Higher structures in homotopy theory

Workshop theme

Homotopy theory has covered a long distance since its origins, the classification of spaces up to homotopy equivalence. Over the years, various kinds of mathematical structures have been investigated from a homotopical perspective, such as equivariant spaces, rings, C^*-algebras, or varieties. Many different approaches of how to formalize what a “homotopy theory” is were proposed, the most prominent ones being the notions of model category and ∞-category.

The relationship between the different ways to formalize a homotopy theory is now well understood; indeed, for comparing different concepts of homotopy theories, one often wants to consider all of them together as another homotopy theory, i.e., a ‘homotopy theory of homotopy theories’. Somewhat surprisingly, most of the concepts organize themselves into a Quillen model category, and the various approaches are Quillen equivalent. After these individual comparison results, Töen was even able to axiomatically characterize a homotopy theory of homotopy theories.

The homotopy theory of homotopy theories is only the first step in a hierarchy of interesting structures, namely the homotopy theoretic approach to higher categories. From this broader perspective, homotopy theories are just (∞, 1)-categories, where the ∞ indatices a structure with higher morphisms of all levels, and the 1 refers to the fact that all 1-morphisms and higher morphisms are weakly invertible. There are now ways to give rigorous meaning to the notion of (∞, n)-categories i.e., where only higher morphisms in level n and above are invertible. Having a rigorous model category of (∞,n)-categories is a cornerstone for the modern approach to topological field theory, thereby unifying categorical considerations with those of homotopy and manifold theory.

This workshop consists of lecture series as well as individual research talks. The introductory series will explain some of the key methods relevant to many parts of the overarching program; they are intended to invite graduate students and postdocs into the field, as well as to strengthen the common ground of the program participants. The individual talks will inform us about recent developments about higher structures in homotopy theory.

Timetable

Monday 2nd July 2018
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from David Abrahams (INI Director)	No Room Required
10:00 to 11:00	Emily Riehl Johns Hopkins University The model-independent theory of (∞,1)-categories (1)	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Thomas Nikolaus Universität Münster Higher categories and algebraic K-theory (1)	Room 1
12:30 to 13:30	Buffet Lunch at INI	No Room Required
13:30 to 14:30	Tobias Dyckerhoff Universität Bonn Higher Segal spaces (1)	Room 1
14:30 to 15:30	John Francis Northwestern University Factorization homology (1)	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Sarah Yeakel University of Maryland, College Park Isovariant homotopy theory	Room 1
17:00 to 18:00	Welcome Wine Reception at INI	No Room Required

Tuesday 3rd July 2018
10:00 to 11:00	Emily Riehl Johns Hopkins University The model-independent theory of (∞,1)-categories (2)	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Thomas Nikolaus Universität Münster Higher categories and algebraic K-theory (2)	Room 1
12:30 to 13:30	Buffet Lunch at INI	No Room Required
13:30 to 14:30	Tobias Dyckerhoff Universität Bonn Higher Segal spaces (2)	Room 1
14:30 to 15:30	John Francis Northwestern University Factorization homology (2)	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Andrew Blumberg University of Texas at Austin Thirteen ways of looking at an equivariant stable category	Room 1

Wednesday 4th July 2018
10:00 to 11:00	Emily Riehl Johns Hopkins University The model-independent theory of (∞,1)-categories (3)	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Thomas Nikolaus Universität Münster Higher categories and algebraic K-theory (3)	Room 1
12:30 to 13:30	Buffet Lunch at INI	No Room Required
13:30 to 17:00	Free Afternoon	No Room Required
19:30 to 22:00	Formal Dinner at Emmanuel College	No Room Required

Thursday 5th July 2018
10:00 to 11:00	Emily Riehl Johns Hopkins University The model-independent theory of (∞,1)-categories (4)	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Thomas Nikolaus Universität Münster Higher categories and algebraic K-theory (4)	Room 1
12:30 to 13:30	Buffet Lunch at INI	No Room Required
13:30 to 14:30	Rune Haugseng Københavns Universitet (University of Copenhagen) Higher categories of higher categories	Room 1
14:30 to 15:30	Christopher Schommer-Pries University of Notre Dame The Relative Tangle Hypothesis	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Kathryn Hess EPFL - Ecole Polytechnique Fédérale de Lausanne A Künneth theorem for configuration spaces of products	Room 1

Friday 6th July 2018
10:00 to 11:00	Claudia Scheimbauer Max-Planck-Institut für Mathematik, Bonn Dualizability in the higher Morita category	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Markus Spitzweck Universität Osnabrück Hermitian K-theory for Waldhausen infinity categories with genuine duality	Room 1
12:30 to 13:30	Buffet Lunch at INI	No Room Required

Organisers

Clark Barwick University of Edinburgh
Julie Bergner University of Virginia
Ieke Moerdijk Universiteit Utrecht
Stefan Schwede Rheinische Friedrich-Wilhelms-Universität Bonn

Speakers

Andrew Blumberg University of Texas at Austin
Tobias Dyckerhoff Universität Bonn
John Francis Northwestern University
Rune Haugseng Københavns Universitet (University of Copenhagen)
Kathryn Hess EPFL - Ecole Polytechnique Fédérale de Lausanne
Jessica Nikolaus Universität Münster
Emily Riehl Johns Hopkins University
Claudia Scheimbauer Max-Planck-Institut für Mathematik, Bonn
Christopher Schommer-Pries University of Notre Dame
Markus Spitzweck Universität Osnabrück
Sarah Yeakel University of Maryland, College Park

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Participants

Fernando Abellan Rheinische Friedrich-Wilhelms-Universität Bonn
Manuel Araujo Max-Planck-Institut für Mathematik, Bonn
Nils Baas Norwegian University of Life Sciences (UMB)
Scott Balchin University of Sheffield
David Barnes Queen's University Belfast
Marco Benini Universität Hamburg
Julie Bergner University of Virginia
Andrew Blumberg University of Texas at Austin
Irina Bobkova University of Rochester
Pierre Cagne Université Paris 7 - Denis-Diderot
Jonathan Campbell Vanderbilt University
Alexander Campbell Macquarie University
Timothy Campion University of Notre Dame
Paolo Capriotti University of Nottingham
Louis Carlier Universitat Autònoma de Barcelona
Alex Cebrian Universitat Autònoma de Barcelona
Boris Chorny Oranim Academic College
Hongyi Chu Université Lille 1
Jack Davies Universität Bonn
Vladimir Dotsenko Trinity College Dublin
Gabriel Drummond-Cole None / Other
Bjorn Ian Dundas NTNU
Tobias Dyckerhoff Universität Bonn
John Francis Northwestern University
Daniel Fuentes-keuthan Johns Hopkins University
Andrea Gagna Aix Marseille Université
Harry Gindi Universität Regensburg
Paul Goerss Northwestern University
John Greenlees University of Sheffield
Rok Gregoric University of Texas at Austin
Jesper Grodal Københavns Universitet (University of Copenhagen)
Araminta Gwynne Massachusetts Institute of Technology
Rune Haugseng Københavns Universitet (University of Copenhagen)
Alice Hedenlund Universitetet i Oslo
Kathryn Hess EPFL - Ecole Polytechnique Fédérale de Lausanne
Asaf Horev Hebrew University of Jerusalem
Gustavo Jasso Rheinische Friedrich-Wilhelms-Universität Bonn
Magdalena Kedziorek EPFL - Ecole Polytechnique Fédérale de Lausanne
Jens Jakob Kjaer University of Notre Dame
Ben Knudsen Harvard University
Edoardo Lanari Macquarie University
Tyler Lawson University of Minnesota
Tobias Lenz Rheinische Friedrich-Wilhelms-Universität Bonn; Max-Planck-Institut für Mathematik, Bonn
Kathryn Lesh Union College
Cynthia Lester University of Oregon
Michael Mandell Indiana University
Lennart Meier Rheinische Friedrich-Wilhelms-Universität Bonn
Ieke Moerdijk Universiteit Utrecht
Lyne Moser EPFL - Ecole Polytechnique Fédérale de Lausanne
Stephen Nand-lal University of Liverpool
Hoang Kim Nguyen Universität Regensburg
Stefano Nicotra University of Liverpool
Jessica Nikolaus Universität Münster
Joost Nuiten Universiteit Utrecht
Viktoriya Ozornova University of Bonn
Simona Paoli University of Leicester
Mark Penney Max-Planck-Institut für Mathematik, Bonn
Thomas Poguntke University of Bonn
Luca Pol University of Sheffield
Kate Ponto University of Kentucky
J.D. Quigley University of Notre Dame
Oscar Randal-Williams University of Cambridge
Emily Riehl Johns Hopkins University
Martina Rovelli Johns Hopkins University
Husney Parvez Sarwar University of Warwick
Christian Sattler Chalmers University of Technology
Claudia Scheimbauer Max-Planck-Institut für Mathematik, Bonn
Alexander Schenkel University of Nottingham
Daniel Scherl University of Vienna
Tomer Schlank Hebrew University of Jerusalem
Marco Schlichting University of Warwick
Christian Schlichtkrull Universitetet i Bergen
Christopher Schommer-Pries University of Notre Dame
Christoph Schrade Universität Regensburg
Stefan Schwede Rheinische Friedrich-Wilhelms-Universität Bonn
XiaoLin Danny Shi Harvard University
Brooke Shipley University of Illinois at Chicago
Markus Spitzweck Universität Osnabrück
Jan Steinebrunner University of Oxford
Karol Szumiło University of Leeds
Markus Szymik Universität Bielefeld
Ulrike Tillmann University of Oxford
Paula Verdugo Universität Osnabrück
Nathalie Wahl Københavns Universitet (University of Copenhagen)
Tashi Walde Rheinische Friedrich-Wilhelms-Universität Bonn
Tim Weelinck University of Edinburgh
Sarah Whitehouse University of Sheffield
Jordan Williamson University of Sheffield
Dylan Wilson University of Chicago
Richard Wong University of Texas at Austin
Lior Yanovski Hebrew University of Jerusalem
Sarah Yeakel University of Maryland, College Park
Inna Zakharevich Cornell University