Mini Course 2016


  1. Course of Michel Brion:

    By a result of Grothendieck, the commutative algebraic groups over a field k are the objects of an abelian category C, with morphisms being the homomorphisms of algebraic k-groups. If k is algebraically closed, then by work of Serre and Oort, the homological dimension of C is 1 in characteristic 0 and 2 in positive characteristics; here the homological dimension is the smallest integer n such that all Ext groups vanish in degrees > n. The lectures will address the category of commutative algebraic groups up to isogeny, defined as the quotient of C by the Serre subcategory F of finite groups. In particular, we will show that the homological dimension of C/F is 1 for any field k. The proof of this uniform result is based on structure theorems for commutative algebraic groups, which take very different forms in characteristic 0 and in positive characteristics.

  2. Course of Bill Jacob:

    Cohomological tools have been crucial to the development of the theory of quadratic forms and division algebras. The cohomological interpretation of the Brauer group and the related descriptions of the graded Witt group as Galois cohomology are basic examples. In 1982 Kazuya Kato was able to resolve the Milnor problem in characteristic two when he computed the graded Witt group as cohomology using a careful analysis of differential forms and this opened up a new techniques. Voevodsky’s subsequent solution to the Milnor problem away from characteristic two required development of substantially different tools. Depending upon the question, the case of finite characteristic can be easier or more difficult, and work in finite characteristic has led to interesting examples and approaches. These lectures will focus on finite characteristic and start by examining recent applications of differential forms to problems involving quadratic forms in characteristic two and division algebras in characteristic p. They will then turn to use of the de Rham Witt complex and groups introduced by Izhboldin in determining Witt and Cohomological kernels under algebraic extensions. The emphasis will be on developing the basic theory, where the machinery gives groups are no longer p-torsion and this feature is extremely helpful to capture arithmetic that otherwise washes out when only using differential forms or cohomology mod p. The lectures will assume familiarity with the basic algebraic theory of quadratic forms, division algebras, the Brauer groups as well as Witt vectors. Much of the recent work described in the lectures is joint with Roberto Aravire and Manuel O’Ryan.

  3. Course of Max Karoubi:

    Hermitian K-theory has at least two origins. Surgery on non simply connected manifolds yields C.T.C. Wall to introduce subtle invariants associated to group algebras, usually noncommutative. The second origin, much older, is the rich theory of quadratic forms over fields.
    The first part of the course is devoted to basic definitions of these invariants and the "higher" ones associated to Quillen's + construction applied to classifying spaces of suitable orthogonal groups.
    Afterwards, we shall make a digression in topological K-theory in order to show that "Hermitian K-theory" is a nice unified framework for the formulation of the 10 fundamental homotopy equivalences of Bott which in turn imply his famous periodicity theorems. Note that the first proof of these equivalences were relying on Morse theory.
    Our more algebraic viewpoint leads to what we call "the fundamental theorem of Hermitian K-theory" which has many applications, besides Bott periodicity. For instance, we generalize to any ring, even noncommutative, the classical Hasse-Witt invariant for quadratic forms over fields.
    At the end we shall give two recent applications of the theory
    1) The analog of Quillen-Lichtenbaum conjecture in the Hermitian framework which is related to a description of the "Hermitian spectrum" as an homotopy fixed spectrum of ℤ/2 acting of the "K-theory spectrum".
    2) A description of the Witt ring associated to a real algebraic variety.

Short talks:

  1. Talk of Adam Chapman:

    We discuss the different types of linkage between two given cyclic division p-algebras of prime degree, the left and the right, which correspond to sharing cyclic Galois and purely inseparable field extensions of the center. We show some connections between them and present some open questions.

  2. Talk of Uriya First:

    Let R be a discrete valuation ring with fraction field F. Two algebraic objects (say, quadratic forms) defined over R are said to birationally isomorphic if they become isomorphic after extending scalars to F. In the case of unimodular quadratic forms, it is a classical result that rational isomorphism is equivalent to isomorphism. This has been recently extended to "almost unimodular" forms by Auel, Parimala and Suresh. We will present further generalizations to hermitian forms over hereditary R-orders and quadratic spaces equipped with a group action ("G-forms"). The results can be regarded as versions of the Grothendieck-Serre conjecture for certain non-reductive groups. We also discuss some surprising connections with Bruhat-Tits Theory and give a general conjecture when dim R=1.

  3. Talk of Matthias Grüninger:

    We use Nagao's Theorem to construct matrix subgroups of dimension $2$ over skew polynomial rings acting sharply transitively on the projective line of the quotient skew field. This way we get new examples of nearfields. Under additional conditions we prove that these nearfields are wild nearfields, by which we mean that they are not Dickson nearfields. To our knowledge these are the first known examples of nearfields of dimension greater than $2$ over their kernel for which it could be certainly proven that they are wild.

  4. Talk of Viacheslav Kopeyko

    Let R be a unitary ring (aka Bak’s form ring). In this talk we will define a transfer map K1 U(R[X]) → K1U(R[Xn]) for any integer n ≥ 2 and will compute the composition of this transfer with group homomorphism K1U(R[Xn]) → K1U(R[X]) induces by the canonical inclusion R[Xn] → R[X]. As consequences we will obtain the unitary K-analogies of Springer’s theorem (from algebraic theory of quadratic form) and Farrell’s theorem (from algebraic K-theory).

  5. Talk of Karsten Naert

    In the late 60s and early 70s, Steinberg and Tits discovered certain abstract groups that seem to be "algebraic groups, defined over two fields at once": the so called groups of mixed type. In this talk, we will make this notion precise, by embedding the category of schemes of characteristic p into a category of mixed schemes. The group objects in this category are what we call mixed algebraic groups and it turns out that their groups of rational points are precisely the abstract groups that are said to be of mixed type. There is also a deeper connection with the twisted groups (Suzuki and Ree groups) that we will explore.

  6. Talk of Cristian-Ioan Popa

    We will explain how to define equivariant hermitian K-Theory groups in a very general setting, formulated using the machinery of stacks and Marco Schlichting's dg (differential graded)catery with weak equivalences and duality formalism.

  7. Talk of Vivek Sadhu

    This is joint work with Charles Weibel. Let f:A → B be a commutative ring extension(or More generally, f:X → S be affine faithful). The relative Cartier divisor group I(f) of f is the group H0(S, f*O×X/OxS). We study the behavior of this group under polynomial and Laurent polynomial extensions. We show that I(f) is a contracted functor in the sense of Bass with contraction H0et(S, f*ℤ/ℤ).