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From 5th to 8th of September 2016, a series of lectures will be given by two speakers to give an exposition of their research. The lectures will take place in room A1 of the CRM.
Tilman Bauer: Dieudonné theory for unstable operations
Ergun Yalçin: Finite group actions on homotopy spheres
Tentative schedule:
Monday 5th
11:30-12:30 Yalçin
15:00-16:00 Bauer
Tuesday 6th
10:00-11:00 Bauer
11:30-12:30 Yalçin
Wednesday 7th
10:00-11:00 Yalçin
11:30-12:30 Bauer
Thursday 8th
10:00-12:30 TBA (question session)
Organizer: Natàlia Castellana
The XI Spanish Group Theory meeting starts on Thursday afternoon in Barcelona, http://www-ma4.upc.edu/~burillo/encuentro/.
Speaker: Tilman Bauer
Title: Dieudoné theory for unstable operations
Abstract:
1. Unstable Adams spectral sequences
Abstract: In this introductory talk, I will discuss the Adams spectral sequence for computing homotopy groups of (or homotopy classes of maps between) spaces. I will touch on the classical unstable Adams spectral sequence based on mod-p cohomology and describe how it can be generalized to more general cohomology theories. I will explain what the analog of unstable algebras and unstable modules over the Steenrod algebra are in a generalized-cohomology setting and how to describe the E_2-term in terms of homological algebra in certain cases.
2. Formal plethories
Abstract: The set of unstable operations for a generalized cohomology theory has a rich algebraic structre which can be thought of as a kind of Hopf algebroid internal to coalgebras. I will explain this structure (called formal plethory) and a point of view on it in algebro-geometric terms. I will then use this to give a more flexible and general description of the Adams E_2-term.
3. Dieudonné theory for unstable operations
Abstract: In this last part of my lecture series, I will concentrate on the particularly important case when the coefficient ring of the cohomology theory is a graded field, such as for the Morava K-theories. In this case, I will construct a Dieudonné correspondence between formal plethories and bialgebras with respect to two different monoidal structures over a certain ring. This allows for hands-on computations with formal plethories, which, time permitting, I will demonstrate.
Speaker: Ergun Yalçin
Title: Finite group actions on homotopy spheres
Abstract: Actions of finite groups on spheres can be studied in various settings,
such as (A) smooth G-actions on a closed manifold homotopy equivalent to a sphere,
(B) finite G-homotopy representations (as defined by tom Dieck), and (C) finite G-CW
complexes homotopy equivalent to a sphere. These three settings generalize the basic
models arising from unit spheres S(V) in orthogonal or unitary G-representations. Recently I wrote a series of papers with Ian Hambleton studying finite group actions on homotopy spheres in the settings (B) and (C) by assuming that the actions have rank 1 isotropy (meaning that the isotropy subgroups of G do not contain Z/p x Z/p, for any prime p). We found some group theoretical constraints imposed on finite groups admitting such actions. In the setting (C) these constraints give a complete characterization of such groups.
In a series of three talks I will give a survey of these results and explain some of the
techniques that are used. In the proofs we use homological algebra over orbit category,
G-equivariant fibrations, G-CW-surgery techniques, and classification theorems from finite
group theory. I will also mention some results from our more recent work with Sune Precht Reeh on representation rings for fusion systems and dimension functions.
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