The Online Algebraic Topology Seminar (OATS) was a research seminar, covering a broad range of topics from algebraic topology.
This page acts as an archive for the Spring 2020 and Autumn 2020 iterations of the seminar series. Titles, abstracts, notes and videos (where available) can be found below.
21st September 2020 @ 1pm UTC:
Title: Nishida Nilpotence
Abstract: In 1973, Nishida proved that every positive degree class in the stable homotopy groups of spheres is nilpotent. We will discuss some modern perspectives on Nishida's original proof. While this will be a mostly expository talk aimed at graduate students, if time permits we will end with a discussion of some open nilpotence questions in motivic stable homotopy theory.
28th September 2020 @ 2pm UTC:
Title: Elliptic cohomology of level n
Abstract: Elliptic genera have played an important role in algebraic topology and algebraic geometry since the 1980s. To every almost-complex manifold they associate a modular form for the congruence subgroups $\Gamma_1(n)$. More recently, elliptic cohomology theories have been built that are natural targets of elliptic genera for families. I will give an overview of these theories and report in particular on certain $C_2$-equivariant refinements.
5th October 2020 @ 2pm UTC:
Viktoriya Ozornova (Ruhr-Universität Bochum)
Title: Models of (infty,2)-categories
Abstract: An (infty,2)-category should be a weak version of a strict 2-category, in which compositions are well-defined, associative and unital up to some higher coherence. There are various models making this precise. In this talk, I will describe a direct comparison between two particular models (which will be introduced), namely Theta_2-spaces and saturated 2-complicial sets. This is joint work in progress with Julie Bergner and Martina Rovelli.
12th October 2020 @ 3pm UTC:
Title: Transfer systems and weak factorization systems
Abstract: N∞ operads over a group G encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of G. In this talk, we will show that when G is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of G. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin–Bearup–Pech–Roitzheim for cyclic groups of squarefree order. We will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg–Hill saturation conjecture.
This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.
19th October 2020 @ 2pm UTC:
Birgit Richter (Universität Hamburg)
Title: Detecting and describing ramification for structured ring spectra
Abstract: This is a report on joint work in progress with Eva Höning.
Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by the spectrum of Kähler differentials. For rings of integers in an extension of number fields it is important to distinguish between tame and wild ramification. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the norm map. We take the latter fact and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. In the talk I will discuss several examples in the context of topological K-theory and modular forms.
26th October 2020 @ 4pm UTC:
Grigory Garkusha (Swansea University)
Title: Motivic Gamma-spaces
Abstract: This is a joint work with Ivan Panin and Paul Arne Østvær. We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines Gamma-spaces and framed correspondences into the concept of motivic Gamma-spaces; these are continuous or enriched functors of two variables that take values in motivic spaces and are equipped with a framing. We craft proofs of our main results by imposing further axioms on motivic Gamma-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, A1- and sigma-invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the A1-homotopy theory of algebraic varieties. As prime examples we discuss the motivic sphere spectrum, algebraic cobordism, motivic cohomology, and Milnor-Witt motivic cohomology.
2nd November 2020 @ 3pm UTC:
Title: Support theory for triangulated categories in algebra and topology
Abstract: We will survey the support theory of triangulated categories through the machinery of tensor-triangulated geometry. We will discuss the stratification theory of Benson—Iyengar—Krause for triangulated categories, the construction by Balmer of the spectrum of a tensor-triangulated category, and the relation between the two. Time permitting, we will discuss a recent application to the category of derived Mackey functors, joint with Beren Sanders.
TUESDAY 10th November 2020 @ 3pm UTC:
Anna Marie Bohmann (Vanderbilt)
Title: Algebraic K-theory for Lawvere theories: assembly and Morita invariance
Abstract: Much like operads and monads, Lawvere theories are a way of encoding algebraic structures, such as those of modules over a ring or sets with a group action. In this talk, we discuss the algebraic K-theory of Lawvere theories, which contains information about automorphism groups of these structures. We'll discuss both particular examples and general constructions in the K-theory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assembly-style maps. This is joint work with Markus Szymik.
16th November 2020 @ 3pm UTC:
Title: Model category structures and spectral sequences
Abstract: I'll discuss a family of model category structures such that weak equivalences are morphisms inducing an isomorphism at a fixed stage of a spectral sequence. The talk will focus on joint work with Xin Fu, Ai Guan and Muriel Livernet, giving such model structures for multicomplexes. A multicomplex (also known as a twisted chain complex) is an algebraic structure generalizing the notion of a chain complex and that of a bicomplex. These structures have arisen in many different places and play an important role in homological and homotopical algebra.
23rd November 2020 @ 3pm UTC:
Title: The coalgebra of chains and the fundamental group
Abstract: Rational homotopy theory tells us that simply connected spaces, up to rational homotopy equivalence, may be classified algebraically by means of rational cocommutative coalgebras (Quillen) or in the finite type case by rational dg commutative algebras (Sullivan). Goerss and Mandell proved versions of these results for fields of arbitrary characteristic by means of simplicial cocommutative coalgebras and E-infinity algebras, respectively. The algebraic structures in these settings are considered up to quasi-isomorphism.
In this talk, I will describe how to extend these results to spaces with arbitrary fundamental group.The key new observation is that the homotopy cocommutative coalgebraic structure of the chains on a space determines the fundamental group in complete generality. The corresponding algebraic notion of weak equivalence between coalgebras is drawn from Koszul duality. The end goal of this program is to completely understand homotopy types in terms of algebraic “chain level” structure. This is joint work with M. Zeinalian and F. Wierstra.
30th November 2020 @ 3pm UTC:
Julie Bergner (University of Virginia)
Title: Variants of the Waldhausen S-construction
Abstract: The S-construction, first defined in the setting of cofibration categories by Waldhausen, gives a way to define the algebraic K-theory associated to certain kinds of categorical input. It was proved by Galvez-Carrillo, Kock, and Tonks that the result of applying this construction to an exact category is a decomposition space, also called a 2-Segal space, and Dyckerhoff and Kapranov independently proved the same result for the slightly more general input of proto-exact categories. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that these results can be maximally generalized to the input of augmented stable double Segal spaces, so that the S-construction defines an equivalence of homotopy theories. In this talk, we'll review the S-construction and the reasoning behind these stages of generalization. Time permitting, we'll discuss attempts to characterize those augmented stable double Segal spaces that correspond to cyclic spaces, which is work in progress with Walker Stern.
20th April 2020 @ 3pm BST:
Ran Levi (University of Aberdeen)
Title: Complexes of Tournaments in Directed Networks
Abstract: Clique graphs whose edges are oriented are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial sets, that we refer to as “tournaplexes", whose simplices are tournaments. In particular, given a directed graph G, we associate with it a “flag tournaplex" which is a tournaplex containing the directed flag complex of G, but also the geometric realisation of cliques that are not directed. We define several types of filtration on tournaplexes, and exploiting persistent homology, we observe that filtered flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of those ideas by applying them to graph data arising from the Blue Brain Project’s digital reconstruction of a rat’s neocortex.
(Slides)
27th April 2020 @3pm BST:
Andy Baker (University of Glasgow)
Title: Fun and games with the Steenrod algebra
Abstract: The mod 2 Steenrod algebra is an important tool in unstable and stable homotopy theory but it is also interesting as a purely algebraic gadget. I will briefly review its algebraic structure and that of some important finite subHopf algebras. Then I will discuss some realisability questions for modules, ranging from classical examples to modules over the E-infinity ring spectra kO and tmf localised at 2. I hope this talk will be accessible to beginners and also have some things to interest experts.
4th May 2020 @ 3pm BST:
Constanze Roitzheim (University of Kent)
Title: Equivariant homotopy commutativity, trees and chicken feet
Abstract: Commutativity up to homotopy can be daunting, and it becomes even more difficult to track when equivariant structures get introduced. In the case of a finite group, however, the options for equivariant homotopy commutativity can be encoded using simple combinatorics, and we will show some examples.
11th May 2020 @ 3pm BST:
Ulrich Pennig (Cardiff University)
Title: Loops, groups, and twists - the role of K-theory in mathematical physics
Abstract: While K-theory has its origins in Grothendieck's formulation and proof of his version of the Riemann-Roch theorem, it now plays a significant role in many diverse branches of mathematics: It provides a fundamental example of a cohomology theory, and it is one of the most important invariants of C*-algebras. In the first half of the talk, I will define the K-groups and discuss some of their applications. In the second half, I will concentrate on equivariant twisted K-theory, which is related to the representation theory of loop groups and the geometry of two-dimensional quantum field theories by a theorem of Freed, Hopkins, and Teleman. I will finish with an outline of joint work with D. Evans, in which we study generalizations of this work to higher twists.
18th May 2020 @ 3pm BST:
Emanuele Dotto (University of Warwick)
Title: Witt vectors with coefficients and characteristic polynomials over non-commutative rings
Abstract: The characteristic polynomial of a matrix with entries in a commutative ring R naturally takes value in the ring of Witt vectors of R. In joint work with Krause, Nikolaus and Patchkoria, we extend the classical Witt vectors construction to allow as input pairs of a ring R and a bimodule M. I will explain how this construction relates to topological Hochschild homology, the Hill-Hopkins-Ravenel norm, and the characteristic polynomial.
25th May 2020 @ 3pm BST:
Henry Adams (Colorado State University)
Title: Vietoris-Rips complexes and Borsuk-Ulam theorems
Abstract: Given a metric space X and a scale parameter r, the Vietoris-Rips simplicial complex VR(X;r) has X as its vertex set, and contains a finite subset as a simplex if its diameter is at most r. Vietoris-Rips complexes were invented by Vietoris in order to define a (co)homology theory for metric spaces, and by Rips for use in geometric group theory. More recently, they have found applications in computational topology for approximating of the shape of a dataset. I will explain how the Vietoris-Rips complexes of the circle, as the scale parameter r increases, obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until they are finally contractible. Only very little is understood about the homotopy types of the Vietoris-Rips complexes of the n-sphere. Knowing the homotopy connectivities of Vietoris-Rips complexes of spheres allows one to prove generalizations of the Borsuk-Ulam theorem for maps from the n-sphere into k-dimensional Euclidean space with k > n. Joint work with John Bush and Florian Frick.
1st June 2020 @ 3pm BST:
Kirsten Wickelgren (Duke University)
Title: There are 160,839<1> + 160,650<-1> 3-planes in a 7-dimensional cubic hypersurface
Abstract: The expression in the title is a bilinear form and it comes from an Euler number in A1-algebraic topology. Such Euler numbers can be constructed with Hochschild homology, self-duality of Koszul complexes, pushforwards in SL_c oriented cohomology theories, and sums of local degrees. We show an integrality result for A1-Euler numbers and apply this to the enumeration of d-planes in complete intersections. Classically such counts are valid over C and sometimes extended to the real numbers, but A1-homotopy theory allows one to perform counts over a large class of fields, and records information about the solutions in bilinear form. The example in the title then follows from work of Finashin--Kharlamov. This is joint work with Tom Bachmann.
8th June 2020 @ 3pm BST:
Kathryn Hess (EPFL)
Title: Calculus from comonads
Abstract: (Joint work with Brenda Johnson.) The many theories of "calculus" introduced in algebraic topology over the past couple of decades--e.g., Goodwillie's calculus of homotopy functors, the Goodwillie-Weiss manifold calculus, the orthogonal calculus, and the Johnson-McCarthy cotriple calculus--all have a similar flavor, though the objects studied and exact methods applied are not the same. We have constructed a relatively simple category-theoretic machine for producing towers of functors from a small category into a simplicial model category, determined conditions under which such tower-building machines constitute a calculus, and showed that this framework encompasses certain well known calculi, as well as providing new classes of examples. The cogs and gears of our machine are cubical diagrams of reflective subcategories and the comonads they naturally give rise to.
In this talk, I will assume no familiarity with comonads and only basic knowledge of simplicial model categories.
15th June 2020 @ 3pm BST:
Markus Szymik (NTNU)
Title: Trigraded spectral sequences for principal fibrations
Abstract: The Leray--Serre and the Eilenberg--Moore spectral sequence are fundamental tools for computing the cohomology of a group or, more generally, of a space. In joint work with Frank Neumann (Leicester), we describe the relationship between these two spectral sequences in the situation when both of them share the same abutment. This talk is an introduction to the topic with many examples. It should be suitable for an audience from graduate students in algebraic topology onward, and I will only assume some casual acquaintance with spectral sequences.