Name: Motivic, Equivariant and Non-commutative Homotopy Theory
Start: 2020-07-06T13:00:00+02:00
End: 2020-07-17T19:00:00+02:00
Location: IHES

Motivic, Equivariant and Non-commutative Homotopy Theory

from Monday, July 6, 2020 (1:00 PM) to Friday, July 17, 2020 (7:00 PM)

Monday, July 6, 2020
1:00 PM Enumerative Geometry and Quadratic Forms (1/3) - Marc LEVINE (Universität Duisburg-Essen)
Enumerative Geometry and Quadratic Forms (1/3)
- Marc LEVINE (Universität Duisburg-Essen)
1:00 PM - 2:00 PM
Room: Marilyn and James Simons Conference Center Enumerative Geometry and Quadratic Forms: Euler characteristics and Euler classes
2:00 PM Discussions / Coffee Break
Discussions / Coffee Break
2:00 PM - 3:30 PM
Room: Marilyn and James Simons Conference Center
3:30 PM Algebraic K-theory and Trace Methods (1/3) - Teena GERHARDT (Michigan State University)
Algebraic K-theory and Trace Methods (1/3)
- Teena GERHARDT (Michigan State University)
3:30 PM - 4:30 PM
Room: Marilyn and James Simons Conference Center Algebraic K-theory is an invariant of rings and ring spectra which illustrates a fascinating interplay between algebra and topology. Defined using topological tools, this invariant has important applications to algebraic geometry, number theory, and geometric topology. One fruitful approach to studying algebraic K-theory is via trace maps, relating algebraic K-theory to (topological) Hochschild homology, and (topological) cyclic homology. In this mini-course I will introduce algebraic K-theory and related Hochschild invariants, and discuss recent advances in this area. Topics will include cyclotomic spectra, computations of the algebraic K-theory of rings, and equivariant analogues of Hochschild invariants.
4:30 PM Discussions / Coffee Break
Discussions / Coffee Break
4:30 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
6:00 PM Exodromy for ℓ-adic Sheaves (1/3) - Clark BARWICK (University of Edinburgh)
Exodromy for ℓ-adic Sheaves (1/3)
- Clark BARWICK (University of Edinburgh)
6:00 PM - 7:00 PM
Room: Marilyn and James Simons Conference Center In joint work with Saul Glasman and Peter Haine, we proved that the derived ∞-category of constructible ℓ-adic sheaves ’is’ the ∞-category of continuous functors from an explicitly defined 1-category to the ∞-category of perfect complexes over ℚℓ. In this series of talks, I want to offer some historical context for these ideas and to explain some of the techologies that go into both the statement and the proof. If time permits, I will also discuss newer work that aims to expand the scope of these results
Tuesday, July 7, 2020
1:00 PM Motives from the Non-commutative Point of View (1/3) - Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)
Motives from the Non-commutative Point of View (1/3)
- Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)
1:00 PM - 2:00 PM
Room: Marilyn and James Simons Conference Center Motives were initially conceived as a way to unify various cohomology theories that appear in algebraic geometry, and these can be roughly divided into two groups: theories of etale type, and theories of cristalline/de Rham type. The obvious unifying feature of all the theories is that they carry some version of a Chern character map from the algebraic K-theory, and there is a bunch of “motivic” conjectures claiming that in various contexts, this map can be refined to some “regulator map” that is not far from an isomorphism. Almost all of these conjectures are still wide open. One observation whose importance was not obvious at first is that K-theory is actually defined in a much larger generality: it makes sense for an associative but not necessarily commutative ring. From the modern of of view, the same should be true for all the theories of de Rham type, with differential forms replaced by Hochschild homology classes, and all the motivic conjectures should also generalize. One prominent example of this is the cyclotomic trace map of B¨okstedt–Hsiang–Madsen that serves as a non-commutative analog of the regulator in the p-adic setting. While the non-commutative conjectures are just as open as the commutative ones, one can still hope that they might be more tractable: after all, if something holds in a bigger generality, its potential proof by necessity should use much less, so it ought to be simpler. In addition to this, non-commutative setting allows for completely new methods. One such is the observation that Hochschild Homology is a two-variable theory: one can define homology groups of an algebra with coefficients in a bimodule. These groups come equipped with certain natural trace-like isomorphisms, and this already allowed one to prove several general comparison results.
2:00 PM Discussions / Coffee Break
Discussions / Coffee Break
2:00 PM - 3:30 PM
Room: Marilyn and James Simons Conference Center
3:30 PM Motivic and Equivariant Stable Homotopy Groups (1/3) - Daniel ISAKSEN (Wayne State University)
Motivic and Equivariant Stable Homotopy Groups (1/3)
- Daniel ISAKSEN (Wayne State University)
3:30 PM - 4:30 PM
Room: Marilyn and James Simons Conference Center I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence and the effective spectral sequence are the key tools. The analysis of these spectral sequences break into three main steps: (1) algebraically compute the E2-page; (2) analyze differentials; (3) resolve hidden extensions. I will demonstrate a variety of techniques for each of these steps. I will make precise the idea that ℂ-motivic stable homotopy theory is a deformation of classical stable homotopy theory. I will discuss some future prospects for homotopical deformation theory in general.
4:30 PM Discussions / Coffee Break
Discussions / Coffee Break
4:30 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
6:00 PM Algebraic K-theory and Trace Methods (2/3) - Teena GERHARDT (Michigan State University)
Algebraic K-theory and Trace Methods (2/3)
- Teena GERHARDT (Michigan State University)
6:00 PM - 7:00 PM
Room: Marilyn and James Simons Conference Center Algebraic K-theory is an invariant of rings and ring spectra which illustrates a fascinating interplay between algebra and topology. Defined using topological tools, this invariant has important applications to algebraic geometry, number theory, and geometric topology. One fruitful approach to studying algebraic K-theory is via trace maps, relating algebraic K-theory to (topological) Hochschild homology, and (topological) cyclic homology. In this mini-course I will introduce algebraic K-theory and related Hochschild invariants, and discuss recent advances in this area. Topics will include cyclotomic spectra, computations of the algebraic K-theory of rings, and equivariant analogues of Hochschild invariants.
Wednesday, July 8, 2020
4:30 PM Exodromy for ℓ-adic Sheaves (2/3) - Clark BARWICK (University of Edinburgh)
Exodromy for ℓ-adic Sheaves (2/3)
- Clark BARWICK (University of Edinburgh)
4:30 PM - 5:30 PM
Room: Marilyn and James Simons Conference Center In joint work with Saul Glasman and Peter Haine, we proved that the derived ∞-category of constructible ℓ-adic sheaves ’is’ the ∞-category of continuous functors from an explicitly defined 1-category to the ∞-category of perfect complexes over ℚℓ. In this series of talks, I want to offer some historical context for these ideas and to explain some of the techologies that go into both the statement and the proof. If time permits, I will also discuss newer work that aims to expand the scope of these results.
5:30 PM Discussions / Coffee Break
Discussions / Coffee Break
5:30 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
6:00 PM Enumerative Geometry and Quadratic Forms (2/3) - Marc LEVINE (Universität Duisburg-Essen)
Enumerative Geometry and Quadratic Forms (2/3)
- Marc LEVINE (Universität Duisburg-Essen)
6:00 PM - 7:00 PM
Room: Marilyn and James Simons Conference Center Computations of Euler Characteristics and Euler Classes
Thursday, July 9, 2020
1:00 PM Motives from the Non-commutative Point of View (2/3) - Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)
Motives from the Non-commutative Point of View (2/3)
- Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)
1:00 PM - 2:00 PM
Room: Marilyn and James Simons Conference Center Motives were initially conceived as a way to unify various cohomology theories that appear in algebraic geometry, and these can be roughly divided into two groups: theories of etale type, and theories of cristalline/de Rham type. The obvious unifying feature of all the theories is that they carry some version of a Chern character map from the algebraic K-theory, and there is a bunch of “motivic” conjectures claiming that in various contexts, this map can be refined to some “regulator map” that is not far from an isomorphism. Almost all of these conjectures are still wide open. One observation whose importance was not obvious at first is that K-theory is actually defined in a much larger generality: it makes sense for an associative but not necessarily commutative ring. From the modern of of view, the same should be true for all the theories of de Rham type, with differential forms replaced by Hochschild homology classes, and all the motivic conjectures should also generalize. One prominent example of this is the cyclotomic trace map of B¨okstedt–Hsiang–Madsen that serves as a non-commutative analog of the regulator in the p-adic setting. While the non-commutative conjectures are just as open as the commutative ones, one can still hope that they might be more tractable: after all, if something holds in a bigger generality, its potential proof by necessity should use much less, so it ought to be simpler. In addition to this, non-commutative setting allows for completely new methods. One such is the observation that Hochschild Homology is a two-variable theory: one can define homology groups of an algebra with coefficients in a bimodule. These groups come equipped with certain natural trace-like isomorphisms, and this already allowed one to prove several general comparison results.
2:00 PM Discussions / Coffee Break
Discussions / Coffee Break
2:00 PM - 3:30 PM
Room: Marilyn and James Simons Conference Center
3:30 PM Algebraic K-theory and Trace Methods (3/3) - Teena GERHARDT (Michigan State University)
Algebraic K-theory and Trace Methods (3/3)
- Teena GERHARDT (Michigan State University)
3:30 PM - 4:30 PM
Room: Marilyn and James Simons Conference Center Algebraic K-theory is an invariant of rings and ring spectra which illustrates a fascinating interplay between algebra and topology. Defined using topological tools, this invariant has important applications to algebraic geometry, number theory, and geometric topology. One fruitful approach to studying algebraic K-theory is via trace maps, relating algebraic K-theory to (topological) Hochschild homology, and (topological) cyclic homology. In this mini-course I will introduce algebraic K-theory and related Hochschild invariants, and discuss recent advances in this area. Topics will include cyclotomic spectra, computations of the algebraic K-theory of rings, and equivariant analogues of Hochschild invariants.
4:30 PM Discussions / Coffee Break
Discussions / Coffee Break
4:30 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
6:00 PM Motivic and Equivariant Stable Homotopy Groups (2/3) - Daniel ISAKSEN (Wayne State University)
Motivic and Equivariant Stable Homotopy Groups (2/3)
- Daniel ISAKSEN (Wayne State University)
6:00 PM - 7:00 PM
Room: Marilyn and James Simons Conference Center I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence and the effective spectral sequence are the key tools. The analysis of these spectral sequences break into three main steps: (1) algebraically compute the E2-page; (2) analyze differentials; (3) resolve hidden extensions. I will demonstrate a variety of techniques for each of these steps. I will make precise the idea that ℂ-motivic stable homotopy theory is a deformation of classical stable homotopy theory. I will discuss some future prospects for homotopical deformation theory in general.
Friday, July 10, 2020
1:00 PM Enumerative Geometry and Quadratic Forms (3/3) - Marc LEVINE (Universität Duisburg-Essen)
Enumerative Geometry and Quadratic Forms (3/3)
- Marc LEVINE (Universität Duisburg-Essen)
1:00 PM - 2:00 PM
Room: Marilyn and James Simons Conference Center Motivic Welschinger invariants
2:00 PM Discussions / Coffee Break
Discussions / Coffee Break
2:00 PM - 3:30 PM
Room: Marilyn and James Simons Conference Center
3:30 PM Motivic and Equivariant Stable Homotopy Groups (3/3) - Daniel ISAKSEN (Wayne State University)
Motivic and Equivariant Stable Homotopy Groups (3/3)
- Daniel ISAKSEN (Wayne State University)
3:30 PM - 4:30 PM
Room: Marilyn and James Simons Conference Center I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence and the effective spectral sequence are the key tools. The analysis of these spectral sequences break into three main steps: (1) algebraically compute the E2-page; (2) analyze differentials; (3) resolve hidden extensions. I will demonstrate a variety of techniques for each of these steps. I will make precise the idea that ℂ-motivic stable homotopy theory is a deformation of classical stable homotopy theory. I will discuss some future prospects for homotopical deformation theory in general.
4:30 PM Discussions / Coffee Break
Discussions / Coffee Break
4:30 PM - 6:00 PM
Room: Marilyn and James Simons Conference Center
6:00 PM Exodromy for ℓ-adic Sheaves (3/3) - Clark BARWICK (University of Edinburgh)
Exodromy for ℓ-adic Sheaves (3/3)
- Clark BARWICK (University of Edinburgh)
6:00 PM - 7:00 PM
Room: Marilyn and James Simons Conference Center In joint work with Saul Glasman and Peter Haine, we proved that the derived ∞-category of constructible ℓ-adic sheaves ’is’ the ∞-category of continuous functors from an explicitly defined 1-category to the ∞-category of perfect complexes over ℚℓ. In this series of talks, I want to offer some historical context for these ideas and to explain some of the techologies that go into both the statement and the proof. If time permits, I will also discuss newer work that aims to expand the scope of these results.
Saturday, July 11, 2020
Sunday, July 12, 2020