arXiv subjects
Publications and source records attributed to Jack Morava.
Notes from a talk at the April 2011 ICMS (Edinburgh) conference on the recent solution of the Kervaire invariant problem. This is an entirely expository account, emphasizing connections with the theory of topological automorphic forms.
In the first part of this note, we show (following Hofbauer and Sigmund) that Dawkins' "Battle of the Sexes" defines an interesting map from a space of economic parameters to psychosocial coordinates. The second part discusses an even more elementary game, but one which is not completely trivial.
Let $A$ be a not necessarily commutative monoid with zero such that projective $A$-acts are free. This paper shows that the algebraic K-groups of $A$ can be defined using the +-construction and the Q-construction. It is shown that these two constructions give the same K-groups. As an immediate application, the homotopy invariance of algebraic K-theory of certain affine $\mathbb{F}_1$-schemes is obtained. From the computation of $K_2(A),$ where $A$ is the monoid associated to a finitely generated abelian group, the universal central extension of certain groups are constructed.
Mass as broken conformal symmetry: the graviton makes better sense as a Goldstone boson associated to the dilaton, than vice versa. The discussion of the Yamabe problem for the interior Schwarzschild solution [§4.2.4] has been sharpened in this revision.
Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of space-time can be naturally formulated. Their string orientations are classified by $H^3$ with coefficients in an extension of the usual group of D-brane charges, by cobordism classes of homology three-spheres with trivial Rokhlin invariant.
The Madsen-Tillmann spectra defined by categories of three- and four-dimensional Spin manifolds have a very rich algebraic structure, whose surface is scratched here.
A category of correspondences based on Waldhausen A-theory has interesting analogies, in the context of differential topology, to categories of mixed Tate motives studied in arithmetic geometry. In particular, the Hopf object S \wedge_A S (regarding A(*) as a kind of local ring over the sphere spectrum) has some similarities to a motivic group for this category; its associated rational Lie algebra is free, on odd-degree generators...
This posting is INVALID and has been WITHDRAWN. Please see N. Kitchloo and JM, Spin cobordism categories in low dimensions, arXiv:0908.3114, for a replacement and revision.
This very speculative sketch suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of `nearby' cycles for stacks or orbifolds.
A formula for the apparent rotation of a relativistically moving object has been known for some time, but it seems not to have been realized that this formula has a very pretty interpretation in terms of formal group laws. Version 2 contains a few rather small additions.
This is a historical talk about the recent confluence of two lines of research in equivariant elliptic cohomology, one concerned with connected Lie groups, the other with the finite case. These themes come together in (what seems to me remarkable) work of N. Ganter, relating replicability of McKay-Thompson series to the theory of exponential cohomology operations.
This is a historical survey, beginning where Atiyah and Sullivan leave off...
We apply a theorem of Wick to rewrite certain classes of exponential measures on random graphs as integrals of Feynman-Gibbs type, on the real line. The analytic properties of these measures can then be studied in terms of phase transitions; spaces of scale-free trees are a particularly interesting example.
We show that the function Li_s(e^x) extends to an entire function of the complex variable s, taking values in tempered distributions in x on the whole real line. That the classical polylogarithm extends similarly, as an entire function taking values in distributions on compactly-supported functions on the positive real axis, is a corollary. We then identify the singularities of Li_s(e^x) in terms of distribution powers of x; this leads to a simple proof of the smoothness of the `modified' polylogarithm of Bloch, Ramakrishnan, Wigner, Wojtkowiak, Zagier, and others.
The second relative homotopy group π_2(V(\C),V(\R)) (of the complex relative to the real points of an algebraic variety) encodes interesting information about the `bubbling' phenomena of quantum cohomology. We consider these groups in particular for genus zero Deligne-Mumford moduli spaces, as well as for certain generalizations introduced more recently by Manin and Losev.
The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A rather striking amount of work on pseudoisotopy theory [Hatcher, Waldhausen, Cohen-Carlsson-Goodwillie-Hsiang-Madsen...] can be formulated as a (generalized) TQFT in this framework. The resulting theory is far from trivial even in the case of Minkowski space, when the relevant three-manifold is the standard sphere.
This is the writeup of an expository talk. It is intended as an introduction to the work of Hopkins, Kuhn, and Ravenel on generalized group characters, which seems to fit very well with the theory of what physicists call higher twisted sectors in the theory of orbifolds.
We construct an S^1-equivariant prospectrum that models the Atiyah dual of a free loop space of a manifold. By applying a suitably completed S^1-equivariant K-theory to the Atiyah dual, we show how to recover the Witten genus of the manifold. The main technical tool is a Tits building for the loop group. We use this building to construct a dualizing spectrum for the loop group and relate it to work of Freed, Hopkins and Teleman.