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# Aui2015Nxj

English

ŏIXV 2016.3.14

2014Nx
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 uҁij Marcos AlexandrinoiInstitute of Mathematics and Statistics, University of Sao Paulo, Brasil) ^Cg Introduction to Polar Foliations 2016N324i؁j (1) 9:30-10:30 (2) 11:00-12:00 2016N325ij (1) 9:30-10:30 (2) 13:00-14:00 ꏊ sw@w e@uA@(F404)
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(1)Lecture 1: Motivations and main definitions (where I should establish the main definitions and provide an overview using the concrete example of conjugation action)
(2)Lecture 2: Differential and geometrical aspects (Palais-Terng problem, slice theorem, equifocality of manifolds)
(3)Lecture 3: Dynamical aspects (Weyl groups their relation with fundamental groups, coxeter orbifold structure of the leaf spaces, and a few words about foliations with no closed leaf)
(4)Lecture 4: Singular theory: level sets and foliations (transnormal and isoparametric maps).
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 uҁij Pierre Baumann (CNRS, the Institut de Recherche Mathematique A vancee, Strasbourg) ^Cg Mirkovic-Vilonen polytopes in finite or affine type 2016N322i΁j (1) 10:30-12:00 (2) 14:00-15:30 (3) 16:00-17:30 2016N323ij (1) 10:30-12:00 (2) 14:00-15:30 (3) 16:00-17:30 ꏊ sww F2K 4uiF205j
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@This series of talks will focus on joint work Joel Kamnitzer(Toronto) and Peter Tingley (Chicago).
@Let us begin with two natural constructions that give birth to convex polytopes, some of the oldest mathematical objects. 1) Given a hamiltonian action of a compact torus T on a compact connected symplectic manifold, the image by the moment map of any closed T-stable subset is a convex polytope. 2) Given an abelian finite length category A and an object X of A, the classes in the Grothendieck group K_0(A) of the subobjects of X are finitely many; the convex hull of these classes in K_0(A) \otimes R is thus a convex polytope, called the Harder-Narasimhan (HN) polytope of X.
@Now we consider the following two constructions, apparently unrelated.Let g be a semisimple complex Lie algebra and let h be the Cartan subalgebra of g. For a dominant integral weight \lambda, we denote by V(\lambda) the irreducible g-module of highest weight \lambda. 1) The geometric Satake correspondence identifies V(\lambda) with the intersection homology of a certain Schubert variety X_\lambda, and certain cycles in X_\lambda, called Mirkovic-Vilonen (MV) cycles, form a basis of this intersection homology. Since there is a hamiltonian action in this situation, we obtain polytopes by taking the image of the MV cycles by the moment map. 2) Let \Lambda be the preprojective algebra built on the Dynkin diagram of g (supposed here to be simply laced). To each finite-dimensional \Lambda-module M, we associate its HN polytope.
@It is remarkable that both constructions yield polytopes whose edges are in root directions. In construction 2, this fact follows from Iyama and Reiten's tilting theory for modules over the preprojective algebra. Further, the polytopes from construction 1 can be obtained from construction 2 by taking M general enough in the representation space of \Lambda. The first lectures will be devoted to the explanation of this coincidence. The main point is that these polytopes can be described by combinatorial data arising from the Kashiwara crystals of the representations V(\lambda).
@The polytopes obtained here are called MV polytopes, because they were first discovered through construction 1. One may wish to extend the definition of these polytopes to the case of an affine Kac-Moody algebra g. We will use construction 2 to achieve this goal. One difficulty to be overcome is that tilting theory has not enough reach to describe the edges that are parallel to the imaginary roots: one must here use other tools like Hall functors, which embed the category of finite-dimensional representations of the preprojective algebra of the Kronecker quiver into 2-Calabi-Yau categories. At the end of the day, one understands that HN polytopes can be decorated by families of partitions. The last lectures will be devoted to a presentation of this construction.
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 uҁij Patrick DehornoyiUniversity of Caen, Francej ^Cg Using selfdistributivity for investigating and ordering braids 2015N518ij@ (1) 10:00-11:00 (2) 11:15-12:15 (3) 14:30-15:30 (4) 15:45-16:45 2015N519i΁j (1) 10:00-11:00 (2) 11:15-12:15 (3) 14:30-15:30 (4) 15:45-16:45 ꏊ wuiwE408j
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@ The connection between the selfdistributive law LD : x(yz) = (xy)(xz) (or its symmetric counterpart) and Reidemeister move III is well-known. So, given an LD-system (or shelf) S, that is, any structure satisfying the LD law, it is natural to try to use S to find topologically interesting results. However, requiring compatibility with Reidemeister moves I and II leads to restricting to LD-systems of a (very) particular type, namely racks and quandles. Otherwise, technical obstructions appear, and seem to discard all topological applications. The aim of the minicourse is to explain how, at least in one case, it is possible to overcome the obstructions and obtain a sort of Reidemeister II compatibility for shelves that are not racks, namely free shelves in that specific case. This approach led to the orderability of Artin's braid groups, and our hope is that the techniques that will be explained can be used again and lead to applications for the (many) known shelves that are neither quandles, nor even racks.

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