All lectures take place in seminar room B 1036, on the 10th floor of N. H. Abel's house.
(I) Speedy review of elliptic cohomology, ending with the statement of the moduli-theoretic interpretation of tmf and an outline of the proof (to be fleshed out in the next two lectures).
(II) Review of the classical theory of p-divisible groups. Definition of derived p-divisible groups. Sketch of the proof of the Serre-Tate theorem.
(III) Artin's representability theorem in derived algebraic geometry, its use in constructing the derived moduli stack, and a sketch of the proof that the derived moduli stack looks nice.
(IV) Higher equivariance; generalizations of the Hopkins-Kuhn-Ravenel character theory. 2-equivariance of elliptic cohomology, and relationship with the Looijenga line bundles on moduli spaces of G-bundles.
(V) Sketches of further applications: construction of the topological Witten genus; calculation of elliptic cohomology at infinity via loop group representations; recognition principles for elliptic cohomology.
This is a continuation of Birgit Richter's talk. In particular, I will give a topological model for the addition and multiplication in the big Witt vectors on a commutative ring with unit and extend that to the ring dual of quasi-symmetric functions.
I describe work in progress to develop certain elements of the theory of crystals in spectral algebraic geometry. These include divided power structures on E∞-algebras, the crystalline ∞-topos, and the ∞-categories of crystals, Dieudonné crystals, and D-crystals. The last of these, D-crystals, are analogues of certain exceptionally well-behaved objects of ordinary algebraic geometry that seem to be almost completely unstudied. There are numerous applications (still in an incomplete state) to K-theory, de Rham cohomology, and p-divisible group theory in spectral algebraic geometry.
This is joint work with André Henriques, where we show that the theory of orbispaces (basically, topological stacks locally of the form X//G for a topological group acting on a topological space) is suitably equivalent to a topological model category of Orb-spaces, or continuous contravariant functors from the topological category of "orbit stacks" to spaces. Our main technical tool is the definition of a fibrant groupoid, or a topological groupoid whose associated sheaf of groupoids is already a stack, and a fibrant replacement functor.
This generalizes the homotopy theory of G-spaces for all topological groups simultaneously, and also admits nice descriptions of orbispace K-theory and orbispace elliptic cohomology (in the spirit of Lurie's recent work).
The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of β-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem.
The classical symmetric L-theory admits an orientation by oriented bordism which is the signature on the coefficients. Away from the prime 2 it coincides with real K-theory. One can ask if Witten's loop space signature allows for a family version which is similar to L-theory and hence provides a geometric description of the classical LRS elliptic cohomology. This talk doesn't answer the question but it generalizes the construction of L-theory to a larger context and thereby produces a machinery for commutative symmetric ring spectra. It gives an idea how such a theory could be built by complexes of vertex algebras. The fundamental classes are provided by the chiral de Rham complexes.
A conformal field theory (CFT) is a monoidal functor from the category whose morphisms are moduli spaces of Riemann surfaces with boundary to the category of vector spaces. Replacing the topological category by a suitable differential graded category one arrives at the notion of a topological conformal field theory (TCFT). It is known that the data of a TCFT is essentially equivalent to that of an algebra over a certain operad which depends on the sort of moduli spaces considered. For example an A∞ algebra with an invariant inner product gives rise to the so-called open TCFT.
Evaluating a given TCFT (or a corresponding operadic algebra) on a given moduli space will give a cohomology class on this moduli space. A construction due to Kontsevich associates a homology class on the moduli space of Riemann surfaces with marked points to an associative algebra with a contractible differential and an invariant scalar product. Using the fact that the moduli spaces are topological manifolds we interpret this construction as a suitable field theory and construct the corresponding modular operad.
A result of S. Stolz and P. Teichner describes an Ω-spectrum for K-theory consisting of certain spaces of euclidean field theories. We interpret these as configuration spaces and introduce a connective version.
I will explain how some elementary algebraic geometry on the stack of formal groups leads to a generalization of recent results of M. Hovey and N. Strickland relating the comodule categories of different Landweber exact theories.
This talk is about a program to construct higher chromatic analogs of tmf. The Landweber Exact Functor Theorem states that a 1-dimensional formal group law over a graded ring R satisfying certain conditions leads to a spectrum E with π*E = R. The FGL associated with a suitable elliptic curve over R has height 2. Applying the LEFT to it leads to elliptic cohomology and eventually to tmf.
An algebraic curve of genus g leads to a g-dimensional formal group law G via its Jacobian. If G has a 1-dimensional summand then we can try to apply the LEFT to it. It is known that certain Artin-Schreier curves have such summands of large height, but they are not Landweber exact. We discuss the problem of deforming them into ones that have exact 1-dimensional formal summands. Tools for studying this problem include the Lagrange Inversion Formula of 1770 and some theorems of Honda of 1970.
It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as a topological model for the ring of quasisymmetric functions. I'll explain how to exploit topology to reprove the Ditters conjecture which says that the ring of quasisymmetric functions is a polynomial ring.
The loop map to BU gives a canonical Thom spectrum over ΩΣ CP∞. This spectrum is highly non-commutative. I'll talk about the homology of its topological Hochschild homology spectrum.
In my talk I would like to explain the general picture of Tannakian duality in a derived setting, generalizing the classical duality between Tannakian categories and affine gerbes. I will give precise statements for special cases, in particular characterizing situations where the Tannakian category can be described by a module category. I will briefly discuss the conjectures in higher category theory which are involved. Finally I will give possible applications for categories of motivic origin.
The étale condition has been important in the theory of structured ring spectra. Often it is used to prove the vanishing of the initial page of an obstruction theory spectral sequence, which then implies the homotopy-discreteness of some space of maps or of algebra structures. In this talk we explore the étale condition and some more direct ways to exploit it.John Rognes, August 30th 2006