exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

bimeasure algebras on locally compact groups for locally compact groups g and h let bm g h denote the banach space of bounded bilinear forms on c0 g c0 h using a consequence of the fundamental inequality of a grothendieck a multiplication and an adjoint operation are introduced on bm g h which generalize the convolution structure of m g h and which make bm g h into a kg2 banach algebra where kg is grothendieck s universal constant various topics relating to the ideal structure of bm g h and the lifting of unitary representations of g h to representations of bm g h are investigated

a solution to an open problem for wave equations with generalized wentzell boundary conditions in this paper we solve an open problem put forward by a favini g r goldstein j a goldstein and s romanelli concerning the mixed problem for wave equations with generalized wentzell boundary conditions as a consequence we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized wentzell boundary conditions

relative quantum field theory we highlight the general notion of a relative quantum field theory which occurs in several contexts one is in gauge theory based on a compact lie algebra rather than a compact lie group this is relevant to the maximal superconformal theory in six dimensions

multi linear measure theory and multiple stochastic integration stochastic integration in a lebesgue stieltjes sense is developed in a general context of multi linear measure theory based on the fundamental grothendieck inequality and factorization theorem

fibration de hitchin et endoscopie we propose a geometric interpretation of the theory of elliptic endoscopy due to langlands and kottwitz in terms of the hitchin fibration as applications we prove a global analog of a purity conjecture due to goresky kottwitz and macpherson for unitary groups this global purity statement has been used in a joint work with g laumon to prove the fundamental lemma over a local fields of equal characteristics

a nonterminating 8 φ 7 summation for the root system c r using multiple q integrals and a determinant evaluation we establish a nonterminating 8 7 summation for the root system cr we also give some important specializations explicitly

twisted fourier mukai transforms for holomorphic symplectic four folds we apply the methods of c a ldararu to construct a twisted fourier mukai transform between a pair of holomorphic symplectic four folds which are fibred by lagrangian abelian surfaces more precisely we obtain an equivalence between the derived category of coherent sheaves on a certain lagrangian fibration and the derived category of twisted sheaves on its mirror partner as a corollary we extend the original fourier mukai transform to degenerations of abelian surfaces another consequence of the general theory is that the holomorphic symplectic four fold and its mirror are connected by a one parameter family of deformations through lagrangian fibrations

le lemme fondamental pondere ii enonces cohomologiques dans cet article on etudie la cohomologie de la fibration de hitchin tronquee introduite dans un article precedent on etend les principaux theoremes de ngo sur la cohomologie de la partie elliptique de la fibration de hitchin comme consequence on obtient une demonstration du lemme fondamental pondere d arthur in this paper we study the cohomology of the truncated hitchin fibration which was introduced in a previous paper we extend ngo s main theorems on the cohomology of the elliptic part of the hitchin fibration as a consequence we get a proof of arthur s weighted fundamental lemma

fourier mukai transforms for gorenstein schemes we extend to singular schemes with gorenstein singularities or fibered in schemes of that kind bondal and orlov s criterion for an integral functor to be fully faithful we also prove that the original condition of characteristic zero cannot be removed by providing a counterexample in positive characteristic we contemplate a criterion for equivalence as well in addition we prove that for locally projective gorenstein morphisms a relative integral functor is fully faithful if and only if its restriction to each fibre is also fully faithful these results imply the invertibility of the usual relative fourier mukai transform for an elliptic fibration as a direct corollary

sweep maps a continuous family of sorting algorithms abstract we define a family of maps on lattice paths called sweep maps that assign levels to each step in the path and sort steps according to their level surprisingly although sweep maps act by sorting they appear to be bijective in general the sweep maps give concise combinatorial formulas for the q t catalan numbers the higher q t catalan numbers the q t square numbers and many more general polynomials connected to the nabla operator and rational catalan combinatorics we prove that many algorithms that have appeared in the literature including maps studied by andrews egge gorsky haglund hanusa jones killpatrick krattenthaler kremer orsina mazin papi vaille and the present authors are all special cases of the sweep maps or their inverses the sweep maps provide a very simple unifying framework for understanding all of these algorithms we explain how inversion of the sweep map which is an open problem in general can be solved in known special cases by finding a bounce path for the lattice paths under consideration we also define a generalized sweep map acting on words over arbitrary alphabets with arbitrary weights which is also conjectured to be bijective

double centralizing theorems for the alternating groups let v n be the n fold tensor product of a vector space v following i schur we consider the action of the symmetric group sn on v n by permuting coordinates in the super z2 graded case v v0 v1 a sign is added these actions give rise to the corresponding schur algebras s sn v here s sn v is compared with s an v the schur algebra corresponding to the alternating subgroup an sn while in the classical signless case these two schur algebras are the same for n large enough it is proved that in the super case where dim v0 dim v1 s an v is isomorphic to the cross product algebra s an v s sn v z2

modular invariant of quantum tori the quantum modular invariant j qt of 2r is defined as a discontin uous pgl2 z invariant multi valued map using the distance to the nearest integer function k k for 2 q it is shown that j qt 1 and for quadratic irrationalities pari gp experiments suggest that j qt is a finite set in the case of the golden mean we produce explicit formulas involving weighted versions of the rogers ramanujan functions for the experimental supremum and infimum we then define a universal modular invariant j a mod c as a continuous and single valued map of ultrasolenoids such that 1 the classical modular invariant is a quotient of the re striction of j to a subsolenoid mod cl a mod fibering over the classical moduli space of elliptic curves and 2 the quantum modular invariant is a quotient of the restric tion of j to a subsolenoid mod qt a mod fibering over the moduli space of elliptic curves equipped with a kronecker foliation

on the geometric langlands conjecture let x be a smooth projective curve and g be a reductive group over a finite field fq the field kx of rational functions on x is what number theorists call a global field and we can do the theory of automorphic functions over it namely we consider the quotient g kx g a here a is the ring of adeles corresponding to kx and study the space of functions on it as a representation of the adele group g a it was essentially an observation of a weil that the quotient g kx g a is closely related to the set of isomorphism classes of principal g bundles on our curve x however g bundles on x possess a richer structure one can view the above set as a set of fq points of an algebraic variety or rather a stack denoted bung moreover instead of studying the vector space of functions on the discrete set of points of bung we will consider the category of adic sheaves on it basically what people call the geometric langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so called hecke functors the latter will be defined in the talk the answer predicted by the geometric langlands conjecture links this spectral decomposition to the moduli stack of local systems on x with respect to the langlands dual group

d equivalence and k equivalence let x and y be smooth projective varieties over they are called d equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories and k equivalent if they are birationally equivalent and the pull backs of their canonical divisors to a common resolution coincide we expect that the two equivalences coincide for birationally equivalent varieties we shall provide a partial answer to the above problem in this paper

noncommutative fourier transforms of bounded bilinear forms and completely bounded multilinear operators abstract let g1 gn be locally compact groups and h a hilbert space any bounded n linear operator c g 1 c g n l h is known to have a unique separately weakly continuous extension gf w g 1 w g n l h the fourier transform of is the function gf g 1 g n l h defined by gf s 1 s n gf 1 s 1 n s n where i g i w g i l h i is the universal representation in the case n 2 l h c characterizations based on the grothendieck pisier haagerup inequality are given for such fourier transforms in terms of weakly harmonizable and hemihomogeneous random fields and continuous unitary jordan representations in the case of arbitrary n attention is confined to completely bounded n linear l h valued operators their fourier transforms are characterized and a convolution operation for them is defined in the case of l h c the completely bounded n linear forms on c g 1 c g n form a commutative banach algebra whose maximal ideal space contains the image of b g1 b gn under a separately continuous injection with a continuous left inverse where b gi is the maximal ideal space of the fourier stieltjes algebra of gi

the local structure of compactified jacobians this paper studies the local geometry of compactified jacobians the main result is a presentation of the completed local ring of the compactified jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve the authors have investigated the geometric and combinatorial properties of these rings in previous work and consequences for compactified jacobians are presented in this paper similar results are given for the local structure of the universal compactified jacobian over the moduli space of stable curves

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

on holomorphic factorization of wzw and coset models it is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the wzw model of two dimensional conformal field theory which previously have been understood primarily by using conformal field theory ward identities we also consider in a similar vein the holomorphic factorization ofg h coset models we discuss theg g model as a topological field theory and comment on a conjecture by spiegelglas

projectively bounded frechet measures a scalar valued set function on a cartesian product of a algebras is a fre chet measure if it is a scalar measure independently in each coordinate a basic question is considered is it possible to construct products of frechet measures that are analogous to product measures in the classical theory a frechet measure is said to be projectively bounded if it satisfies a grothendieck type inequality it is shown that feasibility of products of frechet measures is linked to the projective boundedness property all frechet measures in a two dimensional framework are projectively bounded while there exist frechet measures in dimensions greater than two that are projectively unbounded a basic problem is considered when is a fryechet measure projectively bounded some characterizations are stated applications to harmonic and stochastic analysis are given

ideals associated to deformations of singular plane curves we consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities these loci are the equisingular locus es which parametrizes equisingular or topologically trivial deformations the equigeneric locus eg which parametrizes deformations of constant geometric genus and the equiclassical locus ec which parametrizes deformations of constant geometric genus and class the class of a reduced plane curve is the degree of its dual it was previously known that the tangent space to es corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to eg corresponds to the conductor ideal we show that the support of the tangent cone to ec corresponds to an ideal which we call the equiclassical ideal by studying these ideals we are able to obtain information about the geometry and dimensions of es ec and eg this allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities

mirror symmetry for moduli spaces of higgs bundles via p adic integration we prove the topological mirror symmetry conjecture by hausel thaddeus for smooth moduli spaces of higgs bundles of type sl n and pgl n more precisely we establish an equality of stringy hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties our proof utilises p adic integration relative to the fibres and interprets canonical gerbes present on these moduli spaces as characters on the hitchin fibres using tate duality furthermore we prove for d prime to n that the number of rank n higgs bundles of degree d over a fixed curve defined over a finite field is independent of d this proves a conjecture by mozgovoy schiffmann in the coprime case

generation of localization in a discrete chain with periodic boundary conditions numerical and analytical results we study the generation of localization in a discrete chain composed of n subsystems and periodic boundary conditions strongly localized motions are studied that is time periodic motions where nearly all of the energy is spatially confined to a single subsystem for varying n numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in phase spatially extended solution however in the limit as n rightarrow infty the bifurcation point tends to infinity and a smooth transition from localization to nonlocalization occurs we then present an analytic technique to complement the numerical results it is based on the matching local asymptotic expansions of a solution branch using pade approximants this leads to global analytic representations of the considered solutions valid over the entire range of the control parameter

conjectured statistics for the q t catalan numbers abstract we introduce the distribution function f n q t of a pair of statistics on catalan words and conjecture f n q t equals garsia and haiman s q t catalan sequence c n q t which they defined as a sum of rational functions we show that f n s q t defined as the sum of these statistics restricted to catalan words ending in s ones satisfies a recurrence relation as a corollary we are able to verify that f n q t c n q t when t 1 q we also show the partial symmetry relation f n q 1 f n 1 q by modifying a proof of haiman of a q lagrange inversion formula based on results of garsia and gessel we obtain a q analogue of the general lagrange inversion formula which involves catalan words grouped according to the number of ones at the end of the word

on the e polynomials of a family of text sl _n character varieties we find the e polynomials of a family of twisted character varieties mathcal m text sl n of riemann surfaces by proving they have polynomial count and applying a result of katz regarding the counting functions to count the number of mathbb f q points of these varieties as a function of q we invoke a formula from frobenius our calculations make use of the character tables of text sl n q partially computed by lehrer and a result of hanlon on the mobius function of a fixed subposet of set partitions we compute the euler characteristic of the mathcal m text sl n with these polynomials and show they are connected

equivariant verlinde algebra from superconformal index and argyres seiberg duality in this paper we show the equivalence between two seemingly distinct 2d tqfts one comes from the coulomb branch index of the class ss theory t g on l k 1 s 1 the other is the lglg equivariant verlinde formula or equivalently partition function of lgclgc complex chern simons theory on s 1 we first derive this equivalence using the m theory geometry and show that the gauge groups appearing on the two sides are naturally g and its langlands dual lglg when g is not simply connected we provide a recipe of computing the index of t g as summation over the indices of t g with non trivial background t hooft fluxes where g is the universal cover of g then we check explicitly this relation between the coulomb index and the equivariant verlinde formula for g su 2 or so 3 in the end as an application of this newly found relation we consider the more general case where g is su n or psu n and show that equivariant verlinde algebra can be derived using field theory via generalized argyres seiberg duality we also attach a mathematica notebook that can be used to compute the su 3 equivariant verlinde coefficients

op lax natural transformations twisted quantum field theories and even higher morita categories abstract motivated by the challenge of defining twisted quantum field theories in the context of higher categories we develop a general framework for lax and oplax transformations and their higher analogs between strong n functors we construct a double n category built out of the target n category governing the desired diagrammatics we define op lax transformations as functors into parts thereof and an op lax twisted field theory to be a symmetric monoidal op lax natural transformation between field theories we verify that lax trivially twisted relative field theories are the same as absolute field theories as a second application we extend the higher morita category of e d algebras in a symmetric monoidal n category c to an n d category using the higher morphisms in c

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

twisting derived equivalences we introduce a new method for twisting relative equivalences of derived categories of sheaves on two spaces over the same base the first aspect of this is that the derived categories of sheaves on the spaces are twisted they become derived categories of sheaves on gerbes living over spaces that are locally on the base isomorphic to the original spaces secondly this is done in a compatible way so that the equivalence is maintained we apply this method by proving the conjectures of donagi and pantev on dualities between gerbes on genus one fibrations and comment on other applications to families of higher genus curves we also include a related conjecture in mirror symmetry

some inequalities for the q beta and the q gamma functions via some q integral inequalities some new inequalities for the q gamma the q beta and the q analogue of the psi functions are established via some q integral inequalities

decoupling inequalities for multilinear forms in independent symmetric random variables on considere x 1 x 2 des copies independantes d une suite x x 1 x 2 de variables aleatoires symetriques independantes et m une forme multilineaire symetrique de rang s sur r n dont les composantes a i1 a is relatives a la base standard de r n satisfont a i1 i s 0

the x ray transform and its inversion for the series expansion basis functions in three dimensional tomography a formula for the three dimensional x ray transform and an expression for its inversion have been derived the spherical harmonics and the rotation matrices in terms of euler angles are used in the generalized fourier series expansion some properties of the transformed fourier components are discussed

square boldsymbol q t lattice paths and boldsymbol nabla p_n the combinatorial q t catalan numbers are weighted sums of dyck paths introduced by j haglund and studied extensively by haglund haiman garsia loehr and others the q t catalan numbers besides having many subtle combinatorial properties are intimately connected to symmetric functions algebraic geometry and macdonald polynomials in particular the n th q t catalan number is the hilbert series for the module of diagonal harmonic alternants in 2n variables it is also the coefficient of s 1 n in the schur expansion of e n using q t analogues of labelled dyck paths haglund et al have proposed combinatorial conjectures for the monomial expansion of e n and the hilbert series of the diagonal harmonics modules this article extends the combinatorial constructions of haglund et al to the case of lattice paths contained in squares we define and study several q t analogues of these lattice paths proving combinatorial facts that closely parallel corresponding results for the q t catalan polynomials we also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator in particular we conjecture combinatorial formulas for the monomial expansion of p n the hilbert series p n h 1 n and the sign character p n s 1 n

involutions and higher order automorphisms of higgs bundle moduli spaces this research was partially supported by the spanish mineco under the icmat severo ochoa grant no sev 2011 0087 and grant no mtm2013 43963 p and by the european commission marie curie irses moduli programme pirses ga 2013 612534

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

the representation of the symmetric group on m tamari intervals an m ballot path of size n is a path on the square grid consisting of north and east unit steps starting at 0 0 ending at mn n and never going below the line fx myg the set of these paths can be equipped with a lattice structure called them tamari lattice and denoted by t m n which generalizes the usual tamari lattice tn obtained when m 1 this lattice was introduced by f bergeron in connection with the study of diagonally coinvariant spaces in three sets ofn variables the representation of sn on these spaces is conjectured to be closely related to the natural representation of sn on labelled intervals of them tamari lattice studied in this paper an interval p q of t m n is labelled if the north steps of q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps the symmetric group sn acts on labelled intervals of t m n by permutation of the labels we prove an explicit formula conjectured by f bergeron and the third author for the character of the associated representation of sn in particular the dimension of the representation that is the number of labelledm tamari intervals of sizen is found to be m 1 n mn 1 n 2 these results are new even whenm 1 the form of these numbers suggests a connection with parking functions but our proof is not bijective the starting point is a recursive description of m tamari intervals it yields an equation for an associated generating function which is a refined version of the frobenius series of the representation the form of this equation is highly non standard it involves two additional variables x and y a derivative with respect to y and iterated divided differences with respect tox the hardest part of the proof consists in solving it and we develop original techniques to do so r

an improved sectional anosov closing lemma we prove that all nonwandering points of a sectional anosov flow on a compact 3 manifold can be approximated by periodic points or by points for which the omega limit set is a singularity this improves the closing lemma in morales mich math j 56 1 29 53 2008 we also describe a sectional anosov flow for which the recurrent points are not dense in the nonwandering set

the delta conjecture we conjecture two combinatorial interpretations for the symmetric function eken where f is an eigenoperator for the modified macdonald polynomials defined by bergeron garsia haiman and tesler both interpretations can be seen as generalizations of the shuffle conjecture a statement originally conjectured by haglund haiman remmel loehr and ulyanov and recently proved by carlsson and mellit we show how previous work of the second and third authors on tesler matrices and ordered set partitions can be used to verify several cases of our conjectures furthermore we use a reciprocity identity and llt polynomials to prove another case finally we show how our conjectures inspire 4 variable generalizations of the catalan numbers extending work of garsia haiman and the first author

stacky dualities for the moduli of higgs bundles abstract the central result of this paper is an identification of the shifted cartier dual of the moduli stack m g c of g higgs bundles on c of arbitrary degree modulo shifts by z g with a quotient of the langlands dual stack m g l c via hyperkahler rotation this may equivalently be viewed as the identification of an syz fibration relating hitchin systems for arbitrary langlands dual semisimple groups coupled to nontrivial finite b fields as a corollary certain self dual stacks m g c are observed to exist which i conjecture to be the coulomb branches for the 3d reduction of the 4d n 2 theories of class s

the compactified picard scheme of the compactified jacobian abstract let c be an integral projective curve in any characteristic given an invertible sheaf l on c of degree 1 form the corresponding abel map a l c j which maps c into its compactified jacobian and form its pullback map a l pic j 0 j which carries the connected component of 0 in the picard scheme back to the jacobian if c has at worst double points then a l is known to be an isomorphism we prove that a l always extends to a map between the natural compactifications pic j j and that the extended map is an isomorphism if c has at worst ordinary nodes and cusps

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

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

mirror symmetry langlands duality and the hitchin system among the major mathematical approaches to mirror symmetry are those of batyrev borisov and strominger yau zaslow syz the first is explicit and amenable to computation but is not clearly related to the physical motivation the second is the opposite furthermore it is far from obvious that mirror partners in one sense will also be mirror partners in the other this paper concerns a class of examples that can be shown to satisfy the requirements of syz but whose hodge numbers are also equal this provides significant evidence in support of syz moreover the examples are of great interest in their own right they are spaces of flat slr connections on a smooth curve the mirror is the corresponding space for the langlands dual group pglr these examples therefore throw a bridge from mirror symmetry to the duality theory of lie groups and more broadly to the geometric langlands program