flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

self similar subsets of the cantor set abstract in this paper we study the following question raised by mattila in 1998 what are the self similar subsets of the middle third cantor set c we give criteria for a complete classification of all such subsets we show that for any self similar subset f of c containing more than one point every linear generating ifs of f must consist of similitudes with contraction ratios 3 n n n in particular a simple criterion is formulated to characterize self similar subsets of c with equal contraction ratio in modulus

on the spectra of a class of self affine measures on r2 abstract for i 1 2 let 0 i 1 for an expanding real matrix m d i a g 1 1 2 1 and a three element digit set d z 2 with cardinality d let m d be the self affine measure defined by m d 1 d d d m d m d let f 3 2 1 3 l 1 l 2 t l 1 l 2 n 0 l 1 l 2 2 and z m d x r 2 d d e 2 i d x 0 if a f 3 2 z m d a mod z 2 and x 0 1 2 d d e 2 i d x d 0 0 t then l 2 m d has an exponential orthonormal basis if and only if i 1 3 z 0 for i 1 2

hitchin s self duality equations on complete riemannian manifolds let m be a complete riemannian manifold let g be a real semisimple noncompact algebraic group and let k be its maximal compact subgroup let p be a principal g bundle over m fix a reduction of the structure group g of p to k and consider the space d of smooth g connections on p d is an affine space modeled on the vector space 21 ad p of 1 forms on m with values in the adjoint bundle associated to p for any connection d e d we have a decomposition d d 0 where d is a connection preserving the k structure and 0 is a 1 form on m with values in the orthogonal complement in ad p of ad pk here px is the k bundle defined by the reduction of structure group i f d is a flat connection we have

nonlinear stability of traveling waves to a hyperbolic parabolic system modeling chemotaxis we prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic parabolic system modeling repulsive chemotaxis in contrast to the previous related results where various sm

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

on the long time behavior of type iii ricci flow solutions we show that three dimensional homogeneous ricci flow solutions that admit finite volume quotients have long time limits given by expanding solitons we show that the same is true for a large class of four dimensional homogeneous solutions we give an extension of hamilton s compactness theorem that does not assume a lower injectivity radius bound in terms of riemannian groupoids using this we show that the long time behavior of type iii ricci flow solutions is governed by the dynamics of an mathbb r action on a compact space

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

on one dimensional self similar tilings and pq tiles let b 2 be an integer base d 0 d 1 d b 1 z a digit set and t t b d the set of radix expansions it is well known that if t has nonvoid interior then t can tile r with some translation set j t is called a tile and d a tile digit set there are two fundamental questions studied in the literature i describe the structure of j ii for a given b characterize d so that t is a tile we show that for a given pair b d there is a unique self replicating translation set j c z and it has period b m for some m n this completes some earlier work of kenyon our main result for ii is to characterize the tile digit sets for b pq when p q are distinct primes the only other known characterization is for b p l due to lagarias and wang the proof for the pq case depends on the techniques of kenyon and de bruijn on the cyclotomic polynomials and also on an extension of the product form digit set of odlyzko

on the monodromy of the moduli space of calabi yau threefolds coming from eight planes in mathbb p 3 it is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as zariski open subsets of locally hermitian symmetric domains in the present work we prove that the moduli space of calabi yau threefolds coming from eight planes in mathbb p 3 does not have this property we show furthermore that the monodromy group of a good family is zariski dense in the corresponding symplectic group moreover we study a natural sublocus which we call hyperelliptic locus over which the variation of hodge structures is naturally isomorphic to wedge product of a variation of hodge structures of weight one it turns out the hyperelliptic locus does not extend to a shimura subvariety of type iii siegel space within the moduli space besides general hodge theory representation theory and computational commutative algebra one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of hodge structures

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

polarized variation of hodge structures of calabi yau type and characteristic subvarieties over bounded symmetric domains in this paper we extend the construction of the canonical polarized variation of hodge structures over tube domain considered by gross math res lett 1 1 9 1994 to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of hodge structures which we call characteristic subvarieties we prove that the characteristic subvariety of the canonical polarized variations of hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by mok ann math 125 1 105 152 1987 we verified the generating property of gross for all irreducible bounded symmetric domains which was predicted in gross math res lett 1 1 9 1994

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity abstract this paper establishes the nonlinear stability of traveling wave solutions to a chemotaxis model with singular or logarithmic sensitivity and its transformed parabolic hyperbolic system depending on the parameter signs we discuss the linear instability of traveling wave solutions using the spectral analysis and nonlinear asymptotic stability of traveling wave solutions with zero end state by the weighted energy estimates where the latter result solves the open question left in a previous work li and wang 2009 7

exact hausdorff measure and intervals of maximum density for cantor sets consider a linear cantor set k which is the attractor of a linear iterated function system i f s sjx pjx bj j 1 m on the line satisfying the open set condition where the open set is an interval it is known that k has hausdorff dimension ca given by the equation ftl1 pa 1 and that 7 t k is finite and positive where 7h x denotes hausdorff measure of dimension a we give an algorithm for computing 7 t k exactly as the maximum of a finite set of elementary functions of the parameters of the i f s when pi pm or more generally if log p1 and log pm are commensurable the algorithm also gives an interval i that maximizes the density d i 7 ha k n i ii i the hausdorff measure 7 o k is not a continuous function of the i f s parameters we also show that given the contraction parrameters pj it is possible to choose the translation parameters bj in such a way that 7 h k ikia so the maximum density is one most of the results presented here were discovered through computer experiments but we give traditional mathematical proofs

limits of families of brieskorn lattices and compactified classifying spaces we investigate variations of brieskorn lattices over non compact parameter spaces and discuss the corresponding limit objects on the boundary divisor we study the associated variation of twistors and the corresponding limit mixed twistor structures we construct a compact classifying space for regular singular brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space

polarized mixed hodge structures and the local monodromy of a variation of hodge structure a variation of polarized hodge structures on the complement of a divisor with normal crossings gives rise locally on the divisor to a commuting set ni of nilpotent endomorphisms of the vector space underlying the variation the logarithms of the unipotent parts of the various picard lefschetz transformations these reflect properties of the singularities of the period map associated to the variation cf 5 7 9 and sect 3 for example a variation depending on a single parameter defines asymptotically a mixed hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism n 9 11 this paper is mainly concerned with the properties of the set n arising in the several parameters case in particular theorem 3 3 asserts that all the elements in the open polyhedral cone c spanned over r by the n s determine the same monodromy weight filtration it also describes the relationship between this common filtration and those associated to the faces of the cone this statement was conjectured by deligne based on his analogous result 4 1 9 2 for the q cone associated to those variations which arise from families of polarized non singular algebraic varieties in w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations section2 focuses on the case when each element in such set defines in the sense of 2 4 a polarized mixed hodge structure with a fixed hodge filtration f as is the case for the monodromy cone c of a variation these conditions imply cf 2 16 a generic uniqueness for the monodromy weight filtration we also consider those pola

self similar measures and intersections of cantor sets it is natural to expect that the arithmetic sum of two cantor sets should have positive lebesgue measure if the sum of their dimensions exceeds 1 but there are many known counterexamples e g when both sets are the middle ca cantor set and ca e 1 1 we show that for any compact set k and for a e ag e 0 1 the arithmetic sum of k and the middle ae cantor set does indeed have positive lebesgue measure when the sum of their hausdorff dimensions exceeds 1 in this case we also determine the essential su premum as the translation parameter t varies of the dimension of the intersection of k t with the middle ca cantor set we also establish a new property of the infinite bernoutlli convolutions lj the distributions of random series e 0 a where the signs are chosen independently with probabilities p 1 p let 1 ql q2 2 for p 2 near 2 and for a e a in some nonempty interval iv is absolutely continuous 2 1 and its density is in lil but not in lq2 we also answer a question of kahane concerning the fourier transform of vl 2

generating iterated function systems of some planar self similar sets abstract suppose e e is a planar self similar set where e r is a self similar set satisfying the strong separation condition with 0 dim h e 1 by revealing a relationship between algebraic numbers cos 2 l and sin 2 l with l z we prove that for all generating iterated function systems ifss of e e the associated orthogonal matrices have at most 8 possibilities additionally we investigate all generating ifss of two typical planar self similar sets one has a minimal presentation every other generating ifs is an iteration of this one while the other has not

global generalized solutions to a keller segel system with nonlinear diffusion and singular sensitivity abstract we consider the chemotaxis system with nonlinear diffusion and singular sensitivity u t u m u v v x t 0 v t v u v x t 0 u v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x in a smooth bounded domain r n n 2 in this work it is shown that for all reasonably regular initial data u 0 0 and v 0 0 the corresponding neumann initial boundary value problem possesses a global generalized solution provided that m 1 n 2 2 n

parabolic higgs bundles and teichmuller spaces for punctured surfaces in this paper we study the relation between parabolic higgs vector bundles and irreducible representations of the fundamental group of punctured riemann surfaces established by simpson we generalize a result of hitchin identifying those parabolic higgs bundles that correspond to fuchsian representations we also study the higgs bundles that give representations whose image is contained after conjugation in sl k r we compute the real dimension of one of the components of this space of representations which in the absence of punctures is the generalized teichmuller space introduced by hitchin and which in the case of k 2 is the usual teichmuller space of the punctured surface

infinitesimal variation of hodge structures and the weak global torelli theorem for complete intersections in this paper we will study the weak global torelli theorem for complete intersections of hypersurfaces in a projective space let x be a non singular projective variety of dimension r with an ample line bundle l over the complex number field c then the primitive part hrm x z of the cohomology of x has a polarized hodge structure if the moduli space x of x exists we can define the global period map ho d r from the moduli space x to the classifying space d r of hodge structures attaching the point p in x to the hodge structure of the primitive part hrim xp z of the cohomology group of xp p e 4

the monodromy groups of dolgachev s cy moduli spaces are zariski dense abstract let m n 2 n 2 be the coarse moduli space of cy manifolds arising from a crepant resolution of double covers of p n branched along 2 n 2 hyperplanes in general position we show that the monodromy group of a good family for m n 2 n 2 is zariski dense in the corresponding symplectic or orthogonal group if n 3 in particular the period map does not give a uniformization of any partial compactification of the coarse moduli space as a shimura variety whenever n 3 this disproves a conjecture of dolgachev as a consequence the fundamental group of the coarse moduli space of m ordered points in p n is shown to be large once it is not a point similar zariski density result is obtained for moduli spaces of cy manifolds arising from cyclic covers of p n branched along m hyperplanes in general position a classification towards the geometric realization problem of b gross for type a bounded symmetric domains is given

self similar structure on intersections of triadic cantor sets abstract let c be the triadic cantor set we characterize the all real number such that the intersection c c is a self similar set and investigate the form and structure of the all iterated function systems which generate the self similar set

arakelov inequalities and the uniformization of certain rigid shimura varieties let y be a non singular projective manifold with an ample canonical sheaf and let v be a q variation of hodge structures of weight one on y with higgs bundle e 1 0 e 0 1 coming from a family of abelian varieties if y is a curve the arakelov inequality says that the slopes satisfy e 1 0 e 0 1 1 we prove a similar inequality in the higher dimensional case if the latter is an equality and if the discriminant of e 1 0 or the one of e 0 1 is zero one hopes that y is a shimura variety and v a uniformizing variation of hodge structures this is verified in case the universal covering of y does not contain factors of rank 1 part of the results extend to variations of hodge structures over quasi projective manifolds u

family of curves with large unitary summand in the hodge bundle in this note we construct non isotrivial families of curves of genus g ge 2 where the rank of the unitary summand contained in the hodge bundle can be as large as 2g 1 3 and hence disprove xiao s conjecture for the unitary rank

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

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

harmonic metrics on higgs sheaves and uniformization of varieties of general type we prove a criterion for the existence of harmonic metrics on higgs bundles that are defined on smooth loci of klt varieties as one application we resolve the quasi etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete co compact groups of automorphisms that act freely in codimension one as a further application we establish a nonabelian hodge correspondence on smooth loci of klt varieties

non self similar sets in 0 1 n of arbitrary dimension abstract we consider 0 1 n the unit cube of r n n 1 let s s 1 s m be a finite set of contraction maps from x to itself a non empty subset e of x is an attractor or an invariant set for the iterated function system ifs s if e i 1 m s i e we construct for each s 0 n a nowhere dense perfect set e contained in 0 1 n with hausdorff dimension s which is not an attractor for any iterated function system composed of weak contractions from 0 1 n to itself

locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion abstract we show the existence of locally bounded global solutions to the chemotaxis system u t d u u u v v in 0 v t v u v in 0 u v 0 in 0 u 0 u 0 v 0 v 0 in in smooth bounded domains r n n 2 for d u u m 1 with some 0 provided that m 1 n 4

renormalized radial large data solutions to the higher dimensional keller segel system with singular sensitivity and signal absorption abstract the chemotaxis system u t u u v v v t v u v is considered under homogeneous neumann boundary conditions in the ball b r 0 r n where r 0 and n 2 despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria since its introduction by keller and segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well posedness global existence results in the literature are restricted to spatially one or two dimensional cases so far or alternatively require certain smallness hypotheses on the initial data for all suitably regular and radially symmetric initial data u 0 v 0 satisfying u 0 0 and v 0 0 the present paper establishes the existence of a globally defined pair u v of radially symmetric functions which are continuous in 0 0 and smooth in 0 0 and which solve the corresponding initial boundary value problem for with u 0 v 0 u 0 v 0 in an appropriate generalized sense to the best of our knowledge this in particular provides the first result on global existence for the three dimensional version of involving arbitrarily large initial data

a characterization of certain shimura curves in the moduli stack of abelian varieties let f x y be a semistable family of complex abelian varieties over a curve y of genus g y and smooth over the com

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

an analogue of cobham s theorem for graph directed iterated function systems abstract feng and wang showed that two homogeneous iterated function systems in r with multiplicatively independent contraction ratios necessarily have different attractors in this paper we extend this result to graph directed iterated function systems in r n with contraction ratios that are of the form 1 for integers by using a result of boigelot et al this allows us to give a proof of a conjecture of adamczewski and bell in doing so we link the graph directed iterated function systems to buchi automata in particular this link extends to real numbers we introduce a logical formalism that permits to characterize sets of r n whose representations in base are recognized by some buchi automata this result depends on the algebraic properties of the base being a pisot or a parry number the main motivation of this work is to draw a general picture representing the different frameworks where an analogue of cobham s theorem is known

on the structures of generating iterated function systems of cantor sets abstract a generating ifs of a cantor set f is an ifs whose attractor is f for a given cantor set such as the middle 3rd cantor set we consider the set of its generating ifss we examine the existence of a minimal generating ifs i e every other generating ifs of f is an iterating of that ifs we also study the structures of the semi group of homogeneous generating ifss of a cantor set f in r under the open set condition osc if dim h f 1 we prove that all generating ifss of the set must have logarithmically commensurable contraction factors from this logarithmic commensurability theorem we derive a structure theorem for the semi group of generating ifss of f under the osc we also examine the impact of geometry on the structures of the semi groups several examples will be given to illustrate the difficulty of the problem we study

characterization of calabi yau variations of hodge structure over tube domains by characteristic forms sheng and zuo s characteristic forms are invariants of a variation of hodge structure we show that they characterize gross s canonical variations of hodge structure of calabi yau type over hermitian symmetric tube domains

unitary local systems multiplier ideals and polynomial periodicity of hodge numbers abstract the space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles we show that multiplier ideals provide natural stratifications of this space we prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes generalizing to the quasi projective case results of green lazarsfeld and simpson as an application we show the polynomial periodicity of hodge numbers h q 0 of congruence covers in any dimension generalizing results of e hironaka and sakuma we extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems in particular we obtain a generalization of the polynomial periodicity of betti numbers of unbranched congruence covers due to sarnak adams we derive a geometric characterization of finite abelian covers which recovers the classic one and the one of pardini we use this for example to prove a conjecture of libgober about hodge numbers of abelian covers

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

variations of hodge structure and orbits in flag varieties abstract period domains the classifying spaces for pure polarized hodge structures and more generally mumford tate domains arise as open g r orbits in flag varieties g p we investigate hodge theoretic aspects of the geometry and representation theory associated with these flag varieties in particular we relate the griffiths yukawa coupling to the variety of lines on g p under a minimal homogeneous embedding construct a large class of polarized g r orbits in g p and compute the associated hodge theoretic boundary components an emphasis is placed throughout on adjoint flag varieties and the corresponding families of hodge structures of levels two and four

uniqueness theorems of hermitian metrics of seminegative curvature on quotients of bounded symmetric domains in the theory of locally symmetric spaces of negative ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry and in the hermitian case the complex structure the rigidity theorem of mostow 26 asserts that for a compact locally symmetric riemannian manifold of negative ricci curvature the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds with the obvious exceptions involving compact riemann surfaces the same theorem for locally symmetric riemannian manifolds of finite volume and negative ricci curvature was proved by prasad 27 in case of rank 1 and included in the super rigidity theorem of margulis 13 in the case of rank 2 in the class of compact locally symmetric hermitian manifolds of negative ricci curvature calabi vesentini 5 and borel 3 proved vanishing theorems of certain cohomology groups which imply in particular that for x locally irreducible of complex dimension 2 there exists no non trivial deformation of x as a complex manifold in this direction siu 32 33 proved the strong rigidity of the kdhler manifold x in the sense that any compact kkhler manifold m homotopic to x is necessarily biholomorphic or conjugate biholomorphic to x if m is also locally symmetric by assumption then it follows from the uniqueness of kihler einstein metrics of negative ricci curvature yau 37 that the conjugate biholomorphism is in fact an isometry up to a normalizing constant it is thus natural to ask

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

holomorphic vector bundles on sasakian threefolds and parabolic vector bundles a bijective correspondence is constructed between the parabolic vector bundles with rational parabolic weights on a riemann surface x and the holomorphic vector bundles on quasiregular sasakian threefolds over x this bijective correspondence preserves the stable objects

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

fractafolds based on the sierpinski gasket and their spectra we introduce the notion of fractafold which is to a fractal what a manifold is to a euclidean half space we specialize to the case when the fractal is the sierpinski gasket sg we show that each such compact fractafold can be given by a cellular construction based on a finite cell graph g which is 3 regular in the case that the fractafold has no boundary we show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph using the spectral decimation method of fukushima and shima this enables us to obtain isospectral pairs of nonhomeomorphic fractafolds we also show that although sg is topologically rigid there are fractafolds based on sg that are not topologically rigid

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c