Transchromatic homotopy online conference
We will be holding an online conference on transchromatic homotopy theory via Zoom on Monday, August 9th, 2021. Please register here.
Schedule
15:00-15:10 CEST (9:00 - 9:10 EDT): Welcome and introduction.
15:10-15:50 CEST (9:10 - 9:50 EDT): Eva Höning
Detecting and describing ramification for structured ring spectra
This is joint work with Birgit Richter. In this talk we consider the question of how to transfer the classical ramification theory from algebra to structured ring spectra. We in particular use the Tate construction to study tame and wild ramification for commutative ring spectra, and discuss examples in the context of topological \(K\)-theory and topological modular forms.16:00-16:30 CEST (10:00 - 10:30 EDT): Shachar Carmeli
Chromatic Fourier Transform
In classical Fourier analysis, for a finite abelian group \(M\), the Fourier transform provides an isomorphism between the group algebra of \(M\) over the complex numbers and the algebra of functions on its Pontryagin dual. This Fourier map, which depends on a choice of a complex root of unity, is closely related to the decomposition of complex representations of \(M\) into 1-dimensional representations. In my talk, I will present a work in preparation, joint with Barthel, Schlank, and Yanovski, which generalizes this construction to higher chromatic height. To a "higher root of unity" in a \(T(n)\)-local \(E_\infty\)-ring \(R\) of order \(p^k\), we associate a natural map from the \(R\)-linear group algebra of a \(Z/p^k\)-module spectrum \(M\), to a (shifted version of) the algebra of \(R\)-valued functions on the Brown-Comenetz dual of \(M\). We also provide a categorification, which allows to functorially decompose every \(R\)-linear representation of \(M\) into a colimit of one-dimensional representations. The existence of the higher Fourier map, and its invertibility under suitable primitivity assumptions on the root of unity, allows lifting several results of Hopkins and Lurie regarding the \(K(n)\)-local category to the \(T(n)\)-local one, such as the "affineness" of \(n\)-truncated \(p\)-finite spaces.16:40-17:10 CEST (10:40 - 11:10 EDT): Morgan Opie
Chromatic invariants of vector bundles on projective spaces
In this talk, I will discuss my ongoing work on complex rank 3 topological vector bundles on \(\mathbb{C}P^5\). I will describe a classification of such bundles using twisted, topological modular form-valued invariants. As time allows, I will outline future chromatic directions suggested by this result and by prior work of Atiyah and Rees.17:20-18:00 CEST (11:20 - 12:00 EDT): Dominic Culver
On the motivic Lambda algebra
In classical homotopy theory, an important problem is to compute the Ext groups over the Steenrod algebra since it is the algebraic input for the Adams spectral sequence. An especially useful tool towards this end is the Lambda algebra, which is a DGA given by an explicit presentation whose cohomology is the Adams \(E_2\)-term. As it turns out the Lambda algebra has far reaching applications to other parts of homotopy theory. In this talk, I will describe joint work with William Balderrama and J.D. Quigley where we give a motivic analogue of the Lambda algebra. I will provide examples and, time permitting, discuss applications to finding differentials in the \(\mathbb{R}\)-motivic Adams spectral sequence.18:30-19:00 CEST (12:30 - 13:00 EDT): Chris Lloyd
Calculating the nth Morava \(K\)-theory of the real Grassmannians using \(C_4\)-equivariance
In this talk we will demonstrate how letting the cyclic group of order four act on the real Grassmannians can show that the Atiyah--Hirzebruch spectral sequence calculating their nth Morava \(K\)-theory collapses. This uses chromatic fixed point theory coming from the classification of the equivariant Balmer spectrum of the cyclic groups. This work is joint with Nicholas Kuhn.19:10-19:40 CEST (13:10 - 13:40 EDT): Hana Jia Kong
The homotopy of \(\mathbb{R}\)-motivic image of orthogonal \(j\) spectrum
Bachmann--Hopkins defines the motivic "image of orthogonal \(j\)" spectrum over base fields with characteristic not \(2\). In this talk, I will talk about the effective slice computation of this spectrum over the real numbers. Analogous to the classical story, the result captures a regular pattern that appears in the \(\mathbb{R}\)-motivic stable stems. This is joint work with Eva Belmont and Dan Isaksen..19:50-20:30 CEST (13:50 - 14:30 EDT): Paul Goerss
Picard groups and \(J\)-homomorphisms
Abstract: The classical \(J\)-homomorphism was a mechanism for transferring information from bundle theory to stable homotopy. In this guise, it has been around since the early 1960s, and has re-emerged at various crucial stages in the development of the field. For example, it has recently been applied in several major papers in chromatic homotopy theory, often now combined with a group action. It has been especially effective at producing new and exotic elements in the Picard groups we encounter in this area. In this talk I’d like to take a step back, review some of the classical results and then at how we might use them to conceptualize and generalize some of these important ideas and results. This is an outgrowth of work with Barthel, Beaudry, Bobkova, Henn, Hopkins, Pham, Stojanoska, and probably others I’m forgetting.20:30- CEST (14:30 - EDT): Virtual meet and greet.