Topology Seminar
Upcoming Talks
The seminar will meet at 4:30 on Monday in 2-131 unless otherwise noted.
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If you use a different calendar program, the ics file for this seminar is here:
http://math.mit.edu/topology/topology_seminar.ics
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David Lee (MIT)
$\begingroup $It is a result of Adams and Priddy that $BSU$ admits a unique infinite loop space structure after localization at a prime $p$. On the way to proving this statement, they prove that the $p$-localized connective complex $K$-theory spectrum $ku_{(p)}$ is characterized by its $\mathbb F_p$-cohomology as a Steenrod module. I will talk about a generalization of this part to truncated Brown—Peterson spectra $BP\langle n\rangle$ at odd primes, which can be thought of as higher chromatic analogues of the connective complex $K$-theory. In particular, since the $\mathbb F_p$-cohomology depends only on the $p$-completion, a part of this result is that we can recover the $p$-local homotopy type of $BP\langle n\rangle$ from its $p$-completion. Finally, I will describe some applications and open questions, including the status of the problem at $p=2$.
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J.D. Quigley (University of Virginia)
$\begingroup $Hurewicz homomorphisms allow one to detect nontrivial elements in the stable homotopy groups of spheres by mapping them to the (often simpler) generalized homology groups of spheres. In previous work with Behrens and Mahowald, we computed the image of the 2-local Hurewicz homomorphism for topological modular forms, which allowed us to detect many new infinite families in the stable stems. In this talk, I will explain some recent work with Bhattacharya and Bobkova leveraging these computations, together with computations of the tmf-homology of small projective spaces, to produce additional infinite familes. Time permitting, I will also describe work in progress, building on Bauer’s computation of the tmf-cohomology of the infinite complex projective space, to detect smooth free circle actions on exotic spheres in arbitrarily high dimensions.
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Andy Senger (Harvard University)
$\begingroup $TBA
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Cameron Krulewski (MIT)
$\begingroup $The seminar will meet at 5:30 PM in 2-135.
I will discuss two applications of invertible field theories to quantum field theory. Functorial field theories, which are functors from a bordism category to a target category, are invertible when they factor through the Picard groupoid of the target. After additionally imposing reflection positivity, such theories are classified, due to results of Freed-Hopkins, by Anderson-dual bordism groups.
The first application we study is toward a certain form of spontaneous symmetry breaking. We model three physical processes using a twisted Gysin sequence of Anderson-dual bordism groups. Using generalized Euler classes, we study the \“Smith maps\” of Madsen-Tillmann spectra that underlie the sequence, and use them to draw physical predictions. The second application we study is toward fermionic symmetry-protected topological phases (SPTs). Generalizing work of Freed-Hopkins, we define and compute twisted Atiyah-Bott-Shapiro maps from twisted spin bordism to shifts of K-theory in order to compare two models of SPTs. This talk represents several joint projects with Antolín Camarena, Debray, Devalapurkar, Liu, Pacheco-Tallaj, Sheinbaum, Stehouwer, and Thorngren.
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Melody Chan (Brown University)
$\begingroup $I will discuss aspects of joint work with Brown, Galatius, and Payne. In particular, we identify a Hopf algebraic structure in the weight 0 cohomology with compact supports of the moduli space of abelian varieties, and we deduce exponential growth results as a corollary. A key role is played by the moduli space of tropical abelian varieties, which is stratified by locally symmetric spaces GL_n(Z)\GL_n(R)/O(n). I will try to emphasize aspects of this work not discussed in Galatius' talk in this seminar.
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Ishan Levy (University of Copenhagen)
$\begingroup $TBA
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