Annual Report 2005 - Fields Institute - University of Toronto

The conference featured reports of substantial progress in
many areas. Here are a few highlights relating particularly
to elliptic cohomology and string topology.
Elliptic cohomology mixes topology and number theory,
by associating to a topological space various arithmetic
quantities related to elliptic curves. It connects both of these
subjects to string theory, as it is the natural receptacle of
the one-loop amplitude of various string theories. How it
manages to relate these things is somewhat of a mystery:
Witten once compared the situation to seeing the peaks of a
mountain range, while the valley below is shrouded in mist.
At the conference, Stephan Stolz reported on his work with
Peter Teichner on elliptic cohomology and string theory,
and there were reports on elliptic genera and orbifolds by
Nora Ganter and Hirotaka Tamanoi. Perhaps the most
spectacular event was Jacob Lurie’s announcement of his
invention of “derived algebraic geometry”. This mixture
of algebraic geometry and stable homotopy theory appears
to simultaneously solve a number of the most important
outstanding problems in elliptic cohomology.
The subject of string topology began with the discovery,
by Moira Chas and Dennis Sullivan, of rich structure in
the homology of the free loop space of a manifold. At the
conference, Ralph Cohen reported on his Morse-theoretic
approach to string topology, and Dennis Sullivan reported
on his work on the action of various diffeomorphism
groups (for example, reparametrizations of the loop) on
string topology.
David Ben-Zvi and Paul Goerss gave talks on the Langlands
program, to draw the attention of this community of
researchers to the prominent role in the local Langlands
program of the Lubin-Tate moduli spaces of formal groups
Vassily Gorbounov and Constantin Teleman
T h e m a t i c P r o g r a m s
and the covers of these spaces constructed by Drinfeld.
These are fundamental objects of study in stable homotopy
theory as well, and the situation strongly suggests a relationship
which should be explored.
The conference featured very impressive talks by a number
of graduate students and postdocs, including Mark
Behrens, Alex James Bene, Michael Ching, Nora Ganter,
Veronique Godin, Andre Henriques, Jacob Lurie, Eric
Sharpe, and Andrew Stacey. Along the way, it also celebrated
the influence of Jack Morava, on the occasion of his
60th birthday. The meeting was supported by the Fields and
Perimeter Institutes, the National Science Foundation, and
the Connaught Fund.
Speakers:
Mark Behrens (MIT)
Isogenies of elliptic curves and the K(2)-local sphere
Alex Bene (UCLA)
The locus at hyperelliptic fatgraphs
Michael Ching (MIT)
Operadic bar constructions and the Goodwillie derivatives of
the identity
Ralph Cohen (Stanford)
String topology and Gromov-Witten theory of cotangent
bundles
Christopher Douglas (MIT)
Twisted K-Theory of Lie groups
Nora Ganter (UIUC)
On orbifold genera, Kinl-local spectra, product formulas and
power operations
Veronique Godin (Stanford)
Fat graphs and the mapping class group of a surface with
boundary
Paul Goerss (Northwestern)
Morava modules and local Langlands
Vassily Gorbounov (Kentucky)
Mirror symmetry formula for elliptic genus of some Fano
varieties
Fields Institute 2005 ANNUAL REPORT 21