Mini Course 2015


  1. Course of Jean Fasel:

    In these lectures, we will explain how to classify vector bundles on smooth affine schemes using the machinery of the A1-homotopy category. We will start with a quick introduction on the A1-homotopy category, and then explain the results of Morel on the classification of vector bundles. We will then explain how to extend these results using Postnikov towers, together with the known computations of the second non trivial A1-homotopy sheaf of the affine punctured space of a given dimension. Time permitting, we will also explain how to classify symplectic or orthogonal bundles.

  2. Course of Ulf Rehmann:

    The so called "lower" algebraic K-theory is very closely interrelated with the theory of Chevalley groups and - more general - with the theory of almost simple algebraic groups. The course will give an introduction into these topics.

    Tentative Lecture Titles:

    1. The group structure of SL_n over a field
    2. Chevalley groups over fields
    3. K_1, K_2 related to Chevalley groups
    4. Almost simple algebraic groups over arbitrary fields
    5. K-theoretic results related to simple algebraic groups

    Essential Bibliography:

    [BT] A. Borel, J. Tits, Groupes réductifs Publ. Math. IHES, 27 (1965), p. 55-151 http://www.numdam.org/item?id=PMIHES_1965__27__55_0 Compléments à l'article "Groupes réductifs". Publ. Math. IHES, 41 (1972), p. 253-276 http://www.numdam.org/item?id=PMIHES_1972__41__253_0

    [HO] Hahn, Alexander J.; O'Meara, O. Timothy, The classical groups and $K$-theory Grundlehren der Mathematischen Wissenschaften 291, 1989.

    [KMRT] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, AMS Colloquium Publications, Vol. 44, 1998. cf. http://www.math.uni-bielefeld.de/~rost/BoI.html

    [Mi] J. Milnor, Introduction to Algebraic K-Theory, Annals of Math. Study 72, 1971, Princeton University Press

    [Sp] T.A. Springer, Linear Algebraic Groups, 2nd edition, Birkhäuser, 1998

    [Ti] J. Tits, Classification of algebraic semisimple groups. Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 33–62 Amer. Math. Soc., Providence, R.I., 1966 Proc. Symp. Pure Math 18, 1966, 33-62 http://www.ams.org/books/pspum/009/

  3. Course of Yoav Segev:

    My interest in maps on division rings that resemble valuations started with a conjecture in algebraic group theory. In fact given an algebraic group (or any group G for that matter), the first question to ask is what are the normal subgroups of G. The conjecture that I got interested in is the Margulis-Platonov conjecture which is a conjecture about the normal subgroup structure of certain algebraic groups. The resolution of this conjecture in one important case (the inner forms of type An) was reduced to showing that if D is a finite dimensional division algebra over a global field and ND is a normal subgroup of finite index, then the group D×/N is not (abstractly) simple.

    In trying to show this I was led to consider the commuting graph of the group D×/N, and this graph (miraculously) led to a certain order (via inclusion) of what I sometimes call the shadow of elements of D\ N on N. In trying to better understand this ``miracle'' one is led to the notions of partially preordered and partially ordered groups, the notions of leveled, strongly-leveled, valuation-like and strong-valuation-like -- maps. These are certain maps from N to ``almost'' ordered groups. In my lecture series I will discuss all these notions and show how they reveal certain hidden treasures inside D (where now D can be any division algebra). I will also mention some generalizations, and a certain (notoriously difficult) related open problem.

Short talks:

  1. Talk of Christian Brown:

    Let D be an associative division algebra, σ a ring endomorphism of D and δ a left σ-derivation. We define the Skew Polynomial Ring R=D[t;σ,δ]to be the set of left polynomials a0+a1t+a2t2+...+antn, ai ∈ D, where addition is defined as usual and multiplication is defined by ta=σ(a)t+δ(a). In a little known paper by Petit 1966, these skew polynomial rings were first used in the construction of a nonassociative algebra Sf which we define as follows: Let f ∈ R=D[t;σ,δ] be of degree m and let Rm= { g∈ R : deg(g)< m }. Define a multiplication ο on Rm by a ο b=ab modrf where the juxtaposition ab denotes multiplication in R and modr denotes the remainder on right division by f. Then Rm becomes a nonassociative algebra which, following the work of Lavrauw and Sheekey in 2011, we denote by Sf. In the talk, we look at conditions for Sf to be a division algebra/ring and investigate the automorphisms of Sf.

  2. Talk of Adam Chapman:

    We extend the classical definition of the Clifford algebra of quadratic forms to make it include other types of algebras, such as the algebra determined by a quartic curve of genus one studied by Haile and Han.We provide an upper bound for the degree of its simple homomorphic images and compute its center.Some other related objects are discussed too.
    (This talk is based on a joint work with Danny Krashen and Max Lieblich.)

  3. Talk od Hassan Jolany:

    I start will Calabi's conjecture about finding canonical metrics on Polarized Fano varieties and try to explain Song-Tian theory about the finding of "best" metrics for varieties which do not have definite first Chern class. In final I give a Kobayashi–Hitchin correspondence version and Tian's K-stability in Sasakian geometric setting.

  4. Talk of Mélanie Raczek:

    Given a field F if characteristic different from 2, the Faddeev index of an exponent 2 central simple algebra A over F(t) is the minimum of the index of A⊗F(t)C for algebras C defined over F. The Faddeev index is related to the degree of the ramification sequence of the algebra. We define a quadratic form, called the Bezout form, associated with the ramification sequence. We study its properties and we give an upper bound for the Faddeev index, which is a decreasing function of the Witt index of the Bezout form. In particular, we prove that, if the ramification sequence has degree 4, then the Faddeev index of A is equal to 2 if and only if the Bezout form is isotropic.

  5. Talk of Sergei Sinchuk:

    In mid 70's W. van der Kallen showed for arbitrary commutative ring R that the universal central extension of the elementary group E(n, R) is exactly the Steinberg group St(n, R) provided n ≥ 4. We are going to present some recent results which solve a similar problem for symplectic groups and exceptional Chevalley groups of type El.