De Rham realizations of mixed motives over general base schemes
Friday, 19.4.13, 10:00-11:00, Raum 404, Eckerstr. 1
After briefly reviewing the construction of the stable homotopy category SH(X) of a scheme X and the associated formalism of Grothendieck's six functors, I explain how to construct de Rham realization functors from SH(X) into an ind-completion of the bounded derived category of holonomic DX-modules when X is a smooth, quasi-projective C-scheme. As a corollary, the classical Betti-de Rham comparison theorem furnishes a purely algebraic proof of the Riemann-Hilbert correspondence between the full subcategories of D^bc(X(C),C) and D^bhol(DX)$ spanned by the complexes ``of geometric origin''.\n\n
Quantum product for derived categories
Friday, 10.5.13, 10:00-11:00, Raum 404, Eckerstr. 1
Quantum cohomology ring of a smooth projective variety X is a certain deformation of its usual cohomology ring. This structure was introduced at the begging of 90's motivated by works of string theorists. Later on an analogue of the quantum product was defied in the K-theory. In this talk I will describe a way to define an analogue of the quantum product on the derived category of X.\n\n
Rational Motives over deeper bases
Friday, 31.5.13, 10:00-11:00, Raum 404, Eckerstr. 1
We begin with a brief introduction to motivic homotopy\ntheory and motives. Roughly, one notes that cohomology theories for schemes can be approached in close analogy to those for topological spaces: They factorize through a homotopy category, then through a stable homotopy category where they become representable by some object ("a spectrum"). If this object is a "ring spectrum", they\nfactor further through the category of modules over that ring spectrum - if one starts with motivic cohomology, the latter is the category of motives.\n\nFor a variety of reasons there have been proposed many alternative notions of scheme, e.g. to overcome the asymmetry between the "finite primes" and the "infinite primes" of a number field (\(\bmathbb{F}_1\)-geometry), to construct cohomology theories for spaces ("derived algebraic geometry") or to handle Frobenius lifts ("lambda\nalgebraic geometry"). In this talk we will present a way to construct motives for such alternative schemes: We construct a stable homotopy category and a K-theory spectrum therein, give a rational decomposition, pick a summand (the "alternative Beilinson spectrum") and pass to modules over it. The construction is compatible with base\nchange and admits a different description in terms of the positive rational motivic sphere.
Newton-Okounkov bodies, vanishing sequences, and diophantine approximation
Friday, 21.6.13, 10:00-11:00, Raum 404, Eckerstr. 1
We will study global sections of line bundles on projective varieties. Newton-Okounkov bodies are a useful tool for handling all global sections of all multiples of a given line bundle at the same time via convex geometry in Euclidean spaces. We go one step further and study functions on Newton-Okounkov bodies that come from valuations on the underlying function field. It turns out that a variation of this theme \nhas led McKinnon and Roth to very interesting results in diophantine approximation. This is an account of joint work with Sebastien Boucksom, Catriona Maclean, and Tomasz Szemberg.
Homological smoothness of equivariant derived categories
Friday, 28.6.13, 10:00-11:00, Raum 404, Eckerstr. 1
We introduce the notion of (homological) G-smoothness for a complex G-variety X. If there are only finitely many G-orbits and all stabilizers are connected, we show that X is G-smooth if and only if each orbit is isomorphic to C^n.
Über die Kohomologie endlicher Chevalley-Gruppen in beschreibender Charakteristik
Friday, 5.7.13, 10:00-11:00, Raum 404, Eckerstr. 1
Von grossem Interesse für Algebraiker und Topologen sind die Kohomologieringe der endlichen Gruppen. In meinem Vortrag beschränke ich mich auf die endlichen Chevalley-Gruppen. Im Allgemeinen weiss man nicht viel über diese Ringe. Man kennt nicht einmal in allen Fällen den kleinsten positiven Grad nicht-verschwindender Kohomologieklassen. Quillen war der erste, der dieses Problem in den siebziger Jahren in Angriff nahm.\n\nObwohl man die Kohomologieringe der algebraischen Gruppen und ihrer Frobenius-Kerne besser versteht, gibt es auch hier viele offene Fragen. An Hand von Beispielen erkläre ich neue Methoden, die diese beiden Theorien verbinden und zu bisher nicht bekannten Resultaten für die endlichen Gruppen führen. Dieser Vortrag basiert auf gemeinsamen Arbeiten mit Chris Bendel und Dan Nakano.
Feynman graphs, their motives and torus actions
Friday, 12.7.13, 10:00-11:00, Raum 404, Eckerstr. 1
One of the main problems in Quantum Field Theory is the\ncomputation of the coefficients of the renormalized Dyson series, which appears for instance in the pertubative expansion of the scattering matrix. These coefficients are given by an integral of a differential form which is determined by specifying a certain labeled graph. This integral is called a Feynman amplitude and the labeled\ngraphs are called Feynman diagrams. They come with an algebro-geometric object, the so called graph hypersurface. I will review the basic notions and the general theme of motives associated to Feynman diagrams as it was originated in [Bloch, Esnault, Kreimer: 2006]. Then I will discuss the problem of finding and using \(\bmathbb{G}_m\)-actions\non the graph hypersurfaces to compute the associated motives as well as introducing a class of graphs where the associated graph hypersurface admits a torus action of maximal dimension.\n
Weil-etale cohomology and Zeta functions of arithmetic schemes
Friday, 19.7.13, 10:00-11:00, Raum 404, Eckerstr. 1
We report on joint work with Baptiste Morin on a\ndescription of values\nof Zeta functions of arithmetic schemes in terms of\nWeil-etale\ncohomology complexes, extending the original ideas of\nLichtenbaum and\nGeisser to all arithmetic schemes and all integer\narguments.