Algebraic K-theory, motivic cohomology and motivic homotopy theory

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Workshop theme

Motivic cohomology has its roots going back to theories of algebraic cycles, including such basic topics as enumerative geometry, intersection theory and theories of adequate equivalence relations such as rational, algebraic and numerical equivalence. Algebraic K-theory, beginning with the study of algebraic vector bundles, is in contrast a more recent development. Through Grothendieck's introduction of the theory of Chern classes of algebraic vector bundles and his Riemann-Roch theorem, the two subjects quite quickly became closely intertwined. At the same time, Grothendieck's introduction of theories of motives of smooth projective varieties formalized the link between algebraic cycles and cohomology theories. The search for an extension of a theory of motives to understand cohomology of non-proper varieties, by Beilinson, Bloch, Deligne, Lichtenbaum and many others, eventually led to Voevodsky's development of a triangulated category of (mixed) motives in the 1990s, which made a crucial paradigm shift from cohomology to homology. Parallel to this, ideas for a corresponding homotopy theory were made precise by the work of Morel and Voevodsky, which opened the field of motivic homotopy theory. In this setting, motivic cohomology and algebraic K-theory (for regular schemes) become merely one of the universe of possible motivic cohomology theories represented by objects in the motivic stable homotopy category.

Voevodsky used the triangulted category of motives and motivic homotopy theory in an essential way in his proof of the Milnor conjecture and his contribution to the proof of the Bloch-Kato conjecture. After these stunning successes, the field has had a remarkable further development, leading to a better understanding of the relation of algebraic K-theory with motivic cohomology via Voevodsky's slice tower, opening up new applications of the theory of quadratic forms through Morel's identification of the Grothendieck-Witt ring with the endomorphism ring of the motivic sphere spectrum and acquiring powerful new tools through Ayoub's construction of the Grothendieck six functor formalism for the motivic stable homotopy category. These and other technical advances have enabled computations of basic ``structure constants'' of motivic homotopy theory, such as the homotopy groups of motivic spheres in low degree, yielded new results on splitting of vector bundles on affine varieties, as well as giving striking applications to classical homotopy theory through motivic versions of the Adams and Adams-Novikov spectral sequences. Many new cohomology theories for algebraic varieties, such as algebraic cobordism and other flavors of cobordism, such as symplectic or special linear cobordism, have been analyzed and applied. The problem of recognizing when a space is an infinite loop space, a basic problem in classical homotopy theory, has found a remarkable solution in the motivic setting through the theory of framed cobordism.

At the same time, the motivic approach has a serious, built-in limitation: it can only analyze ``A1-homotopy invariant'' phenomena. A generalization of motivic cohomology through the theory of higher Chow groups with modulus seeks to develop a theory that captures interesting non-A1-invariant phenomena. Parallel to this, the algebraic K-theory of non-regular schemes is rarely A1-homotopy invariant and theories which exploit this fact, such as topological Hochschild homology and topological cyclic homology, have seen major developments in the past few years. The K-theory of non-regular schemes occurs quite naturally in trying to understand the K-theory of regular scheme through deformation theory, which introduces nilpotent schemes in an essential way. Although the deformation theory at a finite level is difficult to capture, recent work has shown that in passing to the limit, one can acheive a quite well-behaved theory. This limit approach has also proved useful in many other settings and has led to the resolution of a number of long-standing conjectures on the K-theory of singular varieties.

This workshop will cover all these areas of recent developments in algebraic K-theory, motivic cohomology and motivic homotopy theory.

Fees

Registration Only

Registration Package: £208
Student Registration Package: £158

The Registration Package includes admission to all seminars, lunches and refreshments on the days that lectures take place (Monday - Friday), wine reception and formal dinner, but does not include other meals or accommodation.

Formal Dinner Only

Formal Dinner: £55

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event. For all remaining participants who would like to attend, such as programme participants, the above charge will apply.

Accommodation

Unfortunately we do not have any accommodation to offer so all successful applicants will need to source their own accommodation.

Please see the Hotels Combined website for a list of local hotels and guesthouses.

Meals

Lunch Lunch timings and location will be confirmed with timetable.

Evening Meal Participants are free to make their own arrangements for dinner.

Formal Dinner The Formal Dinner location and date is to be confirmed.

Participants on the Accommodation Package or Registration Package, including organisers and speakers, are automatically included in this event

Timetable

Monday 13th June 2022
09:00 to 09:55	Registration	No Room Required
09:55 to 10:00	Christie Marr Isaac Newton Institute Welcome Talk	Room 1
10:00 to 11:00	Gunnar Carlsson Stanford University Chair: Marc Levine Representations of Galois groups and Algebraic K-theory of fields	Room 1
11:00 to 11:30	Morning Coffee	Room 1
11:30 to 12:30	Oliver Roendigs Universität Osnabrück Chair: Marc Levine The relation between Quillen K-theory and Milnor K-theory in degree 4	Room 1
12:30 to 13:30	Lunch	No Room Required
13:30 to 14:30	TBA	Room 1
14:30 to 15:30	Parimala Raman Emory University Chair: Marc Levine Pencils of quadrics and period index questions for hyperelliptic curves	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Kyle Ormsby Reed College Chair: Marc Levine Motivic Hochschild homology	Room 1
17:00 to 18:00	Welcome Wine Reception at INI	No Room Required

Tuesday 14th June 2022
10:00 to 11:00	Federico Binda University of Milan; Università degli Studi di Milano Chair: Bruno Kahn On the logarithmic Hochschild-Kostant-Rosenberg theorem	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Adeel Khan Institute of Mathematics, Academia Sinica, Taipei, Taiwan Chair: Bruno Kahn Stacks and equivariant cohomology	Room 1
12:30 to 13:30	Lunch	No Room Required
13:30 to 14:30	TBA	Room 1
14:30 to 15:30	Georg Tamme Johannes Gutenberg-Universität Mainz Chair: James Lewis A version of Vorst's conjecture in positive and mixed characteristic	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Burt Totaro University of California, Los Angeles Chair: James Lewis The Hilbert scheme of points in affine space	Room 1

Wednesday 15th June 2022
10:00 to 11:00	Roy Joshua Ohio State University Motivic Euler characteristics and the Motivic Segal-Becker Theorem	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Tom Bachmann Ludwig-Maximilians-Universität München The slices of KGL over arbitrary bases	Room 1
12:30 to 13:30	Lunch	No Room Required
13:30 to 17:00	Free Afternoon	No Room Required
19:30 to 22:00	Formal Dinner at Robinson College	No Room Required

Thursday 16th June 2022
10:00 to 11:00	Anand Sawant Tata Institute of Fundamental Research Strong A^1-invariance of A^1-connected components of a reductive algebraic group	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Toni Annala University of British Columbia Projective bundle formula for derived cobordism	Room 1
12:30 to 13:30	Lunch	No Room Required
13:30 to 14:30	TBA	Room 1
14:30 to 15:30	Nikita Semenov Ludwig-Maximilians-Universität München Algebraic Morava K-theory and algebraic groups	Room 1
15:30 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Shane Kelly University of Tokyo A nilpotent variant cdh-topology	Room 1

Friday 17th June 2022
10:00 to 11:00	Baptiste Calmès Université d’Artois Hermitian K-theory of stable infinity categories	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Vasudevan Srinivas Tata Institute of Fundamental Research On finite presentation for the tame fundamental group	Room 1
12:30 to 13:30	Lunch	No Room Required
15:30 to 16:00	Afternoon Tea	No Room Required

K-theory, algebraic cycles and motivic homotopy theory

Algebraic K-theory, motivic cohomology and motivic homotopy theory

Organisers

Aravind Asok University of Southern California
Roy Joshua Ohio State University
Marc Levine Universität Duisburg-Essen
Marco Schlichting University of Warwick

Speakers

Toni Annala University of British Columbia
Tom Bachmann Universität Duisburg-Essen
Federico Binda Universität Regensburg; Università degli Studi di Milano
Baptiste Calmès Université d’Artois
Gunnar Carlsson Stanford University
Roy Joshua Ohio State University
Shane Kelly University of Tokyo
Adeel Khan Universität Regensburg
Kyle Ormsby Reed College
Parimala Raman Emory University
Oliver Roendigs Universität Osnabrück
Anand Sawant Tata Institute of Fundamental Research
Nikita Semenov Ludwig-Maximilians-Universität München
Vasudevan Srinivas Tata Institute of Fundamental Research
Georg Tamme
Burt Totaro University of California, Los Angeles

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Participants

Alexey Ananyevskiy None / Other; Steklov Mathematical Institute, Russian Academy of Sciences
Donu Arapura Purdue University
Aravind Asok University of Southern California
Joseph Ayoub Universität Zürich
Ran Azouri Universität Zürich
Tom Bachmann Ludwig-Maximilians-Universität München
Federico Binda University of Milan; Università degli Studi di Milano
Spencer Bloch University of Chicago
Thomas Brazelton University of Pennsylvania
Patrick Brosnan University of Maryland, College Park
Baptiste Calmès Université d’Artois
Gunnar Carlsson Stanford University
Xi Chen University of Alberta
Alessandro D'angelo Universität Duisburg-Essen
Zach Davis Ohio State University
Rob de Jeu Vrije Universiteit Amsterdam
Fred Diamond King's College London
Netan Dogra King's College London
Charles Doran University of Alberta
Elden Elmanto Harvard University
Jean Fasel Université Grenoble Alpes
Pietro Gigli Universität Duisburg-Essen
Phillip Griffiths Institute for Advanced Study, Princeton; University of Miami
Mark Gross University of Cambridge
Gabriela Guzman Polska Akademia Nauk
Jan Hennig Heinrich-Heine-Universität Düsseldorf
José Jaime Hernández Castillo CIMAT (Centro de Investigación en Matemáticas), Mexico
Jens Hornbostel Bergische Universität Wuppertal
William Hornslien Norwegian University of Science and Technology
Jaya Iyer Institute of Mathematical Sciences, Chennai
Roy Joshua Ohio State University
Henry July Bergische Universität Wuppertal
Bruno Kahn Institut de Mathématiques de Jussieu
Mahesh Kakde Indian Institute of Science
Elana Kalashnikov University of Waterloo
Max Karoubi Institut de Mathématiques de Jussieu
Shane Kelly University of Tokyo
Matt Kerr Washington University in St. Louis
Adeel Khan Institute of Mathematics, Academia Sinica, Taipei, Taiwan
Minhyong Kim International Centre for Mathematical Sciences; University of Edinburgh
Hyungseop Kim University of Toronto
Lukas Kofler University of Cambridge
Naomi Kraushar Stanford University
Amalendu Krishna Indian Institute of Science
Marc Levine Universität Duisburg-Essen
Joshua Lieber Max Planck Institute for Mathematics
Paulo Lima-Filho Texas A&M University
Ma Luo University of Oxford
Lorenzo Mantovani Johannes Gutenberg-Universität Mainz
Daniel Marlowe University of Warwick
Keiho Matsumoto Tokyo Institute of Technology
Mona Merling University of Pennsylvania
Ben Moore University of Warwick
Kyle Ormsby Reed College
Owen Patashnick University of Bristol; King's College London
Rina Paucar Rojas Universidad Nacional de Ingeniería
Gregory Pearlstein Texas A&M University; Università di Pisa
Lucas Piessevaux Freie Universität Berlin
Parimala Raman Emory University
Jitendra Rathore Tata Institute of Fundamental Research
Charanya Ravi Universität Regensburg
Oliver Roendigs Universität Osnabrück
Herman Rohrbach Universität Duisburg-Essen
Vivek Sadhu Indian Institute of Science Education and Research (IISER)
Anand Sawant Tata Institute of Fundamental Research
Marco Schlichting University of Warwick
Anthony Scholl University of Cambridge
Nikita Semenov Ludwig-Maximilians-Universität München
Markus Spitzweck Universität Osnabrück
Vasudevan Srinivas Tata Institute of Fundamental Research
Ashar Tafhim Bergische Universität Wuppertal
Georg Tamme Johannes Gutenberg-Universität Mainz
Burt Totaro University of California, Los Angeles
Berkan Uze Bogaziçi Üniversitesi
Charles Weibel Rutgers, The State University of New Jersey
Ursula Whitcher American Mathematical Society
Ben Williams University of British Columbia
Thor Wittich Heinrich-Heine-Universität Düsseldorf
Ferdinando Zanchetta University of Bologna