Cobordism of symplectic manifolds and asymptotic expansions

The cobordism ring of symplectic manifolds defined by V.L. Ginzburg is shown to be isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This suggests an interpretation of the formal group law of complex cobordism, in terms of a composition-law on semiclassical expansions. An appendix discusses related questions about cobordism of toric varieties.Comment: A talk at the conference in honor of S.P. Novikov's 60th birthda

Complex cobordism and algebraic topology

This is a historical survey, beginning where Atiyah and Sullivan leave off...Comment: This is an attempt to supplement recent [BAMS 2004] accounts of Ren\'e Thom's work on cobordism theory with a description of that subject's later evolution -- in particular, its connection with the local Langlands progra

Geometric Tate-Swan cohomology of equivariant spectra

We sketch a quick and dirty geometric approach to the Tate-Swan cohomology of equivariant spectra, illustrating it with conjectural applications to Atiyah-Segal $K$-theory of circle actions, and a possible geometric model for the topological cyclic homology of the sphere spectrum.Comment: \S 2.3 and \S 3.4 revised; references perhaps straightened out. For Graeme Segal and Sir Michael Atiya

A topological group of extensions of \Q by Z\Z

The group of extensions (as in the title), endowed with something like a connection at Archimedean infinity, is isomorphic to the ad\'ele-class group of \Q: which is a topological group with interesting Haar measure.}Comment: For Mike and Takash

The Bagger-Lambert model and Type IIA string theory

We conjecture the existence of a `compactified' version of Fukaya's homology for symplectic manifolds, which carries a canonical 2-Gerstenhaber algebra structure. This may help to understand the 2-Lie algebra structure involved in models for interacting D-branes.Comment: Notes from a talk at the Workshop on Geometry, Topology, and Physics at the University of Pittsburgh, 14-15 May 2014 \[ {\tt http://www.mathematics.pitt.edu/node/1216} \

Categories of orbit types for proper Lie groupoids

It is widely understood that the quotient space of a topological group action can have a complicated combinatorial structure, indexed somehow by the sotropy groups of the action, but how best to record this structure seems unclear. This sketch defines a database category of orbit types for a proper Lie groupoid (based on recent work with roots in the theory of geometric quantization) as an attempt to capture some of this information.Comment: Notes from a talk at the January 2014 AMS Baltimore special session on homotopy theory. Special thanks to the organizer

A theory of base motives

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...Comment: Talk at the conference on p-adic Geometry and Homotopy Theory, Loen, Norway, 8 August 200

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 behaves 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 two-dimensional topological gravity through the homotopy theory of moduli spaces.Comment: This is a version of a talk on 23 June 2001 at the Prague Conference on Quantum Groups and Integrable Systems, published in the Czechoslovak J. Physics 51 (2001

Braids, trees, and operads

The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line, modulo projective equivalence, has a natural compactification (as a space of equivalence classes of trees) which is also (by a theorem of Davis, Januszkiewicz, and Scott) aspherical. The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals. The fundamental groups of these new configuration spaces are not braid groups, but they have many similar formal properties.Comment: Notes from an expository talk at the Gdansk conference on algebraic topology, 5 June 2001

Notes on two elementary evolutionary games

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