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| Oct 24 |
Tue |
Neil Strickland (Sheffield) |
Astronomical Topology Working Group |
| 09:00 |
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Introduction to Chromatic Homotopy
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Hicks Seminar Room J11 |
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Abstract:
This will be an introduction to chromatic homotopy theory, aiming to give the background required to understand the statement of the Telescope Conjecture.
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| Nov 7 |
Tue |
Neil Strickland (Sheffield) |
Astronomical Topology Working Group |
| 09:00 |
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The Telescope Conjecture as Galois theory of ring spectra
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Hicks Seminar Room J11 |
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Abstract:
Let $X$ be a finite $p$-torsion spectrum of type $n$, which means that the Morava $K$-theory $K(n)_{*}(X)$ is nontrivial, but $K(m)_*(X)=0$ for $m\lt n$. By work of Devinatz, Hopkins and smith, there is a map $v\:\Sigma^dX\to X$ for some $d\gt 0$ such that $K(n)_{*}(v)$ is an isomorphism, and this is nearly natural in a certain sense. We can thus form the colimit $v^{-1}X$ of the sequence $X\to\Sigma^{-d}X\to\Sigma^{-2d}X\to\dotsb$. The Telescope Conjecture predicts that this should be the same as the Bousfield localisation $L_{K(n)}(X)$. There is a certain spectrum $E$ called Morava $E$-theory, with a natural action of a group $G$ called the Morava stabiliser group, with the property that $L_{K(n)}(X)$ is the spectrum $(E\wedge X)^{hG}$ of (homotopy) fixed points of the action (by an argument that is not too hard). There is a sense in which $E\wedge X$ is a Galois extension of $L_{K(n)}(X)$ with Galois group $G$, and various other Galois extensions with smaller Galois groups play an important role in the disproof of TC. I will attempt to explain some of these ideas.
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| Nov 14 |
Tue |
Dan Graves (Leeds) |
Astronomical Topology Working Group |
| 09:00 |
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Background from stable homotopy theory |
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Hicks Seminar Room J11 |
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| Nov 21 |
Tue |
Dan Graves (Leeds) |
Astronomical Topology Working Group |
| 09:00 |
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Topological Hochschild homology
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Hicks Seminar Room J11 |
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Abstract:
I'll talk about THH for ring spectra, Tate spectra, the Frobenius and topological cyclic homology with a view towards understanding Proposition 1.1 in the Burklund, Hahn, Levy and Schlank preprint.
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| Nov 28 |
Tue |
TBA |
Astronomical Topology Working Group |
| 09:00 |
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TBA
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Hicks Seminar Room J11 |
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