Lecture 1-Introduction ** Eilenberg-Steenrod axioms in algebraic geometry** Esaga Oberseminar Thursday, April 24, 16:45-17:45

Lecture 2 ** Premotivic categories after Deligne and Ayoub-Voevodsky.** Friday, April 25, 14-16 Uhr, N-U-3.04

Lecture 3 ** ****MGL-modules and Riemann-Roch formulas. **Monday, April 28, 16-18 Uhr, N-U-4.04

Lecture 4 ** ****Modules over K-theory and Beilinson motives.** Tuesday, April 29, 14-16 Uhr, N-U-4.04

Lecture 5 ** Integral motivic complexes. **Wednesday, April 30, 14-16 Uhr, N-U-3.01

** Abstracts**

Lecture 1-**Eilenberg-Steenrod axioms in algebraic geometry.** In this talk I will illustrate how motivic homotopy theory is a deep renewal of the inspiration from algebraic topology to the cohomology theory of algebraic varieties and schemes. Starting from the current state of stable homotopy theory in topology, I will show how one can naturally generalize the classical Eilenberg-Steenrod axioms to get a new axiomatic of a Weil cohomology. This naturally introduces the stable homotopy theory of Morel and Voevodsky, as well as it gives a picture of the central role of motivic cohomology. In the last part of the talk, I will broaden the landscape using the inheritage of Grothendieck 6 functors formalism and explain how an extension of the preceding ideas lead to a natural notion of derived syntomic coefficients. (The first part describes a work obtained in collaboration with D.-C. Cisinski and the second one a work in progress with W. Niziol.)

Lecture 2-**Premotic categories after Deligne and Ayoub-Voevodsky. **I will present the axioms and examples of premotivic categories (abelian, with model structures) and dwell more deeply on the notion of triangulated premotivic categories: the aim is to explain the construction of Deligne of exceptional functors as well as the theorem of Ayoub establishing the complete 6 functors formalism. I will then present more details on the premotivic model category case, giving examples of interesting applications (to projective limits and descent theory), and introduce the main theme of this series of talks:rings and modules over them.

Lecture 3-**MGL-modules and Riemann-Roch formulas. **I will recall basics about oriented ring spectra - the analogs of the classical theory in algebraic topology and explains how to get Riemann-Roch formulas in the algebraic geometrical setting. Thne I will explain how the use of ring spectra allows to get reformulations and extensions of these formulas.

Lecture 4-**Modules over K-theory and Beilinson motives.** This talk will be concentrated on the K-theory ring spectrum KGL, as well as on the two important theorems: the absolute purity and the h-descent theorems. Then I will introduce Beilinson's motivic cohomology, ring spectrum HB, and explain its main property which allows to completely elucidate the Bousfield localization of the stable homotopy category with respect to HB. I will sketch the relation of this localization with respect to motivic complexes and transfers "&agrav; la Voevodsky".

Lecture 5-**Integral motivic complexes. **Integral motivic complexes (if time allows) I will explain the extension of a theorem of Rondigs and Ostvaer comparing (stable) motivic complexes with modules over integral motivic cohomology to the case of an arbitrary equicharacteristic scheme, provided you invert the characteristic in the coefficients and you modified slightly the definition of motivic complexes (when the scheme is singular).