some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

morphisms of f isocrystals and the finite monodromy theorem for unit root f isocrystals we discuss tate type problems for f isocrystals that is the full faithfulness of the natural restriction functors between categories of overconvergent f isocrystals on schemes of positive characteristic we prove it in the cases of unit root f isocrystals using this result we prove that an overconvergent unit root f isocrystal has a finite monodromy

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

frobenius splitting of schubert varieties and linear syzygies introduction let g be a connected simply connected semi simple alge braic group let b be a borel subgroup of g for an ample line bundle l on g b the questions about projective normality of the embedding of g b in the complete linear system of l and the equations defining the embedding were extensively studied ramanathan in 13 proved that the ideal defining the embedding of g b in the complete linear system of l is generated by quadratic equations kempf and ramanathan proved that the equations defining the embedding of g b in the complete linear system of l lr are quadratic and that a schubert variety x is linearly defined with respect to l lr ramanathan had conjectured that the method of frobenius splitting used in 13 can be used to study higher syzygies let r bh g b ljj where the vector mj ranges in nr be the multigraded coordinate ring of g b with respect to l lr in this paper we use frobenius splittings to study the higher syzygies of the irrelevant maximal ideal m of r we prove that m as a module has linear syzygies our methods also give simpler proofs of the results obtained in 6 and 13 let z be a projective variety defined over an algebraically closed field let l l be ample line bundles on z and let m denote the irrelevant maximal ideal of the multi graded ring r h z ljj in the first section we give a criterion for m to have linear syzigies over r we show that if for every n 2 the first cohomology of the n fold product zn with values in certain ideal sheaves tensored with powers of lj vanishes then m has linear syzygies propositions 1 9 and 1 10 ye also give a similar criterion for a homogeneous quotient ring of r to have linear syzygies over r proposition 1 13 in the second section we show that any r ample line bundles l l on g b and their restrictions to a schubert variety x c g b over an algebracally closed field of positive characteristic satisfy the above criteria to prove this result we use the method of frobenius splittings if z is a frobenius split variety and if l is an ample line bundle on z then we have hl z l 0 for 1 further if y is a compatibly split subvariety of z then the restriction

classification of minimal representations of real simple lie groups based on an idea by gan and savin represent theory 9 46 93 2005 we give a classification of minimal representations of connected simple real lie groups not of type a actually we prove that there exist no new minimal representations up to infinitesimal equivalence

graded decomposition numbers for the blob algebra abstract we give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity

representation theory in complex rank i p deligne defined interpolations of the tensor category of representations of the symmetric group s n to complex values of n namely he defined tensor categories rep s t for any complex t this construction was generalized by f knop to the case of wreath products of s n with a finite group generalizing these results we propose a method of interpolating representation categories of various algebras containing s n such as degenerate affine hecke algebras symplectic reflection algebras rational cherednik algebras etc to complex values of n we also define the group algebra of s n for complex n study its properties and propose a schur weyl duality for rep s t

highest weight sl_2 categorifications ii structure theory this paper continues the study of highest weight categorical sl2 actions initi ated in part i we start by refining the definition given there and showing that all examples considered in part i are also highest weight categorifications in the refined sense then we prove that any highest weight sl2 categorification can be filtered in such a way that the suc cessive quotients are so called basic highest weight sl2 categorifications for a basic highest weight categorification we determine minimal projective resolutions of standard objects we use this in particular to examine the structure of tilting objects in basic categorifications and to show that the ringel duality is given by the rickard complex we apply some of these structural results to categories o for cyclotomic rational cherednik algebras we finish by discussing open problems

on the pro semisimple completion of the fundamental group of a smooth variety over a finite field abstract let be the fundamental group of a smooth variety x over f p let q be an algebraic closure of q given a non archimedean place of q prime to p consider the adic pro semisimple completion of as an object of the groupoid whose objects are pro semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component we prove that this object does not depend on if dim x 1 we also prove a crystalline generalization of this fact we deduce this from the langlands conjecture for function fields proved by l lafforgue and its crystalline analog proved by t abe using a reconstruction theorem in the spirit of kazhdan larsen varshavsky we also formulate two related conjecture e 8 1 conjecture e 9 1 each of them is a kind of reciprocity law involving a sum over all l adic cohomology theories including the crystalline theory for l p

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

convex hull results for generalizations of the constant capacity single node flow set for single node flow sets with fixed costs and constant capacities on the inflow and outflow arcs a family of constant capacity flow covers are known to provide the convex hull in different special cases and are conjectured to provide it in the general case here we study more general mixed integer sets for which such single node flow cover inequalities suffice to give the convex hull in particular we consider the case of a path in which each node has one or several incoming and outgoing arcs with constant capacities and fixed costs this can be seen as a lot sizing set with production and sales decisions driven by costs and prices and by the lower and upper bounds on stocks instead of being driven by demands as in the standard lot sizing model the approach we take is classical we characterize the extreme points derive tight extended formulations and project out the additional variables specifically we show that fourier motzkin elimination though far from elegant can be used to carry out the non trivial projections the validity of the conjecture for the single node flow set follows from our results

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

poincare birkhoff witt theorem for quadratic algebras of koszul type abstract in this paper we prove a general poincare birkhoff witt theorem for quadratic koszul algebras the result is similar to that obtained by polischuk and positselsky but the proof is entirely different and uses algebraic deformation theory in particular we obtain a new proof of the classical pbw theorem for lie algebras based on deformation theory

chtoucas de drinfeld et correspondance de langlands on demontre la correspondance de langlands pour gl r sur les corps de fonctions la preuve generalise celle de drinfeld en rang 2 elle consiste a realiser la correspondance en rang r dans la cohomologie l adique des varietes modulaires de chtoucas de drinfeld de rang r

theory of weights in p adic cohomology let k be a finite field of characteristic p 0 we construct a theory of weights for overholonomic complexes of arithmetic d modules with frobenius structure on varieties over k the notion of weight behave like deligne s one in the adic framework first the six operations preserve weights and secondly the intermediate extension of an immersion preserves pure complexes and weights this is joint work with tomoyuki abe

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

path cover and path pack inequalities for the capacitated fixed charge network flow problem capacitated fixed charge network flows are used to model a variety of problems in telecommunication facility location production planning and supply chain management in this paper we investigate capacitated path substructures and derive strong and easy to compute path cover and path pack inequalities these inequalities are based on an explicit characterization of the submodular inequalities through a fast computation of parametric minimum cuts on a path and they generalize the well known flow cover and flow pack inequalities for the single node relaxations of fixed charge flow models we provide necessary and sufficient facet conditions computational results demonstrate the effectiveness of the inequalities when used as cuts in a branch and cut algorithm