**Zeit:** Donnerstags, 14-16 Uhr c.t.

**Ort:** SR E31 Arnimallee 7

Datum | |
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23. April | Laurent Moret-Bailly (Rennes) Title: The topology of torsors over valued fields This is joint work with Philippe Gille and Ofer Gabber; available at http://algebraicgeometry.nl/2014-5/2014-5-025.pdf. Abstract: If K is a valued field (e.g. a local field) and X is a K-variety, the set X(K) has a natural topology induced from the valuation. I will discuss topological properties of the map X(K)--> Y(K) induced by a K-morphism f:X-->Y, in the special case where f is a G-torsor, for an algebraic group G over K. |

28. April (Dienstag), 12:30 UhrRaum: SR 005/A3 |
Bas Edixhoven (Leiden) Title: On Gauss's theorem on sums of 3 squares Abstract: Gauss has shown, for example, that for a positive square free integer n that is 1 modulo 4 the number of solutions in integers of x^2+y^2+z^2=n equals 12 times the class number of the ring Z[t]/(t^2+n). Gauss's proof is long. The aim of the talk is to give a short proof, using the action of the group scheme SO(3). This proof shows in fact that the class group in question naturally acts freely and transitively of the set of solutions modulo SO(3)(Z). This action could be called Gauss composition on spheres. These results are work in progress of Albert Gunawan, PhD student with me and Qing Liu. |

21. Mai, 14 UhrSR140 A7 (!!) |
Andre Chatzistamatiou (MPIM Bonn) Title: Relative de Rham-Witt stratifications and connections Abstract: The ring of Witt vectors of a smooth ring is a rather singular ring. In order to obtain familiar cohomology groups one replaces the de Rham complex by the de Rham-Witt complex. In this talk we will define the category of relative de Rham-Witt stratifications and show that it is equivalent to the category of integrable (relative) de Rham-Witt connections. This is analogous to the equivalence between pd-stratifications and integrable connections for a smooth ring. Our constructions will be relative to the ring of Witt vectors of the integers in contrast to an absolute setup where the ring of integers is the base. |

21. Mai, 15 UhrSR140 A7 (!!) |
Doan Trung Cuong (Institute of Mathematics, VAST) Title: Weierstrass Preparation and u-invariant of some function fields Abstract: In this talk we define a notion of Weierstrass ring extension and present a characterization involving fibers of the extension. This applies to local rings of a projective curve over certain one-dimensional local rings. Some consequences on bounding the u-invariant of some function fields wil be discussed. |

4. Juni | Jakob Stix (Uni Frankfurt) Title: Anabelian geometry with étale homotopy types Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt. |

30. Juni 12:30-14:00 |
Takeshi Saito (Tokyo University) Title: The characteristic cycle and the singular support of an etale sheaf I |

30. Juni 16:00-17:30 |
Takeshi Saito (Tokyo University) Title: The characteristic cycle and the singular support of an etale sheaf II |

2. Juli 14:15-15:15 |
Takeshi Saito (Tokyo University) Title: The characteristic cycle and the singular support of an etale sheaf III |