Derived algebraic geometry and chromatic homotopy theory

Workshop theme

Chromatic homotopy theory is the classical subfield of algebraic topology which uses the theory of smooth one-parameter formal groups to organize calculations and the search for large-scale phenomena in stable homotopy theory. There is a formal group associated to any cohomology theory with Chern classes for complex vector bundles and, remarkably, the information can go the other way: under good conditions we can construct the cohomology theory from the formal group. This allows for a very tight connection between the geometry of formal groups and homotopy theory. For example, the moduli stack of formal groups has a height stratification, and thus induces a corresponding stratification in stable homotopy theory known as the chromatic filtration. Current major projects include calculations at a single height, assembling information from different heights, and the investigation of how various constructions in stable homotopy theory pass between heights.

If algebraic geometry is the study of schemes equipped with a sheaf of rings, then derived algebraic geometry is the study of schemes with sheaves of rings in some category amenable to homotopy theory, such as structured ring spectra. There are very important derived stacks and schemes that live over the moduli stack of formal groups, such as the moduli stack of generalized elliptic curves and certain Shimura varieties. This has led to a fruitful interplay between derived algebraic geometry and chromatic homotopy theory. Examples include the work of Gross and Hopkins establishing deep connections between various types of duality, the Hopkins-Miller work on topological modular forms, and considerable recent work on Picard and Brauer groups and spaces. This has expanded the field in new and unexpected directions.

Applications for this workshop are now closed

Timetable

Monday 24th September 2018
10:00 to 11:00	Agnes Beaudry University of Colorado K(n)-local homotopy from a Galois theory perspective -1	Room 1
11:30 to 12:30	Niko Naumann Universität Regensburg The role of p-divisible groups -1	Room 1
14:30 to 15:30	Christian Ausoni University Paris 13 On the topological Hochschild homology of Johnson-Wilson spectra	Room 1
16:00 to 17:00	Lennart Meier Rheinische Friedrich-Wilhelms-Universität Bonn Topological modular forms with level structure	Room 1

Tuesday 25th September 2018
10:00 to 11:00	Agnes Beaudry University of Colorado Duality and invertibility using finite resolutions - 2	Room 1
11:30 to 12:30	Niko Naumann Universität Regensburg The role of p-divisible groups - 2	Room 1
14:30 to 15:30	Jeremy Hahn The spectrum of units of a height 2 theory	Room 1
16:00 to 17:00	Tomer Schlank Hebrew University of Jerusalem Modes, Ambidexterity and Chromatic homotopy.	Room 1

Wednesday 26th September 2018
10:00 to 11:00	Ben Antieau Derived algebraic geometry I	Room 1
11:30 to 12:30	Nathaniel Stapleton ; Tobias Barthel Københavns Universitet (University of Copenhagen) Transchromatic homotopy theory 1	Room 1

Thursday 27th September 2018
10:00 to 11:00	Ben Antieau Derived algebraic geometry II	Room 1
11:30 to 12:30	Nathaniel Stapleton ; Tobias Barthel Københavns Universitet (University of Copenhagen) Transchromatic homotopy theory 2	Room 1
14:30 to 15:30	Gijs Heuts University of Copenhagen Lie algebras and v_n-periodic spaces	Room 1
16:00 to 17:00	Vesna Stojanoska University of Illinois at Urbana-Champaign Characteristic classes determine dualizing modules	Room 1

Friday 28th September 2018
10:00 to 11:00	Zhouli Xu Motivic Ctau-modules and Stable Homotopy Groups of Spheres	Room 1
11:30 to 12:30	Gereon Quick NTNU Examples of non-algebraic classes in the Brown-Peterson tower	Room 1
14:30 to 15:30	Mark Behrens University of Chicago The tmf-based Adams spectral sequence for Z	Room 1

Derived algebraic geometry and chromatic homotopy theory

Organisers

Mark Behrens University of Chicago
David Gepner Purdue University
Paul Goerss Northwestern University
Michael Hopkins Harvard University
Tyler Lawson University of Minnesota

Speakers

Ben Antieau
Christian Ausoni University Paris 13
Tobias Barthel Københavns Universitet (University of Copenhagen)
Agnes Beaudry University of Colorado
Mark Behrens University of Chicago
Jeremy Hahn
Gijs Heuts University of Copenhagen
Lennart Meier Rheinische Friedrich-Wilhelms-Universität Bonn
Niko Naumann Universität Regensburg
Gereon Quick NTNU
Tomer Schlank Hebrew University of Jerusalem
Nathaniel Stapleton
Vesna Stojanoska University of Illinois at Urbana-Champaign
Zhouli Xu

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Participants

Ben Antieau
Christian Ausoni University Paris 13
Andrew (Andy) Baker University of Glasgow
Scott Balchin University of Warwick
David Barnes Queen's University Belfast
Tobias Barthel Københavns Universitet (University of Copenhagen)
Jonathan Beardsley University of Washington
Agnes Beaudry University of Colorado
Mark Behrens University of Chicago
Nicola Bellumat University of Sheffield
Andrew Blumberg University of Texas at Austin
Anna Marie Bohmann Vanderbilt University
Robert Bruner Wayne State University; Universitetet i Oslo
Michael Ching None / Other
Dominic Culver University of Illinois at Urbana-Champaign
Soren Galatius University of Copenhagen; Københavns Universitet (University of Copenhagen)
David Gepner Purdue University
Paul Goerss Northwestern University
John Greenlees University of Sheffield
Rok Gregoric University of Texas at Austin
Jesper Grodal Københavns Universitet (University of Copenhagen)
Xing Gu University of Melbourne
Jeremy Hahn
Fabian Hebestreit University of Bonn
Daniel Heiss Universität Regensburg
Hans-Werner Henn
Lars Hesselholt Acclarogen Limited
Gijs Heuts University of Copenhagen
Jocelyne Ishak University of Kent
Mohammad Kang Wayne State University
Magdalena Kedziorek Universiteit Utrecht
Tyler Lawson University of Minnesota
Guchuan Li Northwestern University
Michael Mandell Indiana University
Lennart Meier Rheinische Friedrich-Wilhelms-Universität Bonn
Niko Naumann Universität Regensburg
Nikitas Nikandros University of Kent
Robert Oliver Université Paris 13
Viktoriya Ozornova University of Bonn
Martin Palmer-Anghel University of Bonn
Irakli Patchkoria University of Bonn
Viet-Cuong PHAM University of Strasbourg
Piotr Pstragowski Northwestern University
Gereon Quick NTNU
J.D. Quigley University of Notre Dame
Arpon Raksit Stanford University
Oscar Randal-Williams University of Cambridge
George Raptis Universität Regensburg
Doug Ravenel University of Rochester
Birgit Richter Universität Hamburg
Constanze Roitzheim University of Kent
Krishanu Sankar University of British Columbia
Tomer Schlank Hebrew University of Jerusalem
Christian Schlichtkrull Universitetet i Bergen
Stefan Schwede Rheinische Friedrich-Wilhelms-Universität Bonn
XiaoLin Danny Shi Harvard University
Harry Smith University of Illinois at Chicago
Nathaniel Stapleton
Vesna Stojanoska University of Illinois at Urbana-Champaign
Markus Szymik Universität Bielefeld
Hiro Tanaka Harvard University
Elizabeth Tatum University of Illinois at Urbana-Champaign
Ulrike Tillmann University of Oxford
Gabriel Valenzuela Ohio State University
Clément Vitryclement University of Strasbourg
Nathalie Wahl Københavns Universitet (University of Copenhagen)
Jordan Williamson University of Sheffield
Jesse Wolfson University of Chicago
Mingcong Zeng University of Rochester