Monday, 15 May 2023

Topic 1: The local cohomology theorem for group cohomology H^*(BG) by algebraic methods

[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

Introduce duality (Poincare duality, and what you might do for more general graded rings). Introduce the example of the group cohomology ring. In fact (Benson-Carlson) there is a functional equation for the Hilbert series (in the Cohen Macaulay case) and this motivates the duality statement. Reminder of local cohomology and its basic properties.

Forming multiperiodic resolutions in topology and algebra, directly in favourable cases or via support in general. The proof of the local cohomology theorem and the deduction of the functional equation. As a second example (from work with R.R.Bruner), I’d like to mention the bigraded cohomology H^{**}(A) for a finite dimensional Hopf sub algebra of the Steenrod algebra, where the Adams-Margolis criterion is the way the support condition is realised.

Topic 2: Gorenstein duality for C^*(BG)

[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

Differential graded algebras and spectra (description of formal properties rather than a fulll definition and construction). The homotopical Gorenstein condition for a map R—>k. Morita theory (some of this will be covered in Iyengar’s talks). Matlis lifts. Gorenstein duality. Deducing Gorenstein duality from the Gorenstein condition in the orientable case. Gorenstein ascent. C^*(BG)—>k is Gorenstein. Deducing the local cohomology theorem. The ubiquity of other examples and perhaps the connection to Serre duality in derived algebraic geometry.

The following are *not* prerequisites, but the material in the lectures is based on these papers.

A general reference:

J.P.C.Greenlees

``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

Other background:

J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

I will discuss various aspects of the derived category, and associated triangulated categories of interest in commutative algebra.

Resources:

Some familiarity with commutative algebra, and triangulated categories.

Staring from an introduction to p-localisation of spaces and subsequently talking about localisation of spaces and spectra with respect to Homology Theories, we shall have a brief discussion on localisation with respect to a map and it's dual concept cellularisation.

In this talk, we will briefly cover a survey of the representation theory of finite groups, mostly in positive characteristic. We will be discussing some properties of the group algebra and will focus on irreducible and indecomposable modules over the group algebra.

Tuesday, 16 May 2023

The goal of the talks is to introduce some of the foundations of chromatic stable homotopy theory and to explain how these methods are used to study the structure of the stable homotopy category. If time permits, we will look at some more recent results at the end.

Topics will include:

- stable model categories

- left Bousfield localisation,

- p-localisation, p-completion, rationalisation, K-theory,

- E(n) and K(n): construction and properties,

- nilpotence, periodicity, chromatic convergence,

- thick subcategories and the Balmer spectrum.

D. Barnes, C. Roitzheim: Foundations of Stable Homotopy Theory (Cambridge University Press)

I will discuss classifications of thick and localising subcategories for categories of modular representations of finite groups. This requires a certain amount of machinery, for instance a theory of support for triangulated categories. In addition, I plan to talk about various dualities (local and global) which reflect a Gorenstein property of the category of finite group representations.

Resources:

D. J. Benson, S. B. Iyengar, and H. Krause, Representations of finite groups: Local cohomology and support, Oberwolfach Seminars 43, Birkhäuser Verlag, 2012, 111 pp.

H. Krause, Homological Theory of Representations, Cambridge Studies in Advanced Mathematics 195, Cambridge Univers. Press, 2021, 482 pp.

We will discuss the main statements of Grothendieck duality and then briefly explain how certain concrete residue formulas and trace maps arise in the abstract theory.

Resource:

1. Residues and Duality, LNM 20, Springer Verlag, by R Hartshorne.

2. Notes on derived categories and Grothendieck duality, LNM 1960, Springer Verlag, by J Lipman.

3. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, JAMS vol 9, by A Neeman.

In this preparatory talk, we will discuss basics of triangulated and derived categories. We will show that the homotopy category and the derived category of an abelian category are triangulated.

Wednesday, 17 May 2023

I will discuss various aspects of the derived category, and associated triangulated categories of interest in commutative algebra.

Resources:

Some familiarity with commutative algebra, and triangulated categories.

Topic 1: The local cohomology theorem for group cohomology H^*(BG) by algebraic methods

[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

Introduce duality (Poincare duality, and what you might do for more general graded rings). Introduce the example of the group cohomology ring. In fact (Benson-Carlson) there is a functional equation for the Hilbert series (in the Cohen Macaulay case) and this motivates the duality statement. Reminder of local cohomology and its basic properties.

Forming multiperiodic resolutions in topology and algebra, directly in favourable cases or via support in general. The proof of the local cohomology theorem and the deduction of the functional equation. As a second example (from work with R.R.Bruner), I’d like to mention the bigraded cohomology H^{**}(A) for a finite dimensional Hopf sub algebra of the Steenrod algebra, where the Adams-Margolis criterion is the way the support condition is realised.

Topic 2: Gorenstein duality for C^*(BG)

[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

Differential graded algebras and spectra (description of formal properties rather than a fulll definition and construction). The homotopical Gorenstein condition for a map R—>k. Morita theory (some of this will be covered in Iyengar’s talks). Matlis lifts. Gorenstein duality. Deducing Gorenstein duality from the Gorenstein condition in the orientable case. Gorenstein ascent. C^*(BG)—>k is Gorenstein. Deducing the local cohomology theorem. The ubiquity of other examples and perhaps the connection to Serre duality in derived algebraic geometry.

The following are *not* prerequisites, but the material in the lectures is based on these papers.

A general reference:

J.P.C.Greenlees

``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

Other background:

J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

The goal of the talks is to introduce some of the foundations of chromatic stable homotopy theory and to explain how these methods are used to study the structure of the stable homotopy category. If time permits, we will look at some more recent results at the end.

Topics will include:

- stable model categories

- left Bousfield localisation,

- p-localisation, p-completion, rationalisation, K-theory,

- E(n) and K(n): construction and properties,

- nilpotence, periodicity, chromatic convergence,

- thick subcategories and the Balmer spectrum.

D. Barnes, C. Roitzheim: Foundations of Stable Homotopy Theory (Cambridge University Press)

Thursday, 18 May 2023

I will discuss classifications of thick and localising subcategories for categories of modular representations of finite groups. This requires a certain amount of machinery, for instance a theory of support for triangulated categories. In addition, I plan to talk about various dualities (local and global) which reflect a Gorenstein property of the category of finite group representations.

Resources:

D. J. Benson, S. B. Iyengar, and H. Krause, Representations of finite groups: Local cohomology and support, Oberwolfach Seminars 43, Birkhäuser Verlag, 2012, 111 pp.

H. Krause, Homological Theory of Representations, Cambridge Studies in Advanced Mathematics 195, Cambridge Univers. Press, 2021, 482 pp.

Topic 1: The local cohomology theorem for group cohomology H^*(BG) by algebraic methods

[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

Introduce duality (Poincare duality, and what you might do for more general graded rings). Introduce the example of the group cohomology ring. In fact (Benson-Carlson) there is a functional equation for the Hilbert series (in the Cohen Macaulay case) and this motivates the duality statement. Reminder of local cohomology and its basic properties.

Forming multiperiodic resolutions in topology and algebra, directly in favourable cases or via support in general. The proof of the local cohomology theorem and the deduction of the functional equation. As a second example (from work with R.R.Bruner), I’d like to mention the bigraded cohomology H^{**}(A) for a finite dimensional Hopf sub algebra of the Steenrod algebra, where the Adams-Margolis criterion is the way the support condition is realised.

Topic 2: Gorenstein duality for C^*(BG)

[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

Differential graded algebras and spectra (description of formal properties rather than a fulll definition and construction). The homotopical Gorenstein condition for a map R—>k. Morita theory (some of this will be covered in Iyengar’s talks). Matlis lifts. Gorenstein duality. Deducing Gorenstein duality from the Gorenstein condition in the orientable case. Gorenstein ascent. C^*(BG)—>k is Gorenstein. Deducing the local cohomology theorem. The ubiquity of other examples and perhaps the connection to Serre duality in derived algebraic geometry.

The following are *not* prerequisites, but the material in the lectures is based on these papers.

A general reference:

J.P.C.Greenlees

``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

Other background:

J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

We will discuss the main statements of Grothendieck duality and then briefly explain how certain concrete residue formulas and trace maps arise in the abstract theory.

Resource:

1. Residues and Duality, LNM 20, Springer Verlag, by R Hartshorne.

2. Notes on derived categories and Grothendieck duality, LNM 1960, Springer Verlag, by J Lipman.

3. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, JAMS vol 9, by A Neeman.

To begin the talk, we present some stable phenomena that serve as motivation for defining spectra, which subsequently leads to the stable homotopy category. The talk proceeds to explore the various properties and axioms satisfied by the stable homotopy category. We also examine potential candidates for this category, weighing their advantages and disadvantages. Additionally, we discuss modern categories of spectra, which are equivalent to the stable homotopy category. We conclude the talk with a discussion on the Spanier-Whitehead duality for spectra.

Friday, 19 May 2023

The goal of the talks is to introduce some of the foundations of chromatic stable homotopy theory and to explain how these methods are used to study the structure of the stable homotopy category. If time permits, we will look at some more recent results at the end.

Topics will include:

- stable model categories

- left Bousfield localisation,

- p-localisation, p-completion, rationalisation, K-theory,

- E(n) and K(n): construction and properties,

- nilpotence, periodicity, chromatic convergence,

- thick subcategories and the Balmer spectrum.

D. Barnes, C. Roitzheim: Foundations of Stable Homotopy Theory (Cambridge University Press)

"Ambidexterity" refers to situations where we have two functors $F$ and $G$ such that $F$ is both left adjoint and right adjoint to $G$. (Some other talks in the workshop will probably involve cases where a left adjoint is some kind of twist of a right adjoint, but we will be considering cases where there is no twisting.) There are some elementary examples in the classical complex representation theory of finite groups and groupoids, and we will start by reviewing these, because that will make it easier to understand other examples by analogy. We will then discuss ambidexterity in the context of the Morava $K$-theory and $E$-theory of classifying spaces of finite groupoids, which will involve some general theory of formal groups and formal schemes, as well as the generalised character theory of Hopkins, Kuhn and Ravenel. Finally we will generalise from finite groupoids to finite $\infty$-groupoids, or equivalently $\pi$-finite spaces. This will involve some discussion of $\infty$-categories in the sense of Lurie, and the calculation of Morava $E$-theory of Eilenberg-MacLane spaces. If time permits, we will also discuss some interesting applications in chromatic homotopy theory.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)

Ambidexterity in K(n)-Local Stable Homotopy Theory (Hopkins and Lurie, https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

Ambidexterity in Chromatic Homotopy Theory (Carmeli, Schlank and Yanovski, https://arxiv.org/abs/1811.02057)

Quillen's stratification theorem describes the Zariski spectrum of prime ideals of the mod p cohomology of a finite group in terms of the abelian subgroups of the group. In these talks I will explain the key points in the proof, generealizations to other cohomoloogy theories by other authors and finally a recent work (with Barthel, Heard, Naumann and Pol) with shows how Quillen stratification works for the Balmer spectrum in equivariant homotopy theory. If time permits I will analyze the dual situation of costratification.

I. Group cohomology

II. Quillen stratification theorem in mod p cohomology

III. Quillen stratification theorem in equivariant homotopy theory (1)

IV. Quillen stratification theorem in equivariant homotopy theory (2) and costratification (if time permits)

D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971)

A. Mathew, N. Naumann, J. Noel. Derived induction and restriction theory, Geom. Topol 23, 2019.

T. Barthel, N. Castellana, D. Heard, N. Naumann, L. Pol, Quillen stratification in equivariant homotopy theory, ArXiv:2301.02212

Monday, 22 May 2023

I will discuss classifications of thick and localising subcategories for categories of modular representations of finite groups. This requires a certain amount of machinery, for instance a theory of support for triangulated categories. In addition, I plan to talk about various dualities (local and global) which reflect a Gorenstein property of the category of finite group representations.

Resources:

D. J. Benson, S. B. Iyengar, and H. Krause, Representations of finite groups: Local cohomology and support, Oberwolfach Seminars 43, Birkhäuser Verlag, 2012, 111 pp.

H. Krause, Homological Theory of Representations, Cambridge Studies in Advanced Mathematics 195, Cambridge Univers. Press, 2021, 482 pp.

[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

The following are *not* prerequisites, but the material in the lectures is based on these papers.

J.P.C.Greenlees

``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

Quillen's stratification theorem describes the Zariski spectrum of prime ideals of the mod p cohomology of a finite group in terms of the abelian subgroups of the group. In these talks I will explain the key points in the proof, generealizations to other cohomoloogy theories by other authors and finally a recent work (with Barthel, Heard, Naumann and Pol) with shows how Quillen stratification works for the Balmer spectrum in equivariant homotopy theory. If time permits I will analyze the dual situation of costratification.

I. Group cohomology

II. Quillen stratification theorem in mod p cohomology

III. Quillen stratification theorem in equivariant homotopy theory (1)

IV. Quillen stratification theorem in equivariant homotopy theory (2) and costratification (if time permits)

D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971)

A. Mathew, N. Naumann, J. Noel. Derived induction and restriction theory, Geom. Topol 23, 2019.

T. Barthel, N. Castellana, D. Heard, N. Naumann, L. Pol, Quillen stratification in equivariant homotopy theory, ArXiv:2301.02212

In this talk I will present the idea of Landweber exactness and introduce Lubin-Tate's Universal Deformation and consequently show that Lubin-Tate spectra is Landweber exact.

Tuesday, 23 May 2023

"Ambidexterity" refers to situations where we have two functors $F$ and $G$ such that $F$ is both left adjoint and right adjoint to $G$. (Some other talks in the workshop will probably involve cases where a left adjoint is some kind of twist of a right adjoint, but we will be considering cases where there is no twisting.) There are some elementary examples in the classical complex representation theory of finite groups and groupoids, and we will start by reviewing these, because that will make it easier to understand other examples by analogy. We will then discuss ambidexterity in the context of the Morava $K$-theory and $E$-theory of classifying spaces of finite groupoids, which will involve some general theory of formal groups and formal schemes, as well as the generalised character theory of Hopkins, Kuhn and Ravenel. Finally we will generalise from finite groupoids to finite $\infty$-groupoids, or equivalently $\pi$-finite spaces. This will involve some discussion of $\infty$-categories in the sense of Lurie, and the calculation of Morava $E$-theory of Eilenberg-MacLane spaces. If time permits, we will also discuss some interesting applications in chromatic homotopy theory.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)

Ambidexterity in K(n)-Local Stable Homotopy Theory (Hopkins and Lurie, https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

Ambidexterity in Chromatic Homotopy Theory (Carmeli, Schlank and Yanovski, https://arxiv.org/abs/1811.02057)

Quillen's stratification theorem describes the Zariski spectrum of prime ideals of the mod p cohomology of a finite group in terms of the abelian subgroups of the group. In these talks I will explain the key points in the proof, generealizations to other cohomoloogy theories by other authors and finally a recent work (with Barthel, Heard, Naumann and Pol) with shows how Quillen stratification works for the Balmer spectrum in equivariant homotopy theory. If time permits I will analyze the dual situation of costratification.

I. Group cohomology

II. Quillen stratification theorem in mod p cohomology

III. Quillen stratification theorem in equivariant homotopy theory (1)

IV. Quillen stratification theorem in equivariant homotopy theory (2) and costratification (if time permits)

D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971)

A. Mathew, N. Naumann, J. Noel. Derived induction and restriction theory, Geom. Topol 23, 2019.

T. Barthel, N. Castellana, D. Heard, N. Naumann, L. Pol, Quillen stratification in equivariant homotopy theory, ArXiv:2301.02212

We will discuss the main statements of Grothendieck duality and then briefly explain how certain concrete residue formulas and trace maps arise in the abstract theory.

Resource:

1. Residues and Duality, LNM 20, Springer Verlag, by R Hartshorne.

2. Notes on derived categories and Grothendieck duality, LNM 1960, Springer Verlag, by J Lipman.

3. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, JAMS vol 9, by A Neeman.

Wednesday, 24 May 2023

I will discuss various aspects of the derived category, and associated triangulated categories of interest in commutative algebra.

Resources:

Some familiarity with commutative algebra, and triangulated categories.

Resources:

- stable model categories

- left Bousfield localisation,

- p-localisation, p-completion, rationalisation, K-theory,

- E(n) and K(n): construction and properties,

- nilpotence, periodicity, chromatic convergence,

- thick subcategories and the Balmer spectrum.

D. Barnes, C. Roitzheim: Foundations of Stable Homotopy Theory (Cambridge University Press)

Thursday, 25 May 2023

II. Quillen stratification theorem in mod p cohomology

III. Quillen stratification theorem in equivariant homotopy theory (1)

IV. Quillen stratification theorem in equivariant homotopy theory (2) and costratification (if time permits)

A. Mathew, N. Naumann, J. Noel. Derived induction and restriction theory, Geom. Topol 23, 2019.

T. Barthel, N. Castellana, D. Heard, N. Naumann, L. Pol, Quillen stratification in equivariant homotopy theory, ArXiv:2301.02212

"Ambidexterity" refers to situations where we have two functors $F$ and $G$ such that $F$ is both left adjoint and right adjoint to $G$. (Some other talks in the workshop will probably involve cases where a left adjoint is some kind of twist of a right adjoint, but we will be considering cases where there is no twisting.) There are some elementary examples in the classical complex representation theory of finite groups and groupoids, and we will start by reviewing these, because that will make it easier to understand other examples by analogy. We will then discuss ambidexterity in the context of the Morava $K$-theory and $E$-theory of classifying spaces of finite groupoids, which will involve some general theory of formal groups and formal schemes, as well as the generalised character theory of Hopkins, Kuhn and Ravenel. Finally we will generalise from finite groupoids to finite $\infty$-groupoids, or equivalently $\pi$-finite spaces. This will involve some discussion of $\infty$-categories in the sense of Lurie, and the calculation of Morava $E$-theory of Eilenberg-MacLane spaces. If time permits, we will also discuss some interesting applications in chromatic homotopy theory.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)

Ambidexterity in K(n)-Local Stable Homotopy Theory (Hopkins and Lurie, https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

Ambidexterity in Chromatic Homotopy Theory (Carmeli, Schlank and Yanovski, https://arxiv.org/abs/1811.02057)

Resource:

2. Notes on derived categories and Grothendieck duality, LNM 1960, Springer Verlag, by J Lipman.

3. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, JAMS vol 9, by A Neeman.

Friday, 26 May 2023

Resources:

Some familiarity with commutative algebra, and triangulated categories.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)