Seminars & Conferences

Spectral sequences have proved to be the most fruitful way to understand homotopy groups of spectra, particularly the various flavours of Adams spectral sequences. Sometimes, one can identify an Adams spectral sequence for a spectrum X with a more 'ad hoc' spectral sequence constructed specifically for X, one which may a priori be easier to understand. This gives one greater control over the homotopy groups of X.In this talk, I will use synthetic spectra to give much more structured versions of such identifications. Moreover, in cases where such an identification fails, this synthetic approach still goes through. I will show how in certain cases, one can define a synthetic spectrum that captures such an 'ad hoc' spectral sequence, and obtain from it all the benefits one would have had if such an identification had been possible. The main example is a synthetic spectrum Smf of synthetic modular forms.This is joint work with Christian Carrick and Jack Davies.

After a brief introduction to chromatic homotopy theory, I will report on recent advances in our understanding of some K(n)-local invariants using ideas from p-adic geometry. Joint work, partly in progress, with Schlank, Stapleton, and Weinstein.

Pontryagin classes were originally considered as invariants of real vector bundles, but it was realised in the 60s that they can be defined more generally for Euclidean bundles, that is, fibre bundles whose fibres are homeomorphic to Euclidean space. This led to the question whether the well-known vanishing of large-degree Pontryagin classes for small-dimensional vector bundles continues to hold in the setting of Euclidean bundles. Surprisingly, Weiss proved a few years ago that this often fails, even for bundles over spheres. I will explain a strengthening of this result resulting from joint work with A. Kupers: For every k>0, there exists a 6-dimensional Euclidean fibre bundle over a sphere whose kth Pontryagin class is nontrivial.

Nonlinear elliptic PDE problems can be described as zero finding problems of non-linear proper Fredholm mappings $f:H\rightarrow H$, where $H$ is an infinite dimensional Hilbert space. In this talk I will classify these mappings up to homotopy in terms of a non-trivial quotient of the stable homotopy groups of spheres. This is joint work with Lauran Toussaint.

The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope $P$ in $\mathbb{R}^n$, one can cut $P$ into a finite number of smaller polytopes and reassemble these to form $Q$. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut and paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste"). In this talk I will explain the construction that will allow us to speak about the "K-theory of manifolds" spectrum. The zeroth homotopy group of the constructed spectrum recovers the classical groups $\mathrm{SK}_n$. I will show how to relate the spectrum to the algebraic $K$-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further I will describe the connection of our spectrum with the cobordism category.

In the 90s, Cohen- Jones, and Segal asked the question of whether various types of Floer homology theories could be upgraded to the homotopy level by constructing stable homotopy types encoding Floer data. They also sketched how one could construct these Floer homotopy types as (pro)spectra in the situation that the infinite-dimensional manifold involved is “trivially polarized”. It has since been realized that the correct home for Floer homotopy types, in the polarized situation, is twisted spectra. This is a generalization of parametrized spectra that one can roughly think of as sections of bundles of categories whose fibre is the category of spectra. The aim of this talk is to give an introduction of Floer homotopy theory and twisted spectra. I will also outline the construction of a circle equivariant twisted spectrum from Seiberg-Witten Floer data associated to a 3-manifold equipped with a complex spin structure. As there are many moving parts to this (Atiyah-Singer index theory, finite-dimensional approximation, Conley index theory etc.), I will try to keep the talk on a conceptual level that will hopefully be accessible to a large audience. This is joint work in progress with S. Behrens and T. Kragh.

The language of Joyal and Lurie’s $\infty$-categories has now become an indispensable tool in homotopy theory. However, to encode desirable universal properties, just the theory of $\infty$-categories does not always suffice. Sometimes it is necessary to pass to other flavors, like equivariant, internal, or enriched $\infty$-categories. To illustrate, the universal property of G-spectra is best expressed in its incarnation as a G-equivariant $\infty$-category. In this talk, I will give an introduction to a framework that is designed to deal with different generalizations of $\infty$-categories via so-called $\infty$-equipments. These equipments give rise to category theories for their objects, incorporating concepts such as (co)limits and pointwise Kan extensions. There are suitable ambient $\infty$-equipments for enriched, internal, and fibered $\infty$-category theory (and combined flavors), and I would like to highlight some examples.

The tangle hypothesis is a variant of the cobordism hypothesis pertaining to tangles, that is, manifolds and cobordisms between them embedded in Euclidean spaces, and equipped with a suitable type of framing. This elaborate ensemble of geometric data can be assembled into a single higher categorical object, an $E_m$-monoidal $(\infty,n)$-category Tang^{fr}{n,m} where $n$ is the maximal dimension of the underlying cobordisms, and $m$ is the fixed codimension of the Euclidean embeddings. The tangle hypothesis then asserts that Tang^{fr}{n,m} is freely generated as an $E_m$-monoidal $(\infty,n)$-category with duals from a single object. A sketch of proof of the tangle hypothesis appears in Lurie's 2009 text, and a conditional argument reducing the tangle hypothesis to another conjecture was later provided by Ayala and Francis, though a complete and formal proof has not yet appeared. In this talk I will describe work with Joost Nuiten which provides an infinitesimal version of this conjecture. More precisely, applying previous work on Quillen cohomology of higher categories we calculate the cotangent complex of Tang^{fr}{n,m} and show that, in a suitable sense, it is freely generated from a single generator. This can be considered as supporting evidence in the direction of the tangle hypothesis, but also reduces the tangle hypothesis to a statement on the level of $E_m$-monoidal $(n+1,n)$-categories using a form of obstruction theory for higher categories.

The spectral Lie operad is the Koszul dual operad to the cocommutative cooperad in the category of spectra. A spectral Lie algebra is an algebra over the spectral Lie operad. M. Behrens and J. Kjaer constructed so-called Dyer-Lashof-Lie power operations acting on the mod-p homology groups of a spectral Lie algebra. However, they computed relations between these operations only for p=2. In my talk, I will explain how to compute the desired relations for any prime by using functor calculus in the category of simplicial restricted Lie algebras. The latter category might be thought of as an algebraic approximation of the category of spaces, and so, algebraic calculations may also be helpful in understanding of the topological Goodwillie spectral sequence.

Starting with the foundational work of Thomason, there has been an enormous amount of progress in chromatically localized algebraic K-theory. In this talk, I’ll survey some of this progress and explain how one can leverage these techniques to study chromatically localized TR and TC. The crucial input is the close relationship between TR and Bloch’s spectrum of curves on algebraic K-theory as first observed by Hesselholt. Part of this is joint work with Liam Keenan.

My talk concerns bordism rings of compact smooth manifolds equipped with a smooth action by a finite group. I will start by recalling classical results on the subject from the 60's and 70's, mostly due to Conner-Floyd, Boardman, Stong and Alexander. Afterwards I will discuss recent joint work with Stefan Schwede in which we prove an algebraic universal property for the collection of all bordism rings of manifolds with commuting involutions.

Hermitian K-theory is closely related to the classical theory of quadratic forms. We will give an overview of recent calculations of higher Hermitian K-groups of fields and rings of integers in number fields through motivic homotopy theory. The answers involve arithmetic data, while the calculational methods are rooted in homotopy theory. Joint work with Haakon Kolderup, Jonas Kylling, and Oliver Roendigs.

The theory of commutative (a.k.a. $E_\infty$) ring spectra is a natural higher algebraic analog of commutative algebra. However, there are notions from commutative algebra that do not generalize well to the spectral world, as well as notions that admit several reasonable generalizations. One example of the latter phenomenon is the group of units of a commutative ring $R$. The most direct generalization is the units spectrum $gl_1(R)$ of a commutative ring spectrum. However, for various purposes (and especially the formation of quotients of $R$), it is natural to consider a $\mathbb{Z}$-linear variant $\mathbb{G}_m(R)$ called the spectrum of "strict units." The spectrum $\mathbb{G}_m(R)$ admits a natural delooping, namely the strict Picard spectrum, analogous to the Picard group of a commutative ring. In the first part of my talk, I will discuss the theory of spectra and commutative ring spectra, define (strict) units and Picard spectra, and introduce some key ingredients involved in their study. In the second part, I will present the computation of the strict Picard spectrum of the sphere spectrum and its completions, and the strict units of spherical group algebras of finitely generated abelian groups. The second computation is a joint work in preparation with Thomas Nikolaus and Allen Yuan.

In recent work, Guchuan Li, Sarah Peterson, and I have constructed models for $C_{2}$-equivariant analogues of the integral Brown–Gitler spectra. In this talk, I will start by introducing the classical Brown–Gitler spectra, and discussing some of their applications. After that, I will sketch the construction of the $C_{2}$-equivariant integral Brown–Gitler spectra, and discuss the applications we are beginning to study.

Classically the Thom isomorphism relates the cohomology of the Thom space of a vector bundle to the cohomology of its base. The Thom isomorphism for equivariant vector bundles fails in $RO(G)$-graded cohomology, even for $G=C_2$. However, Costenoble–Waner developed an $RO(\Pi)$-graded equivariant cohomology theory, extending the usual representation grading $RO(G)$ to representations of the equivariant fundamental groupoid, and they showed the Thom isomorphism holds in this extended grading. Costenoble recently computed the $RO(\Pi)$-graded cohomology of $B_{C_2}U(1)$, the classifying space for complex $C_2$-line bundles. In this talk I will describe these different gradings and talk about work in progress computing the $RO(\Pi)$-graded cohomology of $B_{C_2}O(1)$, the classifying space for real $C_2$-line bundles. This is joint work with Agnès Beaudry, Chloe Lewis, Sabrina Pauli and Elizabeth Tatum.

Inspired by the work of Lurie and others, Gepner—Meier define families of equivariant cohomology theories based on oriented elliptic curves. By construction, these equivariant elliptic cohomologies are multiplicative, but only in a naïve equivariant sense—there is no obvious construction of norm maps on these theories. In this talk, I will describe how to use a moduli interpretation of the geometric fixed points of these equivariant theories due to Gepner—Meier, to construct what we call "geometric norms". Some applications of these geometric norms will also be discussed. This is joint work-in-progress with William Balderrama and Sil Linskens.

Several structural questions have emerged at least twice in topology: once in chromatic homotopy theory and once in equivariant topology (completions and localization, fracture squares, Balmer spectra, support, telescope conjecture, sheaves, filtrations, ….). In the chromatic world they arise in hard-core form, and in equivariant topology they reach a benign algebraic manifestation in the rational case. My talk is from this gentler world. The overall project is to build an algebraic model for rational G-equivariant cohomology theories for all compact Lie groups G, and when G is small or abelian this has been done. In general, the model is expected to take the form of a category of sheaves of modules over a sheaf of rings over the space of closed subgroups of G. The talk will focus on structural features of the expected model for general G such as those above, and feature recent joint work with Balchin and Barthel.

In this talk, I will explain how to compute the $C_3$-equivariant relative Adams spectral sequence for the Borelification of $tmf(2)$.This yields an entirely algebraic computation of the 3-local homotopy groups of $tmf$. The final answer is well-known of course – the novelty here is that the rASS is completely determined by it $E_1$-page as a cochain complex of Mackey functors. Explicitly, the input consists of the Hopf algebroid structure on $$\mathbb{F}_3 \otimes_{tmf(2)}\mathbb{F}_3$$ modulo transfer, which is deduced from the structure maps on the equivariant dual Steenrod algebra, as well as the knowledge of the homotopy group of the underlying $tmf(2)$ along with the $C_3-$action. Then we construct a bifiltration on $tmf(2)$ and use synthetic arguments to deduce the Adams differentials from the associated square of spectral sequences. The rASS degenerates on $E_{12}$ for tridegree reasons and stabilizes to a periodic pattern that essentially lies within a band of slope 1/4. This is joint work with Jeremy Hahn, Andrew Senger, and Foling Zou.

I will give a leisurely overview of parametrized and higher semiadditivity. In particular I will motivate this concept by giving a variety of examples. As one such example, I will explain how it gives a conceptual interpretation of definitions in (globally) equivariant algebra and homotopy theory. I will then finish by discussing the close connection between generalized semiadditivity and the construction of transfer maps.