, and Robert Ghrist, among others. No prior knowledge of neuroscience will be assumed.

December 22 (10:00 a.m., Tykeson 260). Nathan Dunfield (UIUC). Computing a link diagram from its exterior.
Abstract: A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincare conjecture. This is joint work with Cameron Rudd and Malik Obeidin.  Based on: arXiv:2112.03251.

Winter

January 16. Hokuto Konno (Tokyo). Applications of gauge-theoretic characteristic classes
Abstract: Families of gauge theoretic PDE (e.g. Seiberg-Witten equations) yield characteristic classes of fiber bundles with 4-manifold fiber. These characteristic classes can be used to study diffeomorphism groups of 4-manifolds, and we can reveal interesting phenomena that are special to dimension 4. We present a few instances of such phenomena.

February 6. Collin Litterell (Univ. of Washington).  Tensor triangular geometry of modules over the Steenrod algebra.

Abstract: One result from the seminal work of Devinatz-Hopkins-Smith in stable homotopy theory is the thick subcategory theorem, which describes the global periodic structure of the stable homotopy category. This result inspired many similar theorems in fields such as algebraic geometry and modular representation theory, and the surrounding ideas eventually formed the foundation for the field of tensor triangular geometry. In this talk, we will give a brief overview of these ideas and discuss a new thick subcategory theorem for modules over the mod 2 Steenrod algebra.

February 20. Orsola Capovilla-Searle (UC Davis). Results on exact Lagrangian fillings and cobordisms of Legendrian links

Abstract: An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about new connections between fillings and Newton polytopes, as well as results on distinguishing non-orientable fillings. This is based on joint work with Casals and joint work with Hughes and Weng.

February 27. Eugen Rogozinnikov (Notre Dame). Noncommutative cluster-like coordinates for maximal symplectic representations

Abstract: Representations of the fundamental group of an orientable surface into a Lie group are the main object of higher Teichmüller theory. When the Lie group is Hermitian, the moduli space of such representations has much richer structure, in particular, an important geometric invariant was introduced by Toledo in 1989. When the Toledo invariant is maximal possible, such representations attracted a lot of attention in the last decade (M. Burger, A. Iozzi, A. Wienhard and others) because they naturally generalize Fuchsian representations of the fundamental group of a surface into PSL(2,R). Moreover, maximal representations have particularly nice properties, e.g. they are injective with discrete image.

I will start with a review of Penner and Fock-Goncharov decorated representations into SL(n,R) and their generalization to the first interesting Hermitian Lie group Sp(2n,R). It turns out that there are remarkable coordinates on the space of decorated maximal representations of the fundamental group of a punctured surface into Sp(2n,R). They generalize Penner's Lambda-lengths and exhibit a non-commutative cluster-like structure, in particular, noncommutative Ptolemy relations. If time permits, I will talk about how we can use these coordinates to understand the topology of the space of decorated maximal representations. This is a joint work with D. Alessandrini, O. Guichard and A. Wienhard.

March 5. Jonathan Rosenberg (Maryland). Positive scalar curvature on manifolds with boundary and their doubles.

Abstract:  I will talk about a paper of mine with Shmuel Weinberger that just appeared in PAMQ.  Suppose X is a compact manifold with boundary ∂X, and let M be the double of X along ∂X, i.e., the union of two copies of X joined along their boundaries.  (Note that M is a closed manifold.) We will discuss the interplay between 3 different conditions:

1) X admits a Riemannian metric of positive scalar curvature with positive mean curvature on ∂X,

2) X admits a Riemannian metric of positive scalar curvature with product structure near ∂X, and

3) M admits a Riemannian metric of positive scalar curvature.

We will see that these conditions are closely related but not identical, and will give necessary and sufficient conditions for them in certain cases.

March 19. Ian Zemke (Princeton). Involutive Heegaard Floer homology and Dehn surgery.

Abstract: In this talk, we will give an overview of the homology cobordism group and discuss applications of Heegaard Floer theory to its study. We will focus on applications of Hendricks and Manolescu's involutive Heegaard Floer homology. Of principle interest are techniques for computing involutive Heegaard Floer homology (and interpreting the answer that one gets). 3-manifolds are most naturally described using Dehn surgery on links in S^3, and we will briefly discuss extensions of Hendricks and Manolescu's theory which allows cut-and-paste style computations. The case of Dehn surgery on knots is joint work with Hendricks, Hom and Stoffregen, and the case of Dehn surgery on links is joint work in progress with Hendricks and Stoffregen. We will touch on our general algebraic theory, but will focus on topological examples.