Twists of K-theory and TMF

We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO. We also discuss Poincare duality and umkehr maps in this setting

The sigma orientation for analytic circle-equivariant elliptic cohomology

We construct a canonical Thom isomorphism in Grojnowski's equivariant elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga's weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper3.abs.htm

Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable infinity category parametrized over objects of an infinity topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures. Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.Comment: Submission draft. Various changes, including rewrite in terms of infinity topoi and corrected discussion of functoriality of Atiyah dualit

Core-collapse astrophysics with a five-megaton neutrino detector

The legacy of solar neutrinos suggests that large neutrino detectors should be sited underground. However, to instead go underwater bypasses the need to move mountains, allowing much larger water Čerenkov detectors. We show that reaching a detector mass scale of ~5 Megatons, the size of the proposed Deep-TITAND, would permit observations of neutrino “mini-bursts” from supernovae in nearby galaxies on a roughly yearly basis, and we develop the immediate qualitative and quantitative consequences. Importantly, these mini-bursts would be detected over backgrounds without the need for optical evidence of the supernova, guaranteeing the beginning of time-domain MeV neutrino astronomy. The ability to identify, to the second, every core collapse in the local Universe would allow a continuous “death watch” of all stars within ~5  Mpc, making practical many previously-impossible tasks in probing rare outcomes and refining coordination of multiwavelength/multiparticle observations and analysis. These include the abilities to promptly detect otherwise-invisible prompt black hole formation, provide advance warning for supernova shock-breakout searches, define tight time windows for gravitational-wave searches, and identify “supernova impostors” by the nondetection of neutrinos. Observations of many supernovae, even with low numbers of detected neutrinos, will help answer questions about supernovae that cannot be resolved with a single high-statistics event in the Milky Way

Completions of Z/(p)-Tate cohomology of periodic spectra

We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n-1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie Mahowald's inverse limit of P_{-k} smash SE(n)). We also give a multiplicative splitting of tE(n) after a suitable base extension.Comment: 30 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper8.abs.htm

Operations in complex-oriented cohomology theories related to subgroups of formal groups

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992.Includes bibliographical references (p. 89-91).by Matthew Ando.Ph.D

The Jacobi orientation and the two-variable elliptic genus

We explain the relationship between the sigma orientation and Witten genus on the one hand and the two-variable elliptic genus on the other. We show that if E is an elliptic spectrum, then the Theorem of the Cube implies the existence of canonical SU-orientation of the associated spectrum of Jacobi forms. In the case of the elliptic spectrum associated to the Tate curve, this gives the two-variable elliptic genus. We also show that the two-variable genus arises as an instance of the circle-equivariant sigma orientation.Comment: Revised to better exhibit complex orientation of MSU^(CP^\infty_{-infty}