Topology Seminar
Upcoming Talks
The seminar will meet at 4:30 on Monday in 2131 unless otherwise noted.
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Arpon Raksit (MIT)
$\begingroup $This talk will be about joint work with Jeremy Hahn and Dylan Wilson in which we define a filtration on an arbitrary commutative ring spectrum that we call the "even filtration". I'll introduce the definition, the one method we've come up with for analyzing it, and its relation to other filtrations of interest, in particular motivic filtrations on topological Hochschild homology.
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Nick Kuhn (University of Virginia)
$\begingroup $The study of the action of a finite pgroup $G$ on a finite $G$CW complex $X$ is one of the oldest topics in algebraic topology. In the late 1930's, P. A. Smith proved that if $X$ is mod p acyclic, then so is $X^G$, its subspace of fixed points. A related theorem of Ed Floyd from the early 1950's says that the dimension of the mod p homology of $X$ will bound the dimension of the mod p homology of $X^G$.
The study of the Balmer spectrum of the homotopy category of $G$spectra has lead to the problem of identifying "chromatic" variants of Smith's theorem, with mod p homology replaced by the Morava Ktheories (at the prime p). One such chromatic Smith theorem is proved by Barthel et.al.: if $G$ is a cyclic pgroup and X is $K(n)$ acyclic, then $X^G$ is $K(n1)$ acyclic (and this answers questions like this for all abelian pgroups).
In work with Chris Lloyd, we have been able to show that a chromatic analogue of Floyd's theorem is true whenever a chromatic Smith theorem holds. For example, if $G$ is a cyclic pgroup, then the dimension over $K(n)_\ast$ of $K(n)_\ast(X)$ will bound the dimension over $K(n1)_\ast$ of $K(n1)_\ast(X^G)$.
The proof that chromatic Smith theorems imply the stronger chromatic Floyd theorems uses the representation theory of the symmetric groups.
These chromatic Floyd theorems open the door for many applications. We have been able to resolve open questions involving the Balmer spectrum for the extraspecial 2groups. In a different direction, at the prime 2, we can show quick collapsing of the AHSS computing the Morava Ktheory of some real Grassmanians: this is a nonequivariant result.
In my talk, I'll try to give an overview of some of this.
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Manuel Rivera (Purdue University)

Eoin Mackall (University of Maryland, College Park)

Peter Haine (University of California, Berkeley)

Stephen McKean (Harvard University)

Alexander Kupers (University of Toronto Scarborough)

Allen Yuan (Columbia)
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