uniformization of mathcal g bundles we show some of the conjectures of pappas and rapoport concerning the moduli stack bung of g torsors on a curve c where g is a semisimple bruhat tits groupschemeon c inparticularweprovetheanalogoftheuniformizationtheoremof drinfeld simpson in this setting furthermore we apply this to compute the connected components of these moduli stacks and to calculate the picard group of bung in case g is simply connected

involutions on moduli spaces and refinements of the verlinde formula the moduli space m of semi stable rank 2 bundles with trivial deter minant over a complex curvecarries involutions naturally associated to 2 torsion points on the jacobian of the curve for every lift of a 2 torsion point to a 4 torsion point we define a lift of the involution to the determinant line bundle l we ob tain an explicit presentation of the group generated by these lifts in terms of the order 4 weil pairing this is related to the triple intersections of the components of the fixed point sets in m which we also determine completely using the order 4 weil pairing the lifted involutions act on the spaces of holomorphic sections of powers of l whose dimensions are given by the verlinde formula we compute the characters of these vector spaces as representations of the group generated by our lifts and we obtain an explicit isomorphism as group representations with the combinatorial topological tqft vector spaces of bhmv as an application we describe a brick decomposition with explicit dimension formulas of the verlinde vector spaces we also obtain similar results in the twisted i e degree one case

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

berkovich spaces and tubular descent we consider an algebraic variety xx together with the choice of a subvariety zz we show that any coherent sheaf on xx can be constructed out of a coherent sheaf on the formal neighborhood of zz a coherent sheaf on the complement of zz and an isomorphism between certain representative images of these two sheaves in the category of coherent sheaves on a berkovich analytic space ww which we define

conformal blocks attached to twisted groups geometric representation theory allows to associate to each simple and simply connected algebraic group g over a field and to a positive integer a vector bundle called the sheaf of conformal blocks over m g the stack of curves of genus g this thesis generalizes this construction to the situation in which g is replaced by a twisted group which depends on covering data using the methods described by looijenga in the untwisted case we prove that the main features of the classical sheaf of conformal block still hold in this more general setting in particular we describe the wzw connection the propagation of vacua and the fusion rules

semistable bundles over an elliptic curve abstract we identify the moduli space m n d of semistable bundles of rank n and degree d on an elliptic curve c as a symmetric product of the curve its dimension being the greatest common divisor of n and d under this identification the determinant map det m n d corresponds to the abel jacobi map shc jh so that the fixed determinant moduli space is identified with a projective space we then describe the brill noether loci theta divisors and picard groups and compute the space of theta functions for both the full moduli space and the fixed determinant moduli space of any rank and any degree we extend to elliptic curves a theorem of drezet and narasimhan on the picard group of moduli spaces as well as two formuulas of beauville narasimhan and ramanan on theta functions of level one our results verify in particular for an elliptic curve a formula of e verlinde and an equality first noticed by bott and szenes relating theta functions over the full moduli space and over the fixed determinant moduli space

affine grassmannians and the geometric satake in mixed characteristic we endow the set of lattices in q n p with a reasonable algebro geometric structure as a result we prove the representability of affine grassmannians and establish the geometric satake equivalence in mixed characteristic we also give an application of our theory to the study of rapoport zink spaces

the kauffman skein algebra of a surface at sqrt 1 we study the structure of the kauffman algebra of a surface with parameter equal to sqrt 1 we obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat su 2 connections over the surface we analyse the asymptotics of traces of curve operators in tqft in non standard regimes where the root of unity parametrizing the tqft accumulates to a root of unity we interpret the case of sqrt 1 in terms of parallel transport operators

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

on the coherence conjecture of pappas and rapoport we prove the generalized coherence conjecture proposed by pappas and rapoport as a corollary one of their theorems which describes the geometry of the special bers of the local models for ramied unitary groups holds unconditionally our proof is based on the study of the geometry in particular certain line bundles and adic sheaves of the global schubert varieties which are the equal characteristic counterparts of the local models

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

fusion rings and geometry the algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper a formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings in general and for current algebra theories in particular it is shown that fusion rings lead to a natural orthogonal polynomial structure the rings are expressed through generators and relations the relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties in general the fusion algebras are mapped to affine varieties which are the locus of the relations the connection with modular transformations is investigated in this picture it is explained how chiral algebras arising in n 2 superconformal field theory can be derived from fusion rings in particular it is argued that theories of the type su n k su n 1 are the n 2 counterparts of grassmann manifolds and that there is a natural identification of the chiral fields with schubert varieties which is a graded algebra isomorphism

ideal triangles in euclidean buildings and branching to levi subgroups abstract let g denote a connected reductive group defined and split over z and let m g denote a levi subgroup in this paper we study varieties of geodesic triangles with fixed vector valued side lengths in the bruhat tits buildings associated to g along with varieties of ideal triangles associated to the pair m g the ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex such that other infinite side containing has fixed ideal length and the remaining finite side has fixed length we establish an isomorphism between varieties in the second family and certain varieties in the first family the pair and the triple satisfy a certain relation we apply these results to the study of the hecke ring of g and the restriction homomorphism r g r m between representation rings we deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism

demazure resolutions as varieties of lattices with infinitesimal structure abstract let k be a field of positive characteristic we construct for each dominant cocharacter of the standard maximal torus in sl n a closed subvariety d of the multigraded hilbert scheme of an affine space over k such that the k valued points of d can be interpreted as lattices in k z n endowed with infinitesimal structure the variety d carries a natural sl n k z action moreover for any we construct an sl n k z equivariant universal homeomorphism from d to a demazure resolution of the schubert variety s associated with in the affine grassmannian lattices in d have non trivial infinitesimal structure if and only if they lie over the boundary of the big cell of s

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

cr yamabe constant cr yamabe flow and its soliton abstract on a compact strictly pseudoconvex cr manifold m we consider the cr yamabe constant of its infinite conformal covering by using the maximum principles we then prove a uniqueness theorem for the cr yamabe flow on a complete noncompact cr manifold finally we obtain some properties of the cr yamabe soliton on complete noncompact cr manifolds

characters of topological n 2 vertex algebras are jacobi forms on the moduli space of elliptic supercurves abstract we show that trace functions on modules of topological n 2 super vertex algebras give rise to conformal blocks on elliptic supercurves we show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector bundle of conformal blocks we show that this connection is equivariant with respect to a natural action of the jacobi modular group on the modular parameters and the trace functions in the appendix we prove the convergence of the trace functions

on p adic lattices and grassmannians it is well known that the coset spaces g k z g k z for a reductive group g over a field k carry the geometric structure of an inductive limit of projective k schemes this k ind scheme is known as the affine grassmannian for g from the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form mathcal g mathbf w k 1 p mathcal g mathbf w k where p is a rational prime mathbf w denotes the ring scheme of p typical witt vectors k is a perfect field of characteristic p and mathcal g is a reductive group scheme over mathbf w k the present paper is an attempt to describe which constructions carry over from the function field case to the p adic case more precisely to the situation of the p adic affine grassmannian for the special linear group mathcal g mathbf sl n we start with a description of the r valued points of the p adic affine grassmannian for mathbf sl n in terms of lattices over mathbf w r where r is a perfect k algebra in order to obtain a link with geometry we further construct projective k subvarieties of the multigraded hilbert scheme which map equivariantly to the p adic affine grassmannian the images of these morphisms play the role of schubert varieties in the p adic setting further for any reduced k algebra r these morphisms induce bijective maps between the sets of r valued points of the respective open orbits in the multigraded hilbert scheme and the corresponding schubert cells of the p adic affine grassmannian for mathbf sl n

the index formula for the moduli of g bundles on a curve we prove the formulae conjectured by the first author for the index of k theory classes over the moduli stack of algebraic g bundles on a smooth projective curve the formulae generalize e verlinde s for line bundles and have witten s integrals over the moduli space of stable bundles as their large level limits as an application we prove the newstead ramanan conjecture on the vanishing of high chern classes of certain moduli spaces of semi stable g bundles

a wonderful embedding of the loop group i describe the wonderful compactification of loop groups these compactifications are obtained by adding normal crossing boundary divisors to the group lg of loops in a reductive group g or more accurately to the semi direct product c lg in a manner equivariant for the left and right c lg actions the analogue for a torus group t is the theory of toric varieties for an adjoint group g this is the wonderful compactification of de concini and procesi the loop group analogue is suggested by work of faltings in relation to the compactification of moduli of g bundles over nodal curves

completions of monoid objects and descent results abstract let c 1 be an abelian closed symmetric monoidal category satisfying certain conditions the purpose of this paper is to prove three descent results for schemes over c 1 that are not necessarily noetherian

on the formal arc space of a reductive monoid let x be a scheme of finite type over a finite field k and let cal l x denote its arc space in particular cal l x k x k t using the theory of grinberg kazhdan and drinfeld on the finite dimensionality of singularities of cal l x in the neighborhood of non degenerate arcs we show that a canonical basic function can be defined on the non degenerate locus of cal l x k which corresponds to the trace of frobenius on the stalks of the intersection complex of any finite dimensional model we then proceed to compute this function when x is an affine toric variety or an l monoid our computation confirms the expectation that the basic function is a generating function for a local unramified l function in particular in the case of an l monoid we prove a conjecture formulated by the second author

the generalized verschiebung map for curves of genus 2 let c be a smooth curve and m r c the coarse moduli space of vector bundles of rank r and trivial determinant on c we examine the generalized verschiebung map v r m r c p dashrightarrow m r c induced by pulling back under frobenius our main result is a computation of the degree of v 2 for a general c of genus 2 in characteristic p 2 we also give several general background results on the verschiebung in an appendix

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

the level rank duality for non abelian theta functions we prove that the strange duality conjecture of beauville donagi tu holds for all curves we establish first a more extended level rank duality interesting in its own right from which the standard level rank duality follows by restriction

parameter dependent lyapunov functions for linear systems with constant uncertainties robust stability of linear time invariant systems with respect to structured uncertainties is considered the small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition it is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic lyapunov function which proves robust stability with respect to all allowable time varying perturbations in this technical note we show that if the small gain condition is satisfied with frequency varying scalings then an explicit parameter dependent lyapunov function can be constructed to prove robust stability with respect to constant uncertainties this lyapunov function has a rational quadratic dependence on the uncertainties

foliations in deformation spaces of local g shtukas abstract we study local g shtukas with level structure over a base scheme whose newton polygons are constant on the base we show that after a finite base change and after passing to an etale covering such a local g shtuka is isogenous to a completely slope divisible one generalizing corresponding results for p divisible groups by oort and zink as an application we establish a product structure up to finite surjective morphism on the closed newton stratum of the universal deformation of a local g shtuka similarly to oort s foliations for p divisible groups and abelian varieties this also yields bounds on the dimensions of affine deligne lusztig varieties and proves equidimensionality of affine deligne lusztig varieties in the affine grassmannian

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

affine demazure modules and t fixed point subschemes in the affine grassmannian let g be a simple algebraic group defined over c and t be a maximal torus of g for a dominant coweight of g the t fixed point subscheme formula of the schubert variety formula in the affine grassmannian grg is a finite scheme we prove that for all such if g is of type a or d and for many of them if g is of type e there is a natural isomorphism between the dual of the level one affine demazure module corresponding to and the ring of functions twisted by certain line bundle on grg of formula we use this fact to give a geometrical proof of the frenkel kac segal isomorphism between basic representations of affine algebras of a d e type and lattice vertex algebras

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

frobenius semisimplicity for convolution morphisms this article concerns properties of mixed ell adic complexes on varieties over finite fields related to the action of the frobenius automorphism we establish a fiberwise criterion for the semisimplicity and frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field we conjecture that the direct image of the intersection complex on the domain is always semisimple and frobenius semisimple this conjecture would imply that a strong form of the decomposition theorem of beilinson bernstein deligne gabber is valid over finite fields we prove our conjecture for generalized convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields as a crucial tool we develop a new schematic theory of big cells for loop groups with suitable reformulations the main results are valid over any algebraically closed ground field

conformal blocks verlinde formula and diagram automorphisms abstract the verlinde formula computes the dimension of the space of conformal blocks associated to simple lie algebras and stable pointed curves if a simply laced simple lie algebra admits a nontrivial diagram automorphism then this automorphism acts on the space of conformal blocks naturally we prove an analogue of the verlinde formula for the trace of the diagram automorphism on the space of conformal blocks along the way we get an analogue of the kac walton formula for the trace of the diagram automorphism we also get a twining type formula between the conformal blocks for the pair s l 2 n 1 s p 2 n

asymptotic properties of the quantum representations of the mapping class group we establish two results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat su n bundle with hitchin s connection first we prove that these projective representations lift to asymptotic representations second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion the leading order term of this expansion agrees with the formula derived heuristically by e witten in quantum field theory and the jones polynomial

strange duality of verlinde spaces for g_2 and f_4 we prove that the pull back of the canonical theta divisor for e 8 bundles at level one induces a strange duality between verlinde spaces for g 2 and f 4 at level one on smooth curves of genus g we also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in text pic overline text m g n mathbb q

ls galleries the path model and mv cycles we give an interpretation of the path model of a representation 18 of a complex semisimple algebraic group g in terms of the geometry of its affine grassmannian in this setting the paths are replaced by ls galleries in the affine coxeter complex associated to the weyl group of g to explain the connection with geometry consider a demazure hansen bott samelson desingularization of the closure of an orbit g c t in the affine grassmannian the homology of has a basis given by bia lynicki birula cell s which are indexed by the t fixed points in now the points of can be identified with galleries of a fixed type in the affine tits building associated to g and the t fixed points correspond in this language to combinatorial galleries of a fixed type in the affine coxeter complex we determine those galleries such that the associated cell has a non empty intersection with g c t identified with an open subset of and we show that the closures of the strata associated to ls galleries are exactly the mv cycles 24 which form a basis of the representation v for the langland s dual group g

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights