Tue, 19 Feb 2019
14:15
L4

Arithmetic D-modules over Laurent series fields

Daniel Caro
(Caen)
Abstract

Let k be a characteristic $p>0$ perfect field, V be a complete DVR whose residue field is $k$ and fraction field $K$ is of characteristic $0$. We denote by $\mathcal{E}  _K$ the Amice ring with coefficients in $K$, and by $\mathcal{E} ^\dagger _K$ the bounded Robba ring with coefficients in $K$. Berthelot's classical theory of Rigid Cohomology over varieties $X/k((t))$ gives $\mathcal{E}  _K$-valued objects.  Recently, Lazda and Pal developed a refinement of rigid cohomology,
i.e. a theory of $\mathcal{E} ^\dagger _K$-valued Rigid Cohomology over varieties $X/k((t))$. Using this refinement, they proved a semistable version of the variational Tate conjecture. 

The purpose of this talk is to introduce to a theory of arithmetic D-modules with $\mathcal{E} ^\dagger _K$-valued cohomology which satisfies a formalism of Grothendieck’s six operations. 
 

Tue, 08 May 2018

14:00 - 15:00
L5

Discontinuous Galerkin method for the Oseen problem with mixed boundary conditions: a priori and aposteriori error analyses

Nour Seloula
(Caen)
Abstract

We introduce and analyze a discontinuous Galerkin method for the Oseen equations in two dimension spaces. The boundary conditions are mixed and they are assumed to be of three different types:
the vorticity  and the normal component of the velocity are given on a first part of the boundary, the pressure and the tangential component of the velocity are given on a second part of the boundary and the Dirichlet condition is given on the remainder part . We establish a priori error estimates in the energy norm for the velocity and in the L2 norm for the pressure. An a posteriori error estimate is also carried out yielding optimal convergence rate. The analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini.

Tue, 08 May 2007
17:00
L1

Cluster algebra structures on co-ordinate ring of flag varieties

Prof. Bernard Leclerc
(Caen)
Abstract
  Let G be a complex semisimple algebraic group of type A,D,E. Fomin and Zelevinsky conjecture that the coordinate rings of many interesting varieties attached to G have a natural cluster algebra structure. In a joint work with C. Geiss and J. Schroer we realize part of this program by introducing a cluster structure on the multi-homogeneous coordinate ring of G/P for any parabolic subgroup P of G. This was previously known only for P = B a Borel (Berenstein-Fomin-Zelevinsky) and when G/P is a grassmannian Gr(k,n) (J. Scott). We give a classification of all pairs (G,P) for which this cluster algebra has finite type. Our construction relies on a finite-dimensional algebra attached to G, the preprojective algebra introduced in 1979 by Gelfand and Ponomarev. We use the fact that the coordinate ring of the unipotent radical of P is "categorified" in a natural way by a certain subcategory of the module category of the preprojective algebra.  
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