Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I explain how to define the motive of certain algebraic stacks. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers. If there is time, I will give a conjectural formula for this motive, and explain how this follows from a conjecture on the intersection theory of certain Quot schemes. This is joint work with Simon Pepin Lehalleur.[-]

Moduli spaces of Higgs bundles for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SL_n and PGL_n, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by Groechenig-Wyss-Ziegler and later by Maulik-Shen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen's approach to the Hausel-Thaddeus conjecture, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. The same idea also enables us to prove a motivic chi-independence result. If there is time, I will explain how motivic nearby cycles can be used to specialise these results to positive characteristic. This is joint work with Simon Pepin Lehalleur.[-]

Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to understand. One then construct maps called "cycle class maps" from Chow groups to several cohomological theories.
In this talk, we focus on the case of a variety $X$ over a finite field. In this case, Tate conjecture claims the surjectivity of the cycle class map with rational coefficients; this conjecture is still widely open. In case of integral coefficients, we speak about the integral version of the conjecture and we know several counterexamples for the surjectivity. In this talk, we present a survey of some well-known results on this subject and discuss other properties of algebraic cycles which are either proved or expected to be true. We also discuss several involved methods.[-]

I will discuss the new ?subtle? version of Stiefel-Whitney classes introduced by Alexander Smirnov and me. In contrast to the classical classes of Delzant and Milnor, our classes see the powers of the fundamental ideal, as well as the Arason invariant and its higher analogues, and permit to describe the motives of the torsor and the highest Grassmannian associated to a quadratic form. I will consider in more details the relation of these classes to the J-invariant of quadrics. This invariant defined in terms of rationality of the Chow group elements of the highest Grassmannian contains the most basic qualitative information on a quadric.[-]

A variety X is stably rational if a product of X and some projective space is rational. There exists examples of stably rational non rational complex varieties. In this talk we will discuss recent series of examples of varieties, which are not stably rational and not even retract rational. The proofs involve studying the properties of Chow groups of zero-cycles and the diagonal decomposition. As concrete examples, we will discuss some quartic double solids (C. Voisin), quartic threefolds (a joint work with Colliot-Thélène), some hypersurfaces (Totaro) and others.[-]

Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational nonrational varieties. This problem remained open till 1970th, when three types of such examples were produced: cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin), and some conic bundles (Artin et Mumford). The last examples are even not stably rational. The stable rationality of the first two examples was not known.
In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.[-]

Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational nonrational varieties. This problem remained open till 1970th, when three types of such examples were produced: cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin), and some conic bundles (Artin et Mumford). The last examples are even not stably rational. The stable rationality of the first two examples was not known.
In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.[-]

I will sketch the construction - following ideas of Kontsevich and Nori - of a Tannakian category of exponential motives over a subfield of the complex numbers. It is a universal cohomology theory for pairs of varieties and regular functions, whose de Rham and Betti realizations are given by twisted de Rham and rapid decay cohomology respectively. The upshot is that one can attach to any such pair a motivic Galois group which conjecturally generalizes the Mumford-Tate group of a Hodge structure and, over number fields, governs all algebraic relations between exponential periods. This is a joint work with Peter Jossen (ETH).[-]