Geometric applications of homotopy theory I

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My research is supported by the project Motivic Hopf equations financed by the Research Council of Norway.

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Publications, Preprints, Notes Examples of non-algebraic classes in the Brown-Peterson tower Mathematische Zeitschrift Unstable splittings in Hodge filtered Brown-Peterson cohomology Journal of Homotopy and Related Structures 14 An Abel-Jacobi invariant for cobordant cycles Documenta Mathematica 21 Profinite and discrete G-spectra and iterated homotopy fixed points with Daniel G.

In my talk, I will present a higher categorical analogue of this: the theory of singular support for categories over a scheme, which is important for the local Langlands program. Using the Eisenbud-Khimshiashvili-Levine local degree, which is the A1-local degree of Morel in A1 homotopy theory, we define a degree of a finite map between smooth schemes over k.

Gereon Quick

When the target is appropriately connected, this degree is a bilinear form over k. We discuss some applications to enumerative geometry over non-algebraically closed fields. I will discuss some recent results on the p-adic algebraic K-theory of p-adic rings, obtained using the cyclotomic trace from K-theory to topological cyclic homology. I will report on joint work with Markus Land. To any pullback diagram of ring spectra we associate a new square of ring spectra which becomes cartesian upon applying K-theory, or in fact any localizing invariant.

The new square canonically maps to the original one, and this map is an equivalence in three corners. In the fourth corner, this map is generally not an equivalence. Complex K-theory is a generalized cohomology theory introduced by Atiyah and Hirzebruch, which associates to each finite cell complex X the Grothendieck group KU X of complex vector bundles on X.

However, it also admits a purely algebraic description which makes no mention of vector bundles: it is the complex-oriented cohomology theory associated to the multiplicative formal group over Spec Z. In this talk, I'll discuss a variant of this algebraic picture which can be used to recover equivariant complex K-theory as well as equivariant elliptic cohomology , and explain its relationship with the classical character theory of finite groups and various generalizations thereof.

We show that closed subsets of the character variety of a normal complex variety, which are p-adically integral and Galois invariant, are motivic. Joint with Mortiz Kerz.

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Not really derived, clearly geometric, and p-adic. I will review my geometrization conjecture of the local Langlands correspondence and my ongoing work with Peter Scholze about the construction of local Langlands parameters. The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic K-theory which is notoriously hard to compute, in representation theory it takes the form of classical character theory.

Geometrical Applications of Calculus (1 of 4: An Introduction to the applications of calculus)

Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character.

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In the categorified picture algebraic K-theory is replaced by the category of non-commutative motives. It turns out that the categorified Chern character has many interesting applications.


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For instance we show that the DeRham realisation functor is of non-commutative origin. This is joint work with Michael Larsen and Ayelet Lindenstrauss. However, the existing proofs of this fact are not simple at all.

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Location MSRI: Simons Auditorium Video Abstract I will review the theory of topological group actions on complex-linear categories and their algebraic calculus of characters, with old and new applications to Gromov-Witten theory and dualities in mathematical physics. Supplements Notes 4. Location MSRI: Simons Auditorium Video Abstract I will discuss some recent results on the p-adic algebraic K-theory of p-adic rings, obtained using the cyclotomic trace from K-theory to topological cyclic homology.

Location MSRI: Simons Auditorium Video Abstract We show that closed subsets of the character variety of a normal complex variety, which are p-adically integral and Galois invariant, are motivic. Derived Algebraic Geometry. Characters of categorical representations: theory and applications Constantin Teleman University of California, Berkeley.

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