steenrod operations in chow theory an action of the steenrod algebra is constructed on the mod p chow theory of varieties over a field of characteristic different from p answering a question posed in fulton s intersection theory the action agrees with the action of the steenrod algebra used by voevodsky in his proof of the milnor conjecture however the construction uses only basic functorial properties of equivariant intersection theory

asymptotics of exit times for markov jump processes ii applications to jackson networks we show that a jackson network relaxes exponentially fast to its steady state by giving a lower bound on the cheeger constant for the associated markov process we also give lower bounds on the mean time until some node of the network overflows

on the semi stable coha and its modules arising from smooth models abstract we study a variant of the semi stable cohomological hall algebra which we construct using equivariant chow groups this algebra we call it the semi stable chowha arises as a quotient of the coha smooth models of quiver moduli give rise to modules over the semi stable chowha we prove that these modules are cyclic and we compute a presentation using harder narasimhan methods

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

weight two motivic cohomology of classifying spaces for semisimple groups let f x to y be a torsor for a semisimple group g with y a smooth and geometrically irreducible variety over an arbitrary field we relate the etale motivic cohomology of weight two for x y and g we also compute the etale motivic cohomology groups of degree at most 4 for the classifying space of g this result was used in an another work by the author for the computation of the group of degree 3 cohomological invariants of semisimple groups with coefficients in bbb q bbb z 2

on the first witt index of quadratic forms we prove hoffmann s conjecture determining the possible values of the first witt index of anisotropic quadratic forms of any given dimension the proof makes use of the steenrod type operations on the modulo 2 chow groups constructed by p brosnan

nonabelian localization in equivariant k theory and riemann roch for quotients abstract we prove a localization formula in equivariant algebraic k theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space this extends to nondiagonalizable groups the localization formulas of nielsen diagonalizably linearized coherent sheaves bull soc math france 102 1974 85 97 and thomason une formule de lefschetz en k theorie equivariante algebrique duke math j 68 3 1992 447 462 as an application we give a riemann roch formula for quotients of smooth algebraic spaces by proper group actions this formula extends previous work of toen theoremes de riemann roch pour les champs de deligne mumford k theory 18 1 1999 33 76 and the authors riemann roch for quotients and todd classes of simplicial toric varieties comm algebra 31 2003 3735 3752

localization in equivariant intersection theory and the bott residue formula we prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections and schubert varieties 1 introduction the purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections cf bfq and schubert varieties let t be a split torus acting on a scheme x the 7 equivariant chow groups of x are a module over rt sym t where f is the character group of t the localization theorem states that up to torsion the equivariant chow groups of the fixed locus xt are isomorphic to those of x such a theorem is a hallmark of any equivariant theory the earliest version for equivariant cohomology is due to borel bo subsequently it theory versions were proved by segal se in topological a theory quart qu for actions of a cyclic group and thomason th2 for algebraic theory th2 for equivariant chow groups the localization isomorphism is given by the equivariant push forward induced by the inclusion of xt to x an interesting aspect of this theory is that the push forward is naturally defined on the level of cycles even in the singular case the closest topological analogue of this is equivariant borel moore homology see e g3 for a definition and a similar proof establishes localization in that theory for smooth spaces the inverse to the equivariant push forward can be written explicitly it was realized independently by several authors i n a b b v

orbits in pr n and equivariant quantum cohomology abstract we compute the g l r 1 equivariant chow class of the g l r 1 orbit closure of any point x 1 x n p r n in terms of the rank polytope of the matroid represented by x 1 x n p r using these classes and generalizations involving point configurations in higher dimensional projective spaces we define for each d n matrix m an n ary operation m on the small equivariant quantum cohomology ring of p r which is the n ary quantum product when m is an invertible matrix we prove that m m is a valuative matroid polytope association like the quantum product these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting as an application we compute the number of line sections with given moduli of a general degree 2 r 1 hypersurface in p r generalizing the known case of quintic plane curves

on the geometry of algebraic groups and homogeneous spaces given a connected algebraic group g over an algebraically closed field and a g homogeneous space x we describe the chow ring of g and the rational chow ring of x with special attention to the picard group also we investigate the albanese and the anti affine fibrations of g and x

unramified cohomology of degree 3 and noether s problem let g be a finite group and w be a faithful representation of g over c the group g acts on the field of rational functions c w the question whether the field of invariant functions c w g is purely transcendental over c goes back to emmy noether using the unramified cohomology group of degree 2 of this field as an invariant saltman gave the first examples for which c w g is not rational over c around 1986 bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group g

lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands the study of regularity properties for minimizers of problems of the form min u u dt u e iv q r u a a u b b 1 1 has been undertaken for a long time in calculus of variations and a number of results are by now available on this subject when f t s z is a smooth coercive function problem 1 1 was considered by tonelli see 18 19 who proved that every minimizer u t of the problem 1 1 is locally lipschitz on a relatively open set of full measure in a b under some additional assumptions on f it is also possible to prove global regularity theorems see for instance cesari 3 chap 2 sect 61 in recent years much attention has been devoted to problems of the form l l on the one hand in weakening the hypotheses of tonelli s regularity results and on the other in finding counterexamples of nonregular minimizers for coercive functionals with respect to this last subject we recall that it has been shown see ball and mizel l and clarke and vinter 6 that there exist polynomial functions f t s z such that for suitable c 0 and p

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

equivariant cohomology syzygies and orbit structure let x be a nice space with an action of a torus t we consider the atiyah bredon sequence of equivariant cohomology modules arising from the filtration of x by orbit dimension we show that a front piece of this sequence is exact if and only if the h bt module h t x is a certain syzygy moreover we express the cohomology of that sequence as an ext module involving a suitably defined equivariant homology of x one consequence is that the gkm method for computing equivariant cohomology applies to a poincare duality space if and only if the equivariant poincare pairing is perfect

equivariant cobordism for torus actions abstract we study the equivariant cobordism groups for the action of a split torus t on varieties over a field k of characteristic zero we show that for t acting on a variety x there is an isomorphism t x b t l x as applications we show that for a connected linear algebraic group g acting on a k variety x the forgetful map g x x is surjective with rational coefficients as a consequence we describe the rational algebraic cobordism ring of algebraic groups and flag varieties we prove a structure theorem for the equivariant cobordism of smooth projective varieties with torus action using this we prove various localization theorems and a form of bott residue formula for such varieties as an application we show that the equivariant cobordism of a smooth variety x with torus action is generated by the invariant cobordism cycles in x as b t module

bott s formula and enumerative geometry we outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of bott as an example gromov witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed the results are consistent with predictions made from mirror symmetry compu tations we also compute degrees of some loci in the linear system of plane curves of degrees less than 10 like those corresponding to sums of powers of linear forms and curves carrying inscribed polygons mathematical institute university of oslo p 0 box 1053 n 0316 oslo norway e mail address ellingsr0math uio no mathematical institute university of bergen alleg 55 n 5007 bergen norway e mail address strommeti i uib no this content downloaded from 157 55 39 29 on tue 12 apr 2016 10 23 14 utc all use subject to http about jstor org terms

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

log concavity of characteristic polynomials and the bergman fan of matroids in a recent paper the first author proved the log concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0 answering a long standing conjecture of read in graph theory we extend the proof to all realizable matroids making progress towards a more general conjecture of rota heron welsh our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety the log concavity then follows from an inequality of hodge type

motivic decompositions of twisted flag varieties and representations of hecke type algebras motivated by the motivic galois group and the kostant kumar results on equivariant cohomology of flag varieties we provide a uniform de scription of motivic direct sum decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil hecke algebra h more generally we establish an equivalence between the h motivic subcat egory generated by the motive of e b and the category of projective modules of the associated rational algebra d of push pull operators where e is a tor sor for a split semisimple linear algebraic group g over a field k b is a borel subgroup of g h is an algebraic oriented cohomology theory in the sense of levine morel e g chow ring ch or an algebraic cobordism the algebra d can be think of as an integer analogue of the hopf algebra of the h motivic galois group of e b as an application taking h ch and specializing the coefficients to the finite field fp we obtain that p modular projective representations of d h are generated by an irreducible h module corresponding to the generalized rost voevodsky motive for g p

the integral orbifold chow ring of toric deligne mumford stacks in this paper we study the integral chow ring of toric deligne mumford stacks we prove that the integral chow ring of a semi projective toric deligne mumford stack is isomorphic to the stanley reisner ring of the associated stacky fan the integral orbifold chow ring is also computed our results are illustrated with several examples

algebraic rational cells and equivariant intersection theory we provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action based on this notion we study mathbb q filtrable varieties algebraic varieties where a torus acts with isolated fixed points such that the associated bialynicki birula decomposition consists of algebraic rational cells we show that the rational equivariant chow group of any mathbb q filtrable variety is freely generated by the classes of the cell closures we apply this result to group embeddings and more generally to spherical varieties

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

cohomological donaldson thomas theory of a quiver with potential and quantum enveloping algebras this paper concerns the cohomological aspects of donaldson thomas theory for jacobi algebras and the associated cohomological hall algebra introduced by kontsevich and soibelman we prove the hodge theoretic categorification of the integrality conjecture and the wall crossing formula and furthermore realise the isomorphism in both of these theorems as poincare birkhoff witt isomorphisms for the associated cohomological hall algebra we do this by defining a perverse filtration on the cohomological hall algebra a result of the hidden properness of the semisimplification map from the moduli stack of semistable representations of the jacobi algebra to the coarse moduli space of polystable representations this enables us to construct a degeneration of the cohomological hall algebra for generic stability condition and fixed slope to a free supercommutative algebra generated by a mixed hodge structure categorifying the bps invariants as a corollary of this construction we furthermore obtain a lie algebra structure on this mixed hodge structure the lie algebra of bps invariants for which the entire cohomological hall algebra can be seen as the positive part of a yangian type quantum group

riemann roch for equivariant k theory abstract we show that for a linear algebraic group g acting on a smooth quasi projective scheme x over a field there is a chern character map k i g x r g r g c h x g ch g x i s g s g with rational coefficients which is an isomorphism this establishes the equivariant version of the riemann roch isomorphism between the higher algebraic k theory and the higher chow groups of smooth quasi projective schemes

chern classes of deligne mumford stacks and their coarse moduli spaces let x be a complex projective algebraic variety with gorenstein quotient singularities and mathcal x a smooth deligne mumford stack having x as its coarse moduli space we show that the csm class c sm x coincides with the pushforward to x of the total chern class c t i mathcal x of the inertia stack i mathcal x we also show that the stringy chern class c str x of x whenever it is defined coincides with the pushforward to x of the total chern class c t ii mathcal x of the double inertia stack ii mathcal x some consequences concerning stringy orbifold hodge numbers are deduced

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

quantum riemann roch lefschetz and serre given a holomorphic vector bundle e over a compact kahler manifold x one defines twisted gromov witten invariants of x to be intersection numbers in moduli spaces of stable maps f x with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle h 0 f e h 1 f e using the formalism of quantized quadratic hamiltonians 25 we express the descendant potential for the twisted theory in terms of that for x this result theorem 1 is a consequence of mumford s grothendieck riemann roch theorem applied to the universal family over the moduli space of stable maps it determines all twisted gromov witten invariants of all genera in terms of untwisted invari ants when e is concave and the c equivariant inverse euler class is chosen as the characteristic class the twisted invariants of x give gromov witten invariants of the total space of e nonlinear serre duality 21 23 expresses gromov witten invariants of e in terms of those of the super manifold e it relates gromov witten invariants of x twisted by the inverse euler class and e to gromov witten invariants of x twisted by the euler class and e we derive from theorem 1 nonlinear serre duality in a very general form corollary 2 when the bundle e is convex and a submanifold y x is defined by a global section of e the genus zero gromov witten invariants of e coin cide with those of y we establish a quantum lefschetz hyperplane section principle theorem 2 expressing genus zero gromov witten invariants of a complete intersection y in terms of those of x this extends earlier results 4 9 18 29 33 and yields most of the known mirror formulas for toric complete intersections

framed sheaves on root stacks and supersymmetric gauge theories on ale spaces we develop a new approach to the study of supersymmetric gauge theories on ale spaces using the theory of framed sheaves on root toric stacks which illuminates relations with gauge theories on r 4 and with two dimensional conformal field theory we construct a stacky compactification of the minimal resolution xk of the ak 1 toric singularity c 2 zk which is a projective toric orbifold xk such that xknxk is a zk gerbe we construct moduli spaces of tor sion free sheaves on xk which are framed along the compactification gerbe we prove that this moduli space is a smooth quasi projective variety compute its dimension and classify its fixed points under the natural induced toric action we use this construction to compute the partition functions and correlators of chiral bps operators forn 2 quiver gauge theories on xk with nontrivial holonomies at infinity the partition functions are computed with and without cou plings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas which relate them to the corresponding nekrasov partition functions on the affine toric subsets of xk we compare our new partition functions with previous computations explore their con nections to the representation theory of affine lie algebras and find new constraints on fractional instanton charges in the coupling to fundamental matter we show that the partition functions in the low energy limit are characterised by the seiberg witten curves and in some cases also by suitable blowup equations involving riemann theta functions on the seiberg witten curve with characteristics related to the nontrivial holonomies

equivariant characteristic classes of singular complex algebraic varieties homology hirzebruch characteristic classes for singular varieties have been recently defined by brasselet schuurmann and yokura as an attempt to unify previously known characteristic class theories for singular spaces e g macpherson chern classes baum fulton macpherson todd classes and goresky macpherson l classes in this paper we define equivariant analogues of these classes for singular quasi projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology hirzebruch classes of global quotient varieties we also compute the new classes in the context of monodromy problems e g for varieties that fiber equivariantly in the complex topology over a connected algebraic manifold as another application we discuss atiyah meyer type formulae for twisted hirzebruch classes of global orbifolds 2012 wiley periodicals inc

functorial reconstruction theorems for stacks abstract we study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes this is possible surprisingly often we show that many of the standard examples of moduli stacks are determined by their functors our methods seem to exhibit new anabelian style phenomena in the form of structures in the category of schemes that encode automorphism data in groupoids

motives of deligne mumford stacks abstract for every smooth and separated deligne mumford stack f we associate a motive m f in voevodsky s category of mixed motives with rational coefficients dm eff k q when f is proper over a field of characteristic 0 we compare m f with the chow motive associated to f by toen 2000 31 without the properness condition we show that m f is a direct summand of the motive of a smooth quasi projective variety

the picard group of the universal moduli space of vector bundles on stable curves abstract we construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its picard group as a consequence we obtain an explicit description of the picard groups of the universal moduli stack of vector bundles on smooth curves and of the schmitt s compactification over the stack of stable curves we prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group in particular we give necessary and sufficient conditions for the existence of poincare bundles over the universal curve of an open substack of the rigidification generalizing a result of mestrano ramanan

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

picard group of moduli of hyperelliptic curves the main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves we compute their picard groups giving explicit description of the generators we get an application to the non existence of a tautological family over the coarse moduli space

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group