Derived algebraic geometry and chromatic homotopy theory
Monday 24th September 2018 to Friday 28th 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 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 Etheories. 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 The role of pdivisible groups 1
We will try to give an account of Lurie's treatment of pdivisible 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 On the topological Hochschild homology of JohnsonWilson spectra
Let E(n) denote the nth JohnsonWilson spectrum at an
odd prime p.
The spectrum E(1) coincides with the Adams summand of
plocal topological Ktheory.
McClure and Staffeldt offered an intriguing computation
of THH(E(1)), showing that it splits as a wedge sum of E(1) and a rationalized
suspension of E(1).
In joint work with Birgit Richter, we study the Morava
Ktheories of THH(E(n)), with an aim at investigating if McClureStaffeldt's
splitting in lower chromatic pieces generalizes. Under the assumption that E(2) is
commutative, we show that
THH(E(2)) splits as a wedge sum of E(2) and its lower
chromatic localizations.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Lennart Meier 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 
10:00 to 11:00 
Agnes Beaudry Duality and invertibility using finite resolutions  2
In this talk, I will explain applications of the finite resolutions constructed in the first talk to the study of invertibility and duality in the K(n)local category.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Niko Naumann The role of pdivisible groups  2
We will try to give an account of Lurie's treatment of pdivisible 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 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 pcompletion. I will explain how the AdamsPriddy theorem allows for an identification of sl_1(ku_p), the spectrum of units of pcomplete complex Ktheory. I will then describe work, joint with Andrew Senger, that attempts to similarly understand the spectrum of units of the 2completion 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 Modes, Ambidexterity and Chromatic homotopy.
Spectra can be considered as the universal presentable stable 00category, similarly Set can be considered as the universal presentable 1category and pointed spaces can be considered as the universal pointed presentable. More generally some properties of presentable 00categories can be classified as equivalent to being a module over a universal symmetric monoidal 00category. We call such universal symmetric monoidal 00category "Modes". We describe certain facts about the general theory of modes and present how one can generate new ones from old ones.

INI 1 
10:00 to 11:00 
Ben Antieau 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 ; Nathaniel Stapleton 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 
10:00 to 11:00 
Ben Antieau 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 ; Nathaniel Stapleton Transchromatic homotopy theory 2 
INI 1  
12:30 to 13:30  Buffet Lunch at INI  
14:30 to 15:30 
Gijs Heuts Lie algebras and v_nperiodic spaces 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Vesna Stojanoska Characteristic classes determine dualizing modules
I will address the question of determining the K(n)local SpanierWhitehead dual of the LubinTate 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 SpanierWhitehead duals of spectra like TMF and higher real Ktheories.
This is work in progress, joint with Agnes Beaudry, Paul Goerss, and Mike Hopkins.

INI 1 
10:00 to 11:00 
Zhouli Xu Motivic Ctaumodules and Stable Homotopy Groups of Spheres
I will discuss the equivalence of stable infinity categories, between the motivic Ctaumodules over the complex numbers and the derived category of BP_*BPcomodules. 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 90stem, 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 Examples of nonalgebraic classes in the BrownPeterson 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 Ecohomology of X come from motivic classes. I would like to discuss this question and examples of nonalgebraic classes for the tower of BrownPeterson spectra.

INI 1  
12:30 to 13:30  Buffet Lunch at INI  
14:30 to 15:30 
Mark Behrens The tmfbased Adams spectral sequence for Z
I will describe some specific and qualitative aspects of the tmfbased Adams spectral sequence for the BhattacharyaEgger type 2 spectrum called "Z". This is joint work with Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu.

INI 1 