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Friday, April 27, 2012

Logic Seminar
4:00 pm   in 347 Altgeld Hall,  Friday, April 27, 2012
 Del Edit Copy
Submitted by phierony.
 Eva Leenknegt (Purdue)In search of p-adic minimality: An exploration of weak p-adic structuresAbstract: Consider a structure (F,L), where F is a field and L is a language that is 'related' to the language of rings, in the sense that the Lring-definable subsets of F coincide with the L-definable subsets of F (that is, we require (F,L) to be 'Lring-minimal'). When F is a real closed field, a structure satisfying this property will be o-minimal, so all tools of o-minimality, such as the cell decomposition theorem, are at our disposal when studying such structures. The situation is less clear when F is a p-adic(ally closed) field. If L is an expansion of the ring languages, then (F,L) will be P-minimal, but very little is known in general for weaker structures (reducts of the ring language) When will a reduct L of the ring language give rise to an Lring-minimal structure? In the o-minimal case, the answer is easy: the only requirement is that the order should be definable in L. Our first challenge will be to find a p-adic equivalent of this 'minimal language' (<). Once such a language has been found, one can start constructing examples of weak p-adic Lring- minimal structures. While a general theory is still far away, individual examples show that there are some fundamental differences when comparing p-adic and o-minimal reducts of the ring language. I will give some examples of this. One of the questions that comes up is the existence of cell decomposition: could it possibly be true that every p-adic Lring-minimal language has cell decomposition? I will discuss some (partial) answers to this question, and show how we can use this to study examples of weak structures.

Algebra, Geometry and Combinatoric
4:00 pm   in 341 Altgeld Hall,  Friday, April 27, 2012
 Del Edit Copy
Submitted by darayon2.
 Allen Knutson (Cornell)Manifold atlases consisting of Bruhat cellsAbstract: A Bruhat cell is a finite-dimensional affine space, but comes with many additional structures: a stratification, a torus action, a Poisson structure, a Frobenius splitting, a totally nonnegative part... When we have a manifold with these structures, we can ask whether it can be given an atlas of charts consisting of Bruhat cells. I'll give a construction of a Coxeter diagram D (sometimes) associated to a manifold M with a stratification Y. If the Bruhat atlas program can be carried out for M, it gives a poset antiisomorphism between Y and an order ideal in the Bruhat order W_D. We have checked this combinatorics ex post facto for partial flag manifolds and wonderful compactifications of groups. The full program is complete for the Grassmannian, where the diagram is the affine Dynkin diagram, by work of [K-Lam-Speyer] and [Snider]. For most other examples, D is neither finite nor affine. This work is joint with Xuhua He and Jiang-Hua Lu.