First I state and explain the Nearby Lagrangian conjecture and discuss early results. I will then discus the Viterbo transfer, and how this relates to symplectic homology and Morse theory. Then I will explain Viterbo functoriality and a new consequence of non-orientability of the Maslov-bundle on the symplectic homology of cotangent bundles. I will finish by outlining new results and methods on the Nearby Lagrangian conjecture.
In topology the spectra P(n) are defined as certain quotients of the Brown-Peterson spectra. It is known that they acquire commutative associative ring structures for odd primes. Using methods developed by Elmendorf-Kriz-Mandell-May we construct (not necessarily commutative, associative) ring structures on the corresponding motivic spectra. As application we prove some universality of the motivic Eilenberg MacLane spectrum with finite coefficients.
We discuss inverse limits of Adams spectral sequences, direct limits of A-modules, and inverse limits of A_*-comodules. This is the first part of joint work with Sverre Lunøe-Nielsen on the topological Singer construction.
We review the cohomological Singer construction, which for each A-module M creates an Ext-equivalent A-module R_+(M), and discuss the dual, homological Singer construction, which for (suitable) A_*-comodules M_* creates a completed A_*-comodule R_+(M_*). This is the second part of joint work with Sverre Lunøe-Nielsen on the topological Singer construction.
We review the Tate construction X^{tG} of Greenlees, which for each G-equivariant spectrum X compares the homotopy orbits X_{hG} to the homotopy fixed points X^{hG}, and describe a Tate tower of spectra expressing the Tate construction as an inverse limit. For X bounded below and of finite type we then discuss the (co-)homological Tate spectral sequences computing the continuous (co-)homology of X^{tG}. This is the third part of joint work with Sverre Lunøe-Nielsen on the topological Singer construction.
We first explain the setting of infinity and stable infinity categories. Then we adress the problem of describing extension categories in terms of mapping spaces. Finally we would like to discuss how one possibly can describe 3-extension problems.
In the first part of the talk we give the definition of T-duality triples and motivate it by introducing the T-duality transformation for twisted K-theory. Then we formulate an equivalent notion of T-duality triple in terms of abelian groupstacks. The proof for the equivalence on which we will focus in the second part makes use of a third desciption of T-duality triples in terms of homological algebra data.
There is some interest in the structure of Toric spaces (e.g. Moment angle complexes, Toric manifolds, subspace arrangements, etc.) In recent work with Tony Bahri, Fred Cohen and Sam Gitler we have shown that many of these spaces stably split into canonical pieces. I shall describe how this splitting determines the product structure in the cohomology of a generalized moment angle complex and allows us to compute the KO theory of Davis-Januskiewicz spaces (joint with BBCG , Don Davis and Nigel Ray).
For a symmetric spectrum B we give a C_p-equivariant model B^p for the smash product of p copies of B, and define the topological Singer construction on B to be the Tate construction R_+(B) = (B^p)^{tC_p}. We show that the continuous cohomology of R_+(B) realizes the Singer construction on the A-module H^*(B), when B is bounded below and of finite type mod p. This is the fourth part of joint work with Sverre Lunøe-Nielsen.
The "field with one element" is the name given to a hypothetical mathematical entity which is supposed to lie behind several limit phenomena of geometry/algebra over finite fields and is hoped to provide an explanation of the analogy between function fields and number fields. In this talk we will first present some of the observations which gave rise to the idea of such an entity, then make a quick survey of several proposed definitions of geometry over F_1, and finally take a glimpse into motivic homotopy theory over F_1.
One of the goals of the philosophy of motives is to provide a higher dimensional generalization of the classical Galois theory where number fields and algebraic numbers are replaced by algebraic varieties and their periods. A central place here is occupied by the motivic Galois group, the higher dimensional generalization of the classical Galois group. In the first lecture, I will recall the construction of the triangulated categories of motives and their Betti realizations. In the second lecture, we define the motivic Hopf algebra of a field endowed with a complex embedding and give some of its properties. In the last lecture, we explain the link with periods.
I am experimenting with the philosophy that given a space of a given rank of homology (over F_p), then the fewer cup products there are, the lower the free p-rank of symmetry (assuming always trivial action on homology). For rank 4, if cup products are trivial then p-rank of symmetry is 1, and the only case where it is 2 is the product of spheres case. I have shown that to have p-rank of symmetry 2 the homological rank must be at least 8 and I conjecture that it must be at least 10. I construct geometric examples for every possible algebraic possibility in the rank 4 case.
I'll report on joint work with T. Lawson aiming at using a result of Lurie to make progress on the long-standing problem in homotopy theory asking if BP admits an E_\infty-structure. At present, our results are rather isolated: We can realize BP<2> at 2.
We study the preservation of algebras over colored operads under homotopical localization and colocalization functors in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of colored operads (also called enriched multi-categories) in this context. This is a joint work with C. Casacuberta, I. Moerdijk and R. M. Vogt.