Thomas Agugliaro
Hodge standard conjecture for powers
Abstract: The Hodge standard conjecture predicts positivity of intersection forms on algebraic cycles. It was formulated by Grothendieck in the Sixties, motivated by an intersection theoretic proof of the Weil bound for curves over finite fields. Only recently some progress has been made, based on p-adic Hodge theory. As most conjectures on algebraic cycles, it behaves badly under powers. In this talk, we will investigate this question and prove the conjecture for powers of abelian varieties of dimension 3.
Adam Chapman
Karpenko’s construction of indecomposable algebras and their applications
Abstract: Let p be a prime integer and F be a field over which x^p−1 decomposes into linear factors. The works of Teichmüller and of Merkurjev and Suslin showed that every division F-algebra of exponent p is Brauer equivalent to a tensor product of cyclic algebras of degree p, but this equivalence is not necessarily an isomorphism. And indeed, Karpenko produced examples of indecomposable algebras of exponent p and degree p^n for all cases except p=n=2, to which Albert’s decomposition theorem applies. The significance of Karpenko’s results extends beyond the mere challenge of constructing such algebras, and applications of various kinds appeared in the following years. We will discuss two such applications – in computing the essential dimension of central simple algebras of degree pn and exponent p, and in disproving the excellence of biquadratic extensions of mixed type in characteristic 2. The talk is based on joint works with Kelly McKinnie and with Fatma Kader Bingöl and Ahmed Laghribi.
Vladimir Chernousov
Loop and toral torsors over schemes in positive characteristic
Abstract: We discuss properties of loop and toral torsors defined over schemes in positive characteristic case. Joint work with P. Gille and A. Pianzola.
Stefan Gille
Recent progress on the Gersten conjecture for Witt groups
Abstract: After an overview of the history of the Gersten conjecture for Witt groups I will talk about recent joint work with Ivan Panin, where we proved the conjecture in the case that the regular local ring is essentially smooth over a discrete valuation ring.
Detlev Hoffmann
The quadratic Zariski problem
Abstract: The classical Zariski Problem asks whether two varieties of the same dimension that are stably birationally equivalent are in factbirationally equivalent. While generally this is not true, this is an open problem in the case where both varieties are given by quadrics. Jack Ohm called this the Quadratic Zariski Problem (QZP). No counterexamples are known. However, over certain base fields, e.g. number fields with at most one real place, or for quadrics given by certain types of quadratic forms, e.g. quadratic forms of dimension at most 7, QZP has a positive answer. We give a survey of some of the known results.
Nico Lorenz
Pfister numbers over valued fields
Abstract: Let F be a field of characteristic not 2. The n-th power of the fundamental ideal I^n(F) in the Witt ring of F is the ideal generated by the Witt classes of the n-fold Pfister form. For a given form q whose Witt class lies in I^n(F), we investigate the minimal number k such that there are n-fold Pfister forms p_1, .., p_k whose orthogonal sum is Witt equivalent to q, called the n-Pfister number of q. In this talk, we survey known results for n <= 3. Further we present recent results on the connection between Pfister numbers of a discretely valued field and its residue field, in particular with respect to forms of low dimension and the asymptotic behaviour for growing dimension.
Alexander Merkurjev
Negligible cohomology and cohomological invariants of finite groups
Abstract: We will discuss negligible cohomology in the sense of Serre, cohomological invariants of finite groups and the embedding problem for fields.
Federico Scavia
Brauer classes not split by genus one curves
Abstract: We show that there exist Brauer classes over a field F which are not split by any genus one curve over F. This answers a question of Clark and Saltman. Joint with Zinovy Reichstein.
Stephen Scully
On Symmetric Bilinear Forms of Height 2 in Characteristic 2.
Abstract: The height of a nondegenerate symmetric bilinear form b over a field F is the maximal length of an irredundant splitting tower for b. The anisotropic forms of height 1 over F are the general Pfister forms and their codimension-1 subforms. The problem of classifying the anisotropic forms of height 2 already represents a significant challenge, being closely related to some of the major technical problems in the subject. When the characteristic of F is not 2, nondegenerate symmetric bilinear forms may be identified with nonsingular quadratic forms, enabling one to bring algebraic-geometric tools to bear. Here, past work of Vishik and Karpenko has given partial results supporting a conjectural classification of the height-2 forms due to Kahn. In this talk, I will discuss the special case where F has characteristic 2, which can be approached in a more direct way. Here, additional examples appear, and the problem is more broadly related to that of understanding the forms of low dimension in the powers of the fundamental ideal in the Witt ring.
Pavel Sechin
Descent of motives along functional field extensions
Abstract: How can one check that a motive defined over some field extension k(X) is a pullback of a motive over k? I will give a partial answer to this question, which is enough to categorify Galois cohomology with finite coefficients as invertible motives for Morava K-theory.
Fabio Tanania
The isotropic motivic fundamental group
Abstract: Let X be a smooth variety over a field k, and denote by MBP^iso the isotropic motivic Brown-Peterson spectrum. In this talk, I will present results on the category of cellular MBP^iso -modules (which we also refer to as isotropic Tate motives) over X. In particular, I will show how to equip this category with a motivic t-structure whose heart is a Tannakian category. This result allows us to define a new invariant, called the isotropic motivic fundamental group of X, and to identify isotropic Tate motives over X that lie in the heart with its representations. I will conclude by presenting explicit computations in the cases of the punctured projective line and split tori.
Alexander Vishik
Pure symbols and numerical equivalence of cycles
Abstract: I will discuss the result claiming that numerical equivalence of algebraic cycles with Z/2-coefficients is controlled by pure symbols in Milnor's K-theory mod 2.
Kirill Zaynullin
Rost nilpotence for twisted Milnor hypersurfaces
Abstract: This is a joint project with Charles de Clercq and Evan Marth. We show that the strong Rost nilpotence (in the sense of S. Gille) holds for motives of generic hyperplane sections of twisted Milnor hypersurfaces. Hence, we provide a new family of examples of smooth projective algebraic varieties which satisfy the strong Rost nilpotence principle. As an application, we compute the p-canonical dimension for such varieties.
Maksim Zhykhovich
The J-invariant of symplectic involutions
Abstract: The J-invariant was originally introduced by Vishik for quadratic forms and later generalized to semisimple algebraic groups of inner type by Geldhauser, Petrov, and Zaynullin. The J-invariant of a group G is a discrete invariant which encodes the Chow-motivic decomposition of the variety of Borel subgroups in G.In the case of groups of type C_n, corresponding to symplectic involutions on central simple algebras, however, the J-invariant reflects only the underlying algebra and does not capture information about the involution itself. In this talk, I will explain how the definition of the J-invariant can be modified using normed Chow motives in order to resolve this problem. The talk is based on joint work with Geldhauser and Hendrichs.