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J. Ayoub:
Motivic cohomology, A^1-homotopy theory and algebraic
cobordism.
ABSTRACT:
First, I speak about motivic cohomology and Voevodsky's category of motives.
Then I pass to the non-abelian version, i.e., the basis of Morel-Voevodsky's A^1-homotopy theory.
Then, I describe the stabilization process which yields the stable homotopy category of Tate spectra.
The latter contains an important object: the algebraic cobordism spectrum. If time permits,
I'll mention the approach by Levine and Morel which yields a more geometric algebraic cobordism ring.
L. Barbieri Viale: Sharp Cohomology.
ABSTRACT:
I will introduce formal Hodge structures pointing to "sharp" singular
cohomology of a complex algebraic variety and also "sharp" de Rham
over a field of zero characteristic. Sharp singular cohomology is a
formal Hodge structure containing, in the underlying algebraic
structure, a formal group which is an extension of ordinary singular
cohomology mixed Hodge structure. I take care of introducing 1-motives
and "sharp" realizations. Also "sharp" mixed motives are expected. I
could mention the still conjectural picture for the forthcoming sharp
cohomologies (non homotopical invariant) of schemes.
B. Chiarellotto:
Weil cohomologies and Bloch-Ogus axioms.
System of realizations and compatibilities.
ABSTRACT:
We will deal with "classical" cohomologies theories in algebraic geometry
as crystalline, etale, de Rham, Deligne cohomology, syntomic.
We will focus on their compatibilities
and properties in the framework of the Bloch-Ogus system of axioms.
We ask the students to be familiar with:
-basic arithmetic algebraic geometry
-sheaf theory
-homological algebra (derived functors and basic derived categories theory)
- Chiarellotto Lecture 1
- Chiarellotto Lecture 2
- Chiarellotto Lecture 3
- Chiarellotto Lecture 4
N. Naumann (Regensburg) - G. Powell (Paris 13): Stable Homotopy and Elliptic Cohomology.
ABSTRACT:
1) Foundations of stable homotopy theory: Spectra and the stable
homotopy category; smash product of spectra; (co)homology and Brown
representability. Examples: ordinary cohomology; K-theory; complex
cobordism. Stable cohomology operations.
2) Elliptic cohomology theories: Complex oriented theories, formal
group laws and Quillen's theorem; Landweber's exact functor theorem;
Lubin-Tate spectra and 'classical' elliptic cohomology. The
multiplicative spectra project: the Hopkins-Miller theorem and tmf.
The Witten genus (string orientation).
- Powell Lecture Notes
References:
- Naumann Lecture 1
- Naumann Lecture 2
- Naumann Lecture 3
V. Vologodsky (Oregon): Introduction to Hodge theory.
ABSTRACT:
I will explain basic concepts and results from Deligne's "Theorie de Hodge I,II, III"
Applications will include: the Invariant Cycle Theorem, the existence of virtual Betti numbers, and the Tian-Todorov Theorem.
At the end I will discuss some ideas from Non-ablelian Hodge theory of Simpson and recent applications to Langlands program.