Motivic, Equivariant and Non-commutative Homotopy Theory

6 juil. 2020, 13:00 → 17 juil. 2020, 19:00 Europe/Paris

Marilyn and James Simons Conference Center (IHES)

Watch the videos on Youtube.

----------------------------------------- IMPORTANT INFORMATION ------------------------------------------

>> Due to the Covid-19 pandemic, the 2020 Summer School has been organised through zoom. Mini-courses and talks have been recorded and downloaded on the IHES YouTube Channel.

Organising Committee: Aravind Asok (University of Southern California), Frédéric Déglise (CNRS Dijon), Grigory Garkusha (Swansea University), Paul Arne Østvær (University of Oslo)

Scientific Committee: Eric M. Friedlander (University of Southern California), Haynes R. Miller (MIT Department of Mathematics), Bertrand Toën (CNRS Toulouse)

The IHES 2020 Summer School on "Motivic, Equivariant and Non-commutative Homotopy Theory" will be held from 6 to 17 July 2020.

This school is open to everybody but intended primarily for young participants, including PhD students and postdoctoral fellows.

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The IHES 2020 Summer School will focus on topics in motivic and equivariant homotopy theory, and non-commutative geometry.

These three subjects are currently experiencing a phase of intense growth and development:

• long-standing central conjectures have been solved;
• existing theories are being perfected;
• many new foundational developments are being made on this basis.

It's expected that these developments will spur many further advances and interactions in the near future.

The lecture series and research talks at the IHES Summer School will focus on presenting the latest developments in topics related to categories of motives, calculational and foundational aspects of motivic and equivariant homotopy theory, and the generalisations of these tools and techniques in the setting of non-commutative geometry.

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INVITED SPEAKERS

The Summer School will feature mini-courses by

*  Clark Barwick (University of Edinburgh)
*  Teena Gerhardt (Michigan State University)
*  Daniel Isaksen (Wayne State University)
*  Dmitry Kaledin (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)
*  Marc Levine (Universität Duisburg-Essen)
*  Ivan Panin (St. Petersburg Department of Mathematics)
*  Goncalo Tabuada (MIT/University of Warwick)

as well as research talks by

*  Federico Binda (University of Milan)
*  Tom Bachmann (MIT)
*  Mike Hill (UCLA)
*  Geoffroy Horel (University Paris 13)
*  Alexander Neshitov (Western University)
*  Angélica M. Osorno (Reed College)
*  Marco Robalo (Institut de Mathématiques de Jussieu)
*  Kirsten Wickelgren (Duke University)

 

Both the lecture series and research talks will focus on presenting the latest developments in topics related to categories of motives, calculational and foundational aspects of motivic and equivariant homotopy theory, and the generalisations of these tools and techniques in the setting of non-commutative geometry.

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This is an IHES Summer School organised in partnership with the Research Council of Norway, the Fondation Mathématique Jacques Hadamard and the ANR, and the support of the Société Générale and the ERC.

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Poster IHES Summer School 2020

lundi 6 juillet

1 Enumerative Geometry and Quadratic Forms (1/3)

Enumerative Geometry and Quadratic Forms: Euler characteristics and Euler classes

Orateur: Prof. Marc LEVINE (Universität Duisburg-Essen)

2 Discussions / Coffee Break

3 Algebraic K-theory and Trace Methods (1/3)

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.

Orateur: Prof. Teena GERHARDT (Michigan State University)

4 Discussions / Coffee Break

5 Exodromy for ℓ-adic Sheaves (1/3)

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

Orateur: Prof. Clark BARWICK (University of Edinburgh)

mardi 7 juillet

6 Motives from the Non-commutative Point of View (1/3)

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.

Orateur: Prof. Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)

7 Discussions / Coffee Break

8 Motivic and Equivariant Stable Homotopy Groups (1/3)

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.

Orateur: Prof. Daniel ISAKSEN (Wayne State University)

9 Discussions / Coffee Break

10 Algebraic K-theory and Trace Methods (2/3)

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.

Orateur: Prof. Teena GERHARDT (Michigan State University)

mercredi 8 juillet

11 Exodromy for ℓ-adic Sheaves (2/3)

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.

Orateur: Prof. Clark BARWICK (University of Edinburgh)

12 Discussions / Coffee Break

13 Enumerative Geometry and Quadratic Forms (2/3)

Computations of Euler Characteristics and Euler Classes

Orateur: Prof. Marc LEVINE (Universität Duisburg-Essen)

jeudi 9 juillet

14 Motives from the Non-commutative Point of View (2/3)

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.

Orateur: Prof. Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)

15 Discussions / Coffee Break

16 Algebraic K-theory and Trace Methods (3/3)

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.

Orateur: Prof. Teena GERHARDT (Michigan State University)

17 Discussions / Coffee Break

18 Motivic and Equivariant Stable Homotopy Groups (2/3)

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.

Orateur: Prof. Daniel ISAKSEN (Wayne State University)

vendredi 10 juillet

19 Enumerative Geometry and Quadratic Forms (3/3)

Motivic Welschinger invariants

Orateur: Prof. Marc LEVINE (Universität Duisburg-Essen)

20 Discussions / Coffee Break

21 Motivic and Equivariant Stable Homotopy Groups (3/3)

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.

Orateur: Prof. Daniel ISAKSEN (Wayne State University)

22 Discussions / Coffee Break

23 Exodromy for ℓ-adic Sheaves (3/3)

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.

Orateur: Prof. Clark BARWICK (University of Edinburgh)

lundi 13 juillet

24 Motives from the Non-commutative Point of View (3/3)

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.

Orateur: Prof. Dmitry KALEDIN (Steklov Mathematical Inst. & National Research Univ. Higher School of Economics)

25 Discussions / Coffee Break

26 Noncommutative Counterparts of Celebrated Conjectures (1/3)

Some celebrated conjectures of Beilinson, Grothendieck, Kimura, Tate, Voevodsky, Weil, and others, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of (the majority of) these conjectures remains illusive. The aim of this course, prepared for a broad audience, is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in some new cases.

Orateur: Prof. Gonçalo TABUADA (MIT Department of Mathematics)

27 Discussions / Coffee Break

28 Motivic Realizations of Singularity Categories

In this lecture, we will explain the connection between the (motivic) theory of vanishing cycles and the construction of motivic realizations of singularity categories. We review the results obtained in collaboration with Blanc-Toen-Vezzosi and discuss some of the recent progresses in the field.

Orateur: Prof. Marco Robalo (IMJ-PRG)

mardi 14 juillet

29 A Local Construction of Stable Motivic Homotopy Theory (1/3)

V. Voevodsky [6] invented the category of framed correspondences with the hope to give a new construction of stable motivic homotopy theory SH(k) which will be more friendly for computational purposes. Joint with G. Garkusha we used framed correspondences to develop the theory of framed motives in [4]. This theory led us in [5] to a genuinely local construction of SH(k). In particular, we get rid of motivic equivalences completely.
In my lectures I will recall the definition of framed correspondences and describe the genuinely local model for SH(k) (assuming that the base field k is infinite and perfect). I will also discuss several applications. Let Fr(Y,X) be the pointed set of stable framed correspondences between smooth algebraic varieties Y and X. For the first two applications I choose k = ℂ for simplicity. For further two applications k is any infinite and perfect field.
(1) The simplicial space Fr(𝚫alg,S^1) has the homotopy type of the topological space Ω∞Σ∞(S^1_top). So the topological space Ω^∞S1Σ^∞_S1(S^1_top) is recovered as the simplicial set Fr(𝚫alg,S^1), which is described in terms of algebraic varieties only. This is one of the computational miracles of framed correspondences.
(2) The assignment X ↦ π(Fr(𝚫alg,X⨂S^1)) is a homology theory on complex algebraic varieties. Moreover, this homology theory regarded with ℤ/n-coefficients coincides with the stable homotopies X ↦ π ^S_(X+^S^1_top;ℤ/n) with ℤ/n-coefficients.
The latter result is an extension of the celebrated Suslin–Voevodsky theorem on motivic homology of weight zero to the stable motivic homotopy context.
(3) Another application of the theory is as follows. It turns out that π^s_0,0(X+) = H0(ℤF(𝚫,X)), where (ℤF(𝚫,X)) is the chain complex of stable linear framed correspondences introduced in [4]. For X = G_m^^n this homology group was computed by A. Neshitov as the nth Milnor–Witt group K_n^MW (k) of the base field k recovering the celebrated theorem of Morel.
(4) As a consequence of the theory of framed motives, the canonical morphism of motivic spaces can : C_Fr(X) → Ω^∞ℙ^1 Σ^∞_ℙ^1 (X+) is Nisnich locally a group completion for any smooth simplicial scheme X. In particular, if CFr(X) is Nisnevich locally connected, then the morphism can is a Nisnevich local weak equivalence. Thus in this case C_Fr(X) is an infinite motivic loop space and π_n(C_FR(X)(K)) = π^A1_n,0 (Σ^∞_ℙ^1 (X+))(K).

In my lectures I will adhere to the following references:
[1] A. Ananyevskiy, G. Garkusha, I. Panin, Cancellation theorem for framed motives of algebraic varieties, arXiv:1601.06642
[2] G. Garkusha, A. Neshitov, I. Panin, Framed motives of relative motivic spheres, arXiv:1604.02732v3.
[3] G. Garkusha, I. Panin, Homotopy invariant presheaves with framed transfers, Cambridge J. Math. 8(1) (2020), 1-94.
[4] G. Garkusha, I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), J. Amer. Math. Soc., to appear.
[5] G. Garkusha, I. Panin, The triangulated categories of framed bispectra and framed motives, arXiv:1809.08006.
[6] V. Voevodsky, Notes on framed correspondences, unpublished, 2001, www.math.ias.edu/vladimir/publications

Orateur: Prof. Ivan PANIN (St. Petersburg Department of Steklov Institute of Mathematics)

30 Discussions / Coffee Break

31 Pullbacks for the Rost-Schmid Complex

Let 𝑘 be a perfect field and 𝑀 a strictly homotopy invariant sheaf of abelian groups on Sm_𝑘. The cousin complex can be used to compute the cohomology of a smooth variety 𝑋 over 𝑘 with coefficients in 𝑀. However, if 𝑋 → 𝑌 is a morphism of smooth varieties, there is not in general an induced map on cousin complexes, so computing pullbacks of cohomology classes is difficult. In this talk I will explain how such pullbacks may nonetheless be computed, at least up to choosing a good enough cycle representing the cohomology class (which is always possible in principle, but may be difficult in practice). Time permitting, I will mention applications to the 𝔾_𝑚-stabilization conjecture (which was formulated jointly with Maria Yakerson)

Orateur: Prof. Tom BACHMANN (MIT)

32 Discussions / Coffee Break

33 Fibrant Resolutions of Motivic Thom Spectra

This is a joint work with G. Garkusha. In the talk I will discuss the construction of fibrant replacements for spectra consisting of Thom spaces (suspension spectra of varieties and algebraic cobordism 𝑀𝐺𝐿 being the motivating examples) that uses the theory of framed correspondences. As a consequence we get a description of the infinite loop space of 𝑀𝐺𝐿 in terms of Hilbert schemes

Orateur: Prof. Alexander NESHITOV (Western University)

mercredi 15 juillet

34 A Local Construction of Stable Motivic Homotopy Theory (2/3)

V. Voevodsky [6] invented the category of framed correspondences with the hope to give a new construction of stable motivic homotopy theory SH(k) which will be more friendly for computational purposes. Joint with G. Garkusha we used framed correspondences to develop the theory of framed motives in [4]. This theory led us in [5] to a genuinely local construction of SH(k). In particular, we get rid of motivic equivalences completely.
In my lectures I will recall the definition of framed correspondences and describe the genuinely local model for SH(k) (assuming that the base field k is infinite and perfect). I will also discuss several applications. Let Fr(Y,X) be the pointed set of stable framed correspondences between smooth algebraic varieties Y and X. For the first two applications I choose k = ℂ for simplicity. For further two applications k is any infinite and perfect field.
(1) The simplicial space Fr(𝚫alg,S^1) has the homotopy type of the topological space Ω∞Σ∞(S^1_top). So the topological space Ω^∞S1Σ^∞_S1(S^1_top) is recovered as the simplicial set Fr(𝚫alg,S^1), which is described in terms of algebraic varieties only. This is one of the computational miracles of framed correspondences.
(2) The assignment X ↦ π(Fr(𝚫alg,X⨂S^1)) is a homology theory on complex algebraic varieties. Moreover, this homology theory regarded with ℤ/n-coefficients coincides with the stable homotopies X ↦ π ^S_(X+^S^1_top;ℤ/n) with ℤ/n-coefficients.
The latter result is an extension of the celebrated Suslin–Voevodsky theorem on motivic homology of weight zero to the stable motivic homotopy context.
(3) Another application of the theory is as follows. It turns out that π^s_0,0(X+) = H0(ℤF(𝚫,X)), where (ℤF(𝚫,X)) is the chain complex of stable linear framed correspondences introduced in [4]. For X = G_m^^n this homology group was computed by A. Neshitov as the nth Milnor–Witt group K_n^MW (k) of the base field k recovering the celebrated theorem of Morel.
(4) As a consequence of the theory of framed motives, the canonical morphism of motivic spaces can : C_Fr(X) → Ω^∞ℙ^1 Σ^∞_ℙ^1 (X+) is Nisnich locally a group completion for any smooth simplicial scheme X. In particular, if CFr(X) is Nisnevich locally connected, then the morphism can is a Nisnevich local weak equivalence. Thus in this case C_Fr(X) is an infinite motivic loop space and π_n(C_FR(X)(K)) = π^A1_n,0 (Σ^∞_ℙ^1 (X+))(K).

In my lectures I will adhere to the following references:
[1] A. Ananyevskiy, G. Garkusha, I. Panin, Cancellation theorem for framed motives of algebraic varieties, arXiv:1601.06642
[2] G. Garkusha, A. Neshitov, I. Panin, Framed motives of relative motivic spheres, arXiv:1604.02732v3.
[3] G. Garkusha, I. Panin, Homotopy invariant presheaves with framed transfers, Cambridge J. Math. 8(1) (2020), 1-94.
[4] G. Garkusha, I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), J. Amer. Math. Soc., to appear.
[5] G. Garkusha, I. Panin, The triangulated categories of framed bispectra and framed motives, arXiv:1809.08006.
[6] V. Voevodsky, Notes on framed correspondences, unpublished, 2001, www.math.ias.edu/vladimir/publications

Orateur: Prof. Ivan PANIN (St. Petersburg Department of Steklov Institute of Mathematics)

35 Discussions / Coffee Break

36 Noncommutative Counterparts of Celebrated Conjectures (2/3)

Some celebrated conjectures of Beilinson, Grothendieck, Kimura, Tate, Voevodsky, Weil, and others, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of (the majority of) these conjectures remains illusive. The aim of this course, prepared for a broad audience, is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in some new cases.

Orateur: Prof. Gonçalo Tabuada (MIT Department of Mathematics)

37 Discussions / Coffee Break

38 Equivariant Infinite Loop Space Machines

An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80’s Lewis–May–Steinberger and Shimakawa developed generalizations of the operadic approach and the G-space approach respectively. In this talk I will report on joint work with Bert Guillou, Peter May and Mona Merling on adapting these machines to work multiplicatively and on understanding their categorical input

Orateur: Prof. Angélica M. Osorno (Reed College)

jeudi 16 juillet

39 Knots and Motives

The pure braid group is the fundamental group of the space of configurations of points in the complex plane. This topological space is the Betti realization of a scheme defined over the integers. It follows, by work initiated by Deligne and Goncharov, that the pronilpotent completion of the pure braid group is a motive over the integers (what this means precisely is that the Hopf algebra of functions on that group can be promoted to a Hopf algebra in an abelian category of motives over the integers). I will explain a partly conjectural extension of that story from braids to knots. The replacement of the lower central series of the pure braid group is the so-called Vassiliev filtration on knots. The proposed strategy to construct the desired motivic structure relies on the technology of manifold calculus of Goodwillie and Weiss.

Orateur: Prof. Geoffroy HOREL (University Paris 13)

40 Discussions / Coffee Break

41 Noncommutative Counterparts of Celebrated Conjectures (3/3)

Some celebrated conjectures of Beilinson, Grothendieck, Kimura, Tate, Voevodsky, Weil, and others, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of (the majority of) these conjectures remains illusive. The aim of this course, prepared for a broad audience, is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in some new cases.

Orateur: Prof. Gonçalo TABUADA (MIT Department of Mathematics)

42 Discussions / Coffee Break

43 Integrality Results for 𝔸^1-Euler Numbers and Arithmetic Counts of Linear Subspaces of Complete Intersections

𝔸^1-Euler numbers can be constructed with Hochschild homology, self-duality of Koszul complexes, pushforwards in 𝑆𝐿_𝑐 oriented cohomology theories, and sums of local degrees. We show an integrality result for 𝔸^1-Euler numbers and apply this to the enumeration of 𝑑-planes in complete intersections. Classically such counts are valid over the complex numbers and sometimes extended to the real numbers. 𝔸^1-homotopy theory allows one to perform counts over arbitrary fields, and records information about the arithmetic and geometry of the solutions with bilinear forms. For example, it then follows from work of Finashin–Kharlamov that there are 160;839⟨1⟩+160;650⟨-1⟩ 3-planes in any 7-dimensional cubic hypersurface when these 3-planes are counted with an appropriate weight. This is joint work with Tom Bachmann.

Orateur: Prof. Kirsten WICKELGREN (Duke University)

vendredi 17 juillet

44 A Local Construction of Stable Motivic Homotopy Theory (3/3)

V. Voevodsky [6] invented the category of framed correspondences with the hope to give a new construction of stable motivic homotopy theory SH(k) which will be more friendly for computational purposes. Joint with G. Garkusha we used framed correspondences to develop the theory of framed motives in [4]. This theory led us in [5] to a genuinely local construction of SH(k). In particular, we get rid of motivic equivalences completely.
In my lectures I will recall the definition of framed correspondences and describe the genuinely local model for SH(k) (assuming that the base field k is infinite and perfect). I will also discuss several applications. Let Fr(Y,X) be the pointed set of stable framed correspondences between smooth algebraic varieties Y and X. For the first two applications I choose k = ℂ for simplicity. For further two applications k is any infinite and perfect field.
(1) The simplicial space Fr(𝚫alg,S^1) has the homotopy type of the topological space Ω∞Σ∞(S^1_top). So the topological space Ω^∞S1Σ^∞S1(S^1_top) is recovered as the simplicial set Fr(𝚫alg,S^1), which is described in terms of algebraic varieties only. This is one of the computational miracles of framed correspondences.
(2) The assignment X ↦ π(Fr(𝚫alg,X⨂S^1)) is a homology theory on complex algebraic varieties. Moreover, this homology theory regarded with ℤ/n-coefficients coincides with the stable homotopies X ↦ π ^S(X+^S^1_top;ℤ/n) with ℤ/n-coefficients.
The latter result is an extension of the celebrated Suslin–Voevodsky theorem on motivic homology of weight zero to the stable motivic homotopy context.
(3) Another application of the theory is as follows. It turns out that π^s_0,0(X+) = H0(ℤF(𝚫,X)), where (ℤF(𝚫,X)) is the chain complex of stable linear framed correspondences introduced in [4]. For X = G_m^^n this homology group was computed by A. Neshitov as the nth Milnor–Witt group K_n^MW (k) of the base field k recovering the celebrated theorem of Morel.
(4) As a consequence of the theory of framed motives, the canonical morphism of motivic spaces can : C_Fr(X) → Ω^∞ℙ^1 Σ^∞_ℙ^1 (X+) is Nisnich locally a group completion for any smooth simplicial scheme X. In particular, if CFr(X) is Nisnevich locally connected, then the morphism can is a Nisnevich local weak equivalence. Thus in this case C_Fr(X) is an infinite motivic loop space and π_n(C_FR(X)(K)) = π^A1_n,0 (Σ^∞_ℙ^1 (X+))(K).

In my lectures I will adhere to the following references:
[1] A. Ananyevskiy, G. Garkusha, I. Panin, Cancellation theorem for framed motives of algebraic varieties, arXiv:1601.06642
[2] G. Garkusha, A. Neshitov, I. Panin, Framed motives of relative motivic spheres, arXiv:1604.02732v3.
[3] G. Garkusha, I. Panin, Homotopy invariant presheaves with framed transfers, Cambridge J. Math. 8(1) (2020), 1-94.
[4] G. Garkusha, I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), J. Amer. Math. Soc., to appear.
[5] G. Garkusha, I. Panin, The triangulated categories of framed bispectra and framed motives, arXiv:1809.08006.
[6] V. Voevodsky, Notes on framed correspondences, unpublished, 2001, www.math.ias.edu/vladimir/publications

Orateur: Prof. Ivan PANIN (St. Petersburg Department of Steklov Institute of Mathematics)

45 Discussions / Coffee Break

46 Triangulated Categories of Log Motives over a Field

In this talk I will sketch the construction and highlight the main properties of a new motivic category for logarithmic schemes, log smooth over a ground field k (without log structure). This construction is based on a new Grothendieck topology (called the “dividing topology”) and on the principle that homotopies should be parametrised by the affine line with compactifying log structure. The resulting category logDM shares many of the fundamental properties of Voevodsky’s DM, that can be faithfully embedded inside it, and can be used to represent cohomology theories that are not A^1-homotopy invariant (like Hodge cohomology or Hodge-Witt cohomology). If time permits, we will discuss some conjectures relating the étale version of our category with integral coefficients with the Milne-Ramachandran category of integral étale motivic complexes. This is a joint work with D. Park (Zurich) and P.-A.Østvær (Oslo).

Orateur: Prof. Federico BINDA (University of Milan)

47 Discussions / Coffee Break

48 Real and Hyperreal Equivariant and Motivic Computations

Foundational work of Hu—Kriz and Dugger showed that for Real spectra, we can often compute as easily as non-equivariantly. The general equivariant slice filtration was developed to show how this philosophy extends from $C_2$-equivariant homotopy to larger cyclic $2$-groups, and this has some fantastic applications to chromatic homotopy. This talk will showcase how one can carry out computations, and some of the tools that make these computations easier.

The natural source for Real spectra is the complex points of motivic spectra over $\mathbb R$, and there is a more initial, parallel story here. I will discuss some of how the equivariant shadow can show us structure in the motivic case as well.

Orateur: Prof. Mike HILL (UCLA)