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Summer School Interactions between Algebra, Equivariance, and Homotopy Theory (June 24-28, 2024)
Aim and Scope
The aim of the summer school is to bring
together researchers in algebra and homotopy theory, especially those with a link to equivariant
methods, and seek to encourage collaboration and interaction between these different, yet
deeply related fields. A key focus of this summer school is on early career researchers.
Invited Speakers
- John Greenlees "Commutative algebra and equivariant cohomology theories",
- Natalia Castellana "Algebraic models for the mod p homotopy type of classifying spaces",
- Magdalena Kędziorek "Equivariant commutativity and norms",
- Srikanth Iyengar "Dualisable objects in local algebra and modular representation theory".
There will also be a series of contributed talks by participants.
Practical Information
City of Regensburg: Regensburg is a Unesco World Heritage site that is famous for its well-preserved medieval city center and its beautiful Gothic cathedral.
Further information about Regensburg can be found here.
Internet: Access to eduroam and BayernWLAN is available throughout the Mathematics Building.
Accomodation: Regensburg is a tourist destination and we encourage guests to book their rooms as early as possible. Here is a list of hotels in the area:
- Hotel Münchner Hof (In the city center, one must take a bus to the University.)
- Hotel Kaiserhof am Dom (In the city center, one must take a bus to the University.)
- Hotel Jakob (In the city center, one must take a bus to the University.)
- Hotel Apollo (Near the University, but limited eating options nearby.)
- Hotel Wiendl (Between the city center and the University.)
- Hotel Central (Between the city center and the University, one must take a bus to the University.)
Family Friendly Campus: Our UR family service offers various rooms for families and services. If you need further information look here.
Venue
All lectures and research talks are in the lecture hall M311, at the 3rd floor of the Mathematics building of Regensburg University (Attention: not the department "Mathematik und Informatik" of the OTH).
One can reach the University of Regensburg by following the instructions here and see here for maps of the campus. Many of the participants will probably arrive by bus at the Central Bus Station of the university which is close to the math building, a bit more towards North (=top of the plan).
Inside of the Mathematics building, use the staircase next to the main entrance and go up to the top.
Program and Schedule
The schedule of the conference will follow a month before the conference.
Titles and Abstracts
Kamil Rychlewicz -- Equivariant cohomology theories as rings of functions
The classical Poincaré-Hopf theorem shows how to recover the Euler characteristic of a compact manifold from isolated zeros of the vector field. The work of Carrell-Liebermann and Akyildiz-Carrell extended that result, showing that in case of a smooth projective complex variety, one can recover the whole cohomology ring from the ring of functions of the zero scheme of the vector field. Theorem of Brion-Carrell extends it to equivariant cohomology for an action of a Borel of SL_2. Together with Tamas Hausel, we recently generalized the result to any reductive group, or its Borel, acting with a single zero of a regular nilpotent. The spectrum of the equivariant cohomology then shows up as a geometrically constructed zero scheme of a vector field. I will report on that result, and sketch a further circle of ideas which allows to view more general cohomology rings, e.g. for spherical varieties or certain singular varieties, as rings of functions on potentially non-affine variety.
Marius Verner Bach Nielsen -- K(n)-local homotopy in synthetic spectra
Using deformation theory one can categorify the E-Adams spectral sequence based on an Adams type spectrum E. For K(n)-local spectra one often uses the K(n)-local E_n-Adams spectral sequence, which does not directly admit a categorification. In this talk, we will propose a solution to this, by using a localization of E_n-based synthetic spectra. As a proof of concept, we compute the synthetic homotopy groups of the K(1)-local sphere. This is joint work with Torgeir Aambø.
Torgeir Aambø -- K(n)-local deformation theory
Pstragowski’s category of hypercomplete E_n-based synthetic spectra, Syn_E, acts as a one-parameter deformation between Sp_n and the derived category of E_*E-comodules. However, there is a more fundamental building block inside Sp_n — the category of K(n)-local spectra — and we can ask whether this also has an associated deformation. But, K(n)-based synthetic spectra deform to the wrong category of comodules, and hence can be interpreted as an incorrect deformation for Sp_{K(n)}. In this talk, we investigate a K(n)-local analog of Syn_E and prove that it has the correct one-parameter deformation properties. This is joint work with Marius Nielsen.
Maxime Wybouw -- Minimal models in diagrams of chain complexes
A classical theorem by Kadeishvili states that the information of the quasi-isomorphism type of an associative differential graded algebra over a field can be encoded as a minimal A_\infty structure on its homology algebra. In this talk, I will discuss generalizations of this result that allow to encode commutative multiplications and to work over more general ground rings. A motivating example is the cochain algebra of a space. One main tool is to work with diagrams of chain complexes indexed by finite sets and injections.
Conference Poster
You can download the conference poster here.
Organizers
Conference Picture
Sponsors of the conference
This conference is funded by SFB 1085 "Higher Invariants" and Foundation Compositio Mathematica
