Jack Morava

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 the Mahowald inverse limit lim ←− (P−k ∧ ΣE(n))). We also give a multiplicative splitting of tE(n) after a suitable base extension. AMS Classification numbers Primary: 55N22, 55P60 Secondary: 14L05

Elliptic Cohomology: Thom prospectra for loopgroup representations

This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new local- ization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

Elliptic Cohomology: The motivic Thom isomorphism

The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Delignes ideas on motivic Galois groups.

An algebraic analog of the Virasoro group

The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it acts as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of 2D topological gravity.

Homotopy-theoretically enriched categories of noncommutative motives

AbstractWaldhausen’s K-theory of the sphere spectrum (closely related to the algebraic K-theory of the integers) is naturally augmented as an S0-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification of the rationalization of this (covariant) dual with the Hopf algebra of functions on the motivic group for their category of mixed Tate motives over . This paper argues that the rationalizations of categories of noncommutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over . We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over but over the sphere ring-spectrum S0. Mathematics subject classification: 11G, 19F, 57R, 81T

On gauge theories of mass

Abstract The classical Einstein–Hilbert action in general relativity extends naturally to a blow-up (in the sense of algebraic geometry) of the usual space of pseudo-Riemannian metrics; this presents the metric tensor g i k as a kind of Goldstone boson associated to the real scalar field defined by its determinant. This seems to be quite compatible with the Higgs mechanism in the standard model of particle physics.

Defining modularity: A comment on “The emergence of modularity in biological systems” by Lorenz, Jeng, and Deem

The importance of modularity in evolution presented in this review [1] is extremely compelling; to a mathematician, it cries out for new developments and new thinking in statistical mechanics. Part of the great impact of Shannon’s work in communication theory was a consequence of the generality of his notion of entropy. Physicists [2] now have a deep understanding of phase transitions [§2.2.4], and the modern theory of large deviations [3] interprets the rate at which a system explores its neighborhood in phase space (more precisely, the Legendre transform of its Helmholtz free energy) as a kind of entropy very familiar in biological terms; but it’s difficult to apply such ideas to evolutionary theory, in part because modularity, robustness, hierarchy and similar concepts tend to be defined heuristically, example by example. Finding general, flexible definitions for such notions is an extremely important theoretical problem. Describing a biological system as a network is a kind of reductionism, but an immense amount of information is available in such terms; which, unfortunately, we lack the technical tools to analyse effectively. One obstacle is that biology presents us with many kinds of networks: metabolism, for example, involves cyclic structures, while gene (and perhaps protein-interaction) networks [§3.2] are highly directed. It’s unwise to ignore these differences; algorithms like Google’s PageRank (which turns a directed graphs into a probability space) might be very useful in the latter case, but probably not in the former. I hope the ideas in this paper will stimulate interest in algorithms to estimate modularity (and related quantities) from networks and more general data clouds. I also can’t help but think historical linguistics could provide a useful perspective on these questions. In the 1830’s, von Humboldt [4] noted the tendency of some languages toward isolative, and others toward agglutinative structures; since then it’s become clear that language moves in both directions. I wonder if this might be an example of modularity, and if something like Eq. (12) of the current paper (which allows for both its increase and decrease) might be relevant. There is an enormous amount of such data available for crunching. . .

Topology, Geometry and Quantum Field Theory: Heisenberg groups and algebraic topology

We study the Madsen-Tillmann spectrum

Forms ofK-theory

\C P^\infty_{-1}

A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space

as a quotient of the Mahowald pro-object