Time | Speaker | Title | Resources | |
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09:30 to 10:45 | Constanze Roitzheim (University of Kent, U.K.) |
Introduction to Chromatic Homotopy Theory III 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: D. Barnes, C. Roitzheim: Foundations of Stable Homotopy Theory (Cambridge University Press) |
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11:15 to 12:30 | Neil Strickland (University of Sheffield, UK) |
Ambidexterity I "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) |
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14:30 to 15:45 | Natalia Castellana (Universitat Autonoma de Barcelona, Spain) |
Quillen stratification theorem(s)
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 D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971) |
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16:15 to 17:30 | Aryaman Maithani (University of Utah, USA) | Prep 5- Gorenstein Rings |

Time | Speaker | Title | Resources | |
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09:30 to 10:45 | Neil Strickland (University of Sheffield, UK) |
Ambidexterity II "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) |
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11:15 to 12:30 | Natalia Castellana (Universitat Autonoma de Barcelona, Spain) |
Quillen stratification theorem(s)
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 D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971) |
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14:30 to 15:45 | Suresh Nayak (ISIBC, India) |
Residues and Trace maps in Grothendieck Duality 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. |
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16:15 to 17:30 | - | Discussion/Problem |

Time | Speaker | Title | Resources | |
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09:30 to 10:45 | Natalia Castellana (Universitat Autonoma de Barcelona, Spain) |
Quillen stratification theorem(s)
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 D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971) |
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11:15 to 12:30 | Neil Strickland (University of Sheffield, UK) |
Ambidexterity III "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) |
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14:30 to 15:45 | Suresh Nayak (ISIBC, India) |
Residues and Trace maps in Grothendieck Duality 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. |
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16:15 to 17:30 | - | Discussion/Problem |

Time | Speaker | Title | Resources | |
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09:30 to 10:45 | Srikanth B. Iyengar (University of Utah, USA) |
Duality and stratification in commutative algebra IV 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. |
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11:15 to 12:30 | Neil Strickland (University of Sheffield, UK) |
Ambidexterity IV K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109) |
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14:30 to 15:45 | - | Discussion/Problem |