skip to content
 

Timetable (HHHW03)

Derived algebraic geometry and chromatic homotopy theory

Monday 24th September 2018 to Friday 28th September 2018

Monday 24th September 2018
09:30 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Agnes Beaudry (University of Colorado)
K(n)-local homotopy from a Galois theory perspective -1
The goal of these two talks is to give an overview of a perspective on K(n)-local homotopy theory that lends itself to computations. I will introduce some of the key players of the workshop such as the Morava K and E-theories. The introduction will emphasize the Galois theory inherent to the situation and be designed to lead us to consider finite resolutions of the K(n)-local sphere. I will then explain how these perspectives have helped us gain a better understanding on problems in chromatic homotopy theory such as the study of invertible elements and chromatic reassembly.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Niko Naumann (Universität Regensburg)
The role of p-divisible groups -1
We will try to give an account of Lurie's treatment of p-divisible groups in spectral algebraic geometry, to the extend available by the date of the workshop.
INI 1
12:30 to 13:30 Buffet Lunch at INI
14:30 to 15:30 Christian Ausoni (University Paris 13)
tba
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Lennart Meier (Rheinische Friedrich-Wilhelms-Universität Bonn)
Topological modular forms with level structure
Topological modular forms with level structure are spectra associated with moduli of elliptic curves with extra structure. These come in a huge variety. We will report on progress on the conjecture that all of these spectra split additively into a few simple pieces. Another goal is to obtain an understanding of connective variants of TMF with level structure. We will present some constructions and a conjecture on their associated stacks. This work is partially joined with Viktoriya Ozornova.
INI 1
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 25th September 2018
10:00 to 11:00 Agnes Beaudry (University of Colorado)
K(n)-local homotopy from a Galois theory perspective - 2
The goal of these two talks is to give an overview of a perspective on K(n)-local homotopy theory that lends itself to computations. I will introduce some of the key players of the workshop such as the Morava K and E-theories. The introduction will emphasize the Galois theory inherent to the situation and be designed to lead us to consider finite resolutions of the K(n)-local sphere. I will then explain how these perspectives have helped us gain a better understanding on problems in chromatic homotopy theory such as the study of invertible elements and chromatic reassembly.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Niko Naumann (Universität Regensburg)
The role of p-divisible groups - 2
We will try to give an account of Lurie's treatment of p-divisible groups in spectral algebraic geometry, to the extend available by the date of the workshop.
INI 1
12:30 to 13:30 Buffet Lunch at INI
14:30 to 15:30 Jeremy Hahn (Massachusetts Institute of Technology); (Harvard University)
The spectrum of units of a height 2 theory
The space BSU admits two infinite loop space structures, one arising from addition of vector bundles and the other from tensor product. A surprising fact, due to Adams and Priddy, is that these two infinite loop spaces become equivalent after p-completion. I will explain how the Adams-Priddy theorem allows for an identification of sl_1(ku_p), the spectrum of units of p-complete complex K-theory. I will then describe work, joint with Andrew Senger, that attempts to similarly understand the spectrum of units of the 2-completion of tmf_1(3). Our computations seem suggestive of broader phenomena, and I will include discussion of several open questions.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Tomer Schlank (Hebrew University of Jerusalem)
Modes, Ambidexterity and Chromatic homotopy.
Spectra can be considered as the universal presentable stable 00-category, similarly Set can be considered as the universal presentable 1-category and pointed spaces can be considered as the universal pointed presentable. More generally some properties of presentable 00-categories can be classified as equivalent to being a module over a universal symmetric monoidal 00-category. We call such universal symmetric monoidal 00-category "Modes". We describe certain facts about the general theory of modes and present how one can generate new ones from old ones.
INI 1
Wednesday 26th September 2018
10:00 to 11:00 Ben Antieau (University of Illinois at Chicago)
Derived algebraic geometry I
In this series of lectures I will outline the major features of derived algebraic geometry in both the connective and nonconnective settings. The connective setting is what is closer to classical algebraic geometry and the study of geometric moduli spaces, for example moduli of objects in derived categories; the nonconnective setting is what is needed for applications to topological modular forms.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Tobias Barthel (Københavns Universitet (University of Copenhagen)); Nathaniel Stapleton (University of Kentucky)
Transchromatic homotopy theory 1
INI 1
12:30 to 13:30 Buffet Lunch at INI
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Trinity College
Thursday 27th September 2018
10:00 to 11:00 Ben Antieau (University of Illinois at Chicago)
Derived algebraic geometry II
In this series of lectures I will outline the major features of derived algebraic geometry in both the connective and nonconnective settings. The connective setting is what is closer to classical algebraic geometry and the study of geometric moduli spaces, for example moduli of objects in derived categories; the nonconnective setting is what is needed for applications to topological modular forms.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Tobias Barthel (Københavns Universitet (University of Copenhagen)); Nathaniel Stapleton (University of Kentucky)
Transchromatic homotopy theory 2
INI 1
12:30 to 13:30 Buffet Lunch at INI
14:30 to 15:30 Gijs Heuts (University of Copenhagen)
Lie algebras and v_n-periodic spaces
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Vesna Stojanoska (University of Illinois at Urbana-Champaign)
Characteristic classes determine dualizing modules
I will address the question of determining the K(n)-local Spanier-Whitehead dual of the Lubin-Tate spectrum, equivariantly with respect to the action of the Morava stabilizer group. A dualizing module can be constructed abstractly, and we use characteristic classes to relate it to a certain representation sphere, at least when we restrict the action to a finite subgroup. As a consequence in specific examples, explicit calculations of characteristic classes also give explicit formulas for the Spanier-Whitehead duals of spectra like TMF and higher real K-theories. This is work in progress, joint with Agnes Beaudry, Paul Goerss, and Mike Hopkins.
INI 1
Friday 28th September 2018
10:00 to 11:00 Zhouli Xu (Massachusetts Institute of Technology)
Motivic Ctau-modules and Stable Homotopy Groups of Spheres
I will discuss the equivalence of stable infinity categories, between the motivic Ctau-modules over the complex numbers and the derived category of BP_*BP-comodules. As a consequence, the motivic Adams spectral sequence for Ctau is isomorphic to the algebraic Novikov spectral sequence. This isomorphism of spectral sequences allows computations of classical stable stems at least to the 90-stem, with ongoing computations into even higher dimensions. I will also discuss the situation in the real motivic world, and some connections to the new Doomsday Conjecture, if time permits. This is joint work with Mark Behrens, Bogdan Gheorghe, Dan Isaksen and Guozhen Wang.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Gereon Quick (NTNU)
Examples of non-algebraic classes in the Brown-Peterson tower
It is a classical problem in algebraic geometry to decide whether a class in the singular cohomology of a smooth complex variety X is algebraic, that is if it can be realized as the fundamental class of an algebraic subvariety of X. One can ask a similar question for motivic spectra: Given a motivic spectrum E, which classes in the topological E-cohomology of X come from motivic classes. I would like to discuss this question and examples of non-algebraic classes for the tower of Brown-Peterson spectra.
INI 1
12:30 to 13:30 Buffet Lunch at INI
14:30 to 15:30 Mark Behrens (University of Chicago)
The tmf-based Adams spectral sequence for Z
I will describe some specific and qualitative aspects of the tmf-based Adams spectral sequence for the Bhattacharya-Egger type 2 spectrum called "Z". This is joint work with Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu.
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons