Elliptic Festival: Mini Workshop and Lectures
We are excited to have Zhen Huan visiting from Sun Yat-Sen University this month. To celebrate this, we are stepping up our elliptic activities, starting with a Mini-Workshop (Monday afternoon and Tuesday morning this coming week) and followed by a lunch-time lecture series by Zhen Huan, on Wednesdays and Fridays in 213 for the next two weeks.
A gentle introduction, suitable for advanced students, to the topics of elliptic cohomology (Nora), its formulation in the Hopkins-Lurie world (Gufang) applications to geometric representation theory and Nakajima quiver varieties (Gufang) new geometric definitions (Matt Spong or Nora) orbifold loop spaces and other basic notions for quasi-ellptic cohomology (Zhen).
Format: a gentle introduction to this circle of ideas in 30-40 minute talks with many breaks and opportunity to discuss and ask questions, laying the foundations needed to follow the lunch-time lectures.
Time and Place: Monday 2-- 5, we have the Greg Hjorth Room until 4:15 then Belz.
Tuesday Belz room from 10 to 12.
Wednesdays and Fridays 12:30 to 2:00 in 213, starting this coming Wednesday, Nov 7.
Topics: For the first two lectures, I will use the board and talk about
* Review: elliptic cohomology and Tate K-theory.
* Introduction to quasi-elliptic cohomology.
* The loop space construction of quasi-elliptic cohomology.
* The power operation of quasi-elliptic cohomology; Classification of the finite subgroups of the Tate curve.
* Quasi-theories: a generalization of the idea of quasi-elliptic cohomology.
Quasi-elliptic cohomology is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most ellipitc cohomology theories. Quasi-elliptic cohomology is closely related to Tate K-theory. It can be interpreted by orbifold loop spaces and expressed in terms of equivariant K-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal.
For the third and fourth lecture, I will use the slides and talk about
* Review: model category; history of equivariant homotopy theory and global homotopy theory.
* orthogonal G-spectrum of quasi-elliptic cohomology.
* Almost global homotopy theory that I construct recently where global quasi-elliptic cohomology resides. Some quasi-theories can be globalized in it.
Many important equivariant theories naturally exist not only for a particular group, but in a uniform way for a family of groups. Prominent examples are equivariant stable homotopy theory, equivariant K-theory and equivariant bordism. This observation led to the birth of global homotopy theory. Globalness is a measure of the naturalness of a cohomology theory. Schwede developed a modern approach of it by global orthogonal spectra, which is inspired by Greenlees and May.
So far several models of global homotopy theory have been established with different motivations and advantages, including Bohmann's model, Gepner's model and Rezk's model. We construct a new global homotopy theory, almost global homotopy theory, which is equivalent to the previous models. But with it we can show that some quasi-theories can be globalized if the original cohomology theory can be globalized. Especially, quasi-elliptic cohomology can be globalized in it. This leads to the conjecture that the globalness of a cohomology theory is determined by the formal component of its divisible group; when the etale component varies, the globalness does not change.
Please contact Nora Ganter for further details.