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.