# Timetable (NSTW04)

## Elliptic Cohomology and Chromatic Chenomena \& Higher Chromatic Phenomena

Sunday 8th December 2002 to Saturday 21st December 2002

 08:15 to 09:00 RegistrationSession: Elliptic cohomology and chromatic phenomena 09:00 to 10:00 Topological aspects of elliptic cohomologySession: Elliptic cohomology and chromatic phenomena INI 1 10:00 to 11:00 Two-vector bundles and elliptic objectsSession: Elliptic cohomology and chromatic phenomena We (Baas, Dundas, Rognes) define a two-vector bundle over a base space X as a kind of family of two-vector spaces (in the sense of Kapranov and Voevodsky) parametrized over X. The rank 1 case recovers the notion of a Dixmier-Douady gerbe over X. The equivalence classes of two-vector bundles over X form an abelian monoid, whose Grothendieck group completion is the zero-th generalized cohomology group of X represented by the algebraic K-theory of the symmetric bimonoidal category of two-vector spaces. We argue that this K-theory agrees with the algebraic K-theory of the S-algebra (= A-infinity ring spectrum) representing connective topological K-theory, which by explicit computation (Ausoni, Rognes) is a connective, integrally defined form of elliptic cohomology, i.e., has chromatic complexity two. Two-vector bundles are thus geometric objects over X which provide the "cycles" for an elliptic cohomology theory at X. We also wish to indicate how a two-vector bundle over X leads to a virtual (anomaly) vector bundle over the free loop space of X, and an associated action functional for compact oriented surfaces over X. INI 1 11:00 to 11:30 CoffeeSession: Elliptic cohomology and chromatic phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic cohomology and chromatic phenomena 15:00 to 15:30 TeaSession: Elliptic cohomology and chromatic phenomena 15:30 to 16:30 S Stolz ([Notre Dame])The spinor bundle on the free loop spaceSession: Elliptic cohomology and chromatic phenomena The spinor bundle $S(E)$ associated to an even dimensional real vector bundle $E$ with spin structure has (at least) two roles in life: from a homotopy theory point of view it represents the $K$-theory Euler class of $E$; from a geometric/analytic point of view, the Dirac operator acts on the sections of the spinor bundle $S(TX)$ associated to the tangent bundle of a spin manifold $X$. Analogously, it is believed that the {\it spinor bundle} or {\it Fock space bundle} $\mathcal F(E)\to LX$ over the free loop space $LX$ associated to an even dimensional vector bundle $E\to X$ with string structure' plays a similar dual role: it should represent the Euler class of $E$ in $tmf^*(X)$, and there should be a Dirac-Witten operator' acting on the sections of $\mathcal F(TX)$, whose $S^1$-equivariant index is the Witten genus of $X$. The main result of this joint work with Peter Teichner is that the spinor bundle $\mathcal F(E)$ can be equipped with additional structures we call conformal connection' and fusion'. We speculate that vector bundles over $LX$ equipped with these two structures represent elements in $tmf^*(X)$. INI 1 17:30 to 18:30 Wine \& Beer ReceptionSession: Elliptic cohomology and chromatic phenomena 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic cohomology and chromatic phenomena
 09:00 to 10:00 I Grojnowski ([Cambridge])Hilbert schemes and integrable system: the cohomology ring of a compactification of configuration spaceSession: Elliptic cohomology and chromatic phenomena INI 1 10:00 to 11:00 The tangent complex for the moduli stack of formal groupsSession: Elliptic cohomology and chromatic phenomena The main purpose of this talk is to explain and explore the object in the title and to outline why it might be useful. In particular, I hope to organize the following questions: when can (chromatic-type) homology theories be realized by structured ring spectra and, if they can, what can you say about maps between them? Both problems can be formulated in terms of an Andre'-Quillen cohomology calculation, which is where the tangent complex comes in. With any luck, I will get to the point where I can talk about some of the applications to elliptic spectra arising from the moduli stack of elliptic curves. I am, of course, following closely in the footsteps of others, in particular of Mike Hopkins and Haynes Miller. INI 1 11:00 to 11:30 CoffeeSession: Elliptic cohomology and chromatic phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic cohomology and chromatic phenomena 15:00 to 15:30 TeaSession: Elliptic cohomology and chromatic phenomena 15:30 to 16:30 G Segal ([Oxford])What is an elliptic object?Session: Elliptic cohomology and chromatic phenomena INI 1 16:30 to 17:30 G Mason ([California])Orbifold conformal field theory and cohomology of the monsterSession: Elliptic cohomology and chromatic phenomena We explain what is known and what is conjecture concerning rational orbifold models (ie 'good' vertex operator algebras and their automorphism groups) and the passage to the associated topological field theory. We explain how such theories may be used to detect group cohomology, and illustrate with the example of the Monster simple group, where there is a surprising prediction. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic cohomology and chromatic phenomena
 09:00 to 10:00 Algebraic groups and equivariant cohomology theoriesSession: Elliptic cohomology and chromatic phenomena Equivariant cohomology theories E_G^*(.) are represented by G-spectra. When G is a torus and the theories are rational, there is a complete and calculable algebraic model A(G) of rational G-spectra (based on the idea that E_G^*(.) is built from its behaviour at each isotropy group). This model is formally very similar to a category of sheaves on an algebraic group C. Based on this, one can investigate the properties of cohomology theories E_G^*(.) based on the group C. In several cases of interest (including the case when C is an elliptic curve over a field of characteristic zero and G is the circle group) these properties are sufficient to determine a _construction_ of the cohomology theory using the model A(G). INI 1 10:00 to 11:00 D Freed ([Texas])Loop groups and twisted K-theorySession: Elliptic cohomology and chromatic phenomena INI 1 11:00 to 11:30 CoffeeSession: Elliptic cohomology and chromatic phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic cohomology and chromatic phenomena 15:00 to 15:30 TeaSession: Elliptic cohomology and chromatic phenomena 15:30 to 16:30 On the M-theory action on a manifold with boundarySession: Elliptic cohomology and chromatic phenomena INI 1 16:30 to 17:30 Gerbes of chiral differential operatorsSession: Elliptic cohomology and chromatic phenomena In this talk we give a complete classification of a certain important class of vertex algebras, the so called algebras of chiral differential operators (cdo for short). These were introduced by Beilinson and Drinfeld, motivated by the ideas from the conformal field theory. It turned out, that "almost classical" structure of what we call the vertex algebroid controls the world of cdo. This structure consists of two part. The first is a structure of an algebroid Lie, and the other is its extension, both derived from the major identities held in chiral algebras. These extra structures can be fit into a complex of vector spaces which is a direct generalization of the De Rham-Chevalley complex for Lie algebras. The classes of equivalences of cdo are in one to one correspondence with the third cohomology group of this complex. We also provide the description of each isomorphism class. This allows us to study the sheafication of a an important class of cdo over a given manifold X. This is a cdo defined by the Heisenberg algebra and the Clifford algebra, or the free bosons and the free fermions. It turns out that there is a characteristic class which is an obstraction to glueing these local pieces into a sheaf. It has two components, first, is the Atiyah class, and the other is the Simons term. There is a projection of this class into cohomology of X, and the image of this projection is the second component of the Chern character of X. We also describe these algebras for specific classes of manifolds, namely algebraic groups and homogeneous spaces and a conjectural connection of these algebras defined for hypersurfaces to the Vafa orbifold models. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic cohomology and chromatic phenomena
 09:00 to 10:00 Equivariant veresion of elliptic spectraSession: Elliptic cohomology and chromatic phenomena Let C be an elliptic curve over an affine scheme S, and let E be an even periodic ring spectrum whose associated formal group is the formal completion of C. This makes E an elliptic spectrum''. Now let A be a finite abelian group. We will describe what it means for an A-equivariant ring spectrum EA to be an equivariant version'' of E, in terms of the theory of equivariant formal groups. We will show how to construct EA when E is K(n)-local for some n. We will then give a method for recovering the general case from the K(n)-local case. The method always produces an A-spectrum EA, but it may not be well-defined or have a ring structure. We will describe some cases in which one can get around these problems. INI 1 10:00 to 11:00 K(1)-local topological modular forms, the Witten orientation and E_\infty cellular structuresSession: Elliptic cohomology and chromatic phenomena We describe the construction and the orientation of the topological modular forms spectrum tmf in the K(1)-local E_\infty category at the prime 2. We use the theory of theta algebras to decompose tmf and the bordism theory MO into E_infty cells: MO splits into the cone T_\zeta and a free part. Moreover, as was shown by Mike Hopkins, the spectrum tmf is obtained from T_\zeta by attaching one more cell. With this information we give an explicit E_\infty orientation which refines the Witten genus for families of O manifolds. INI 1 11:00 to 11:30 CoffeeSession: Elliptic cohomology and chromatic phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic cohomology and chromatic phenomena 15:00 to 15:30 TeaSession: Elliptic cohomology and chromatic phenomena 15:30 to 16:30 Hecke operators and logarithmic cohomology operationsSession: Elliptic cohomology and chromatic phenomena We describe how the theory of unstable power operations gives rise to an action of Hecke operators'' on Morava's cohomology theory $E=E_n$; when $n=2$ the theory $E$ is a version of elliptic cohomology completed at a prime p, and such power operations correspond to Hecke operators on modular forms of $p$-power degree. We then interpret the formula for a certain natural logarithmic cohomology operation on $E$ in terms of such operators. INI 1 16:30 to 17:30 The elliptic genus of a singular varietySession: Elliptic cohomology and chromatic phenomena I will begin by describing how the elliptic genus of a manifold can be characterised among all characteristic numbers by its "rigidity" properties. Next, I will give my characterization of (one version of) the elliptic genus by its invariance under "flops", a class of surgeries that comes up naturally in algebraic geometry. Borisov and Libgober proved a stronger invariance property, which allowed them to define the elliptic genus for a large class of singular complex spaces. To find even stronger invariance properties of the elliptic genus, we can try to define the elliptic genus for singular real spaces; I will discuss some calculations which support this possibility. INI 1 19:45 to 00:00 Conference Dinner at Trinity HallSession: Elliptic cohomology and chromatic phenomena
 09:00 to 10:00 Elliptic genera and Thom classesSession: Elliptic cohomology and chromatic phenomena I will survey some features of elliptic genera from the mathematics and physics literature and explain how they look from the point of view of homotopy theory. In particular, I shall explain how features of the "sigma orientation" and its equivariant analogue relate to modularity, rigidity, the two-variable elliptic genus, orbifold elliptic genera, and discrete torsion. INI 1 10:00 to 11:00 Complex orientations and Motivic Galois theorySession: Elliptic cohomology and chromatic phenomena Grothendieck's program for an anabelian geometry suggests that the Galois group Gal(Q=Q) possesses very interesting pronilpotent representations, associated to a free Lie algebra on generators conjecturally identied with the values of the zeta function at odd positive integers > 1. Some such automorphism algebra acts on Kontsevich's deformation quantization of Poisson manifolds, and there are rea- sons for thinking there are similar actions on algebras of asymptotic expansions for geometric heat kernels [math.SG/9908070] dened via Ginzburg's cobordism of symplectic manifolds. This is all pretty hypothetical, but there is an interesting concrete representa- tion of such a free Lie algebra in the group of formal dieomorphisms of the line, closely related to a genus associated to the Gamma function by Kontsevich [math.QA/9904055, x4.6]; it is a deformation of the A^-genus, connected to the the- ory of quasisymmetric functions [math.AG/9908065]. This genus seems to be worth investigating, whether one believes any of the conjectures above, or not; this talk is an introduction to its properties. 1 INI 1 11:00 to 11:30 CoffeeSession: Elliptic cohomology and chromatic phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic cohomology and chromatic phenomena 15:00 to 15:30 TeaSession: Elliptic cohomology and chromatic phenomena 15:30 to 16:30 Cohomology of the stable mapping class groupSession: Elliptic cohomology and chromatic phenomena INI 1 16:30 to 17:30 Discussion SessionSession: Elliptic cohomology and chromatic phenomena INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic cohomology and chromatic phenomena
 10:00 to 11:00 A note on Hopkins-Kuhn-Ravenel character Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 11:00 to 11:30 Coffee at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic Cohomology and Chromatic Phenomena 14:00 to 15:00 Spin bordism, contact structure and the cohomology of p-groups? Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 15:00 to 15:30 Tea at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic Cohomology and Chromatic Phenomena
 10:00 to 11:00 Resolutions of the K(2) - local sphere Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 11:00 to 11:30 Coffee at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic Cohomology and Chromatic Phenomena 15:00 to 15:30 Tea at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 15:30 to 16:30 What are the elliptic objects of K3 cohomology Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic Cohomology and Chromatic Phenomena
 10:00 to 11:00 Towards higher chromatic analogues of elliptic cohomology: curves with high formal group laws. Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena Elliptic cohomology is related to v_2-periodicity because the formal group law associated with a certain elliptic curve has height 2. It is known that no elliptic curve will give greater height. In this talk I will display for each prime p and each i>0 a curve C(p,i) whose Jacobian has a 1-dimensional formal summand of height i(p-1). Both the curve and this formal group admit an action by a finite subgroup of the Morava stabilizer group containing an element of order p, which is maximal when p does not divide i. 11:00 to 11:30 Coffee at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic Cohomology and Chromatic Phenomena 15:00 to 15:30 Tea at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 15:30 to 16:30 Hierarchy of morava K-theories? Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic Cohomology and Chromatic Phenomena
 10:00 to 11:00 Gamma-cohomology and $E-infinity$ structures on some periodic spectra Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 11:00 to 11:30 Coffee at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic Cohomology and Chromatic Phenomena 15:00 to 15:30 Tea at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 15:30 to 16:30 Cooperations in elliptic homology and E(n) homology Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic Cohomology and Chromatic Phenomena
 10:00 to 11:00 E-infinity maps between Spanier-Whitehead duals Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena 11:00 to 11:30 Coffee at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 12:30 to 13:30 Lunch at Wolfson CourtSession: Elliptic Cohomology and Chromatic Phenomena 15:00 to 15:30 Tea at the Isaac Newton InstituteSession: Elliptic Cohomology and Chromatic Phenomena 15:30 to 16:30 An idea of homotopical algebraic geometry Talk being held in the CMS - Wolfson RoomSession: Elliptic Cohomology and Chromatic Phenomena Motivated by "Derived deformation theory" and "brave new algebra," we will present an approach to algebraic geometry in homotopical contexts. We will show some applications of this theory, e.g. to a definition of etale K-theory of "commutative" ring spectra. 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)Session: Elliptic Cohomology and Chromatic Phenomena