applications of the complex interpolation method to a von neumann algebra non commutative lp spaces non commutative lp spaces 1 p associated with a von neumann algebra are considered the paper consists of two parts in part i by making use of the complex interpolation method non commutative lp spaces are defined as interpolation spaces between the von neumann algebra in question and its predual also all expected properties such as duality and uniform convexity are proved in the frame of interpolaton theory and relative modular theory in part ii these lp spaces are compared with haagerup s lp spaces based on this comparison a non commutative analogue of the classical stein weiss interpolation theorem is obtained

gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

hamilton s gradient estimates and a monotonicity formula for heat flows on metric measure spaces abstract in this paper we extend the hamilton s gradient estimates hamilton 1993 and a monotonicity formula of entropy ni 2004 for heat flows from smooth riemannian manifolds to non smooth metric measure spaces with appropriate riemannian curvature dimension condition

congruences between derivatives of abelian l functions at s 0 let k k be a finite abelian extension of global fields we prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s 0 of derivatives of the dirichlet l functions associated to k k we also show that these congruences provide a universal approach to the refined abelian stark conjectures formulated by inter alia stark gross rubin popescu and tate we thereby obtain the first proofs of amongst other things the rubin stark conjecture and the refined class number formulas of both gross and tate for all extensions k k in which k is either an abelian extension of or is a function field

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

the arithmetic of function fields 2 the cyclotomic theory contents i the ciassical rheoy qf cutx es orer a finite jeld 2 the cfclotomic construction for number jelds and function jields 2 1 number fields 2 2 function fields 3 complex arld p adic zeta functions 3 1 the complex theory 3 2 the p adic theory 4 classical iwasabila rheory 4 1 kummer s criterion 4 2 the theoq of the tower 5 two variable zera functions 5 1 the theory at infinity 5 2 special values 6 the theory for futzction fields 6 1 the basic set up 6 2 the kummer criterion 6 3 the theory of the tower 7 l nirs 7 i number fields 7 2 function fields references

stark units in positive characteristic we show that the module of stark units associated to a sign normalized rank one drinfeld module can be obtained from anderson s equivariant a harmonic series we apply this to obtain a class formula a la taelman and to prove a several variable log algebraicity theorem generalizing anderson s log algebraicity theorem we also give another proof of anderson s log algebraicity theorem using shtukas and obtain various results concerning the module of stark units for drinfeld modules of arbitrary rank

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

arithmetic of function field units we introduce the module of stark units associated to a drinfeld module defined over the ring of integers of a global function field we investigate the arithmetic properties of these objects and as an application we prove a discrete analogue for taelman s class modules of certain conjectures formulated by greenberg for cyclotomic fields

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

the canonical trace and the noncommutative residue on the noncommutative torus using a global symbol calculus for pseudodifferential operators on tori we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols we characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed resp any non integer order on the grounds of this uniqueness result we prove that in the commutative setup this canonical trace on the noncommutative torus reduces to kontsevich and vishik s canonical trace which is thereby identified with a discrete sum a similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace class operators generalises fathizadeh and wong s characterisation in so far as it includes the case of operators of fixed integer order by means of the canonical trace we derive defect formulae for regularized traces the conformal invariance of the function at zero of the laplacian on the noncommutative torus is then a straightforward consequence

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

proof of an exceptional zero conjecture for elliptic curves over function fields based on the analogy between number fields and function fields of one variable over finite fields we formulate and prove an analogue of the exceptional zero conjecture of mazur tate and teitelbaum for elliptic curves defined over function fields the proof uses modular parametrization by drinfeld modular curves and the theory of non archimedean integration as an application we prove a refinement of the birch swinnerton dyer conjecture if the analytic rank of the elliptic curve is zero

characterization of su q l 1 equivariant spectral triples for the odd dimensional quantum spheres the quantum group suq l 1 has a canonical action on the odd dimensional sphere s 2l 1 q all odd spectral triples acting on the l2 space of s 2l 1 q and equivariant under this action have been characterized this characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial k homology class these generalize the results of chakraborty pal for suq 2 ams subject classification no 58b34 46l87 19k33

universal gauss thakur sums and l series we prove that the maximal abelian extension tamely ramified at infinity of the rational function field over mathbb f q is generated by the values at the points in the algebraic closure of mathbb f q of the higher derivatives of the so called anderson and thakur function omega we deduce a similar property for the special values of the higher derivatives of a new kind of l series introduced by the second author

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

symmetries of levy processes on compact quantum groups their markov semigroups and potential theory quantum markov semigroups qms i e strongly continuous semigroups of unital completely positive maps on compact quantum groups are studied we show that translation invariant qmss on the universal or reduced c algebra of a compact quantum group are in one to one correspondence with levy processes on its hopf algebra we use the theory of levy processes on involutive bialgebras to characterize symmetry properties of the associated qms it turns out that the qms is self adjoint resp kms symmetric if and only if the generating functional of the levy process is invariant under the antipode resp the unitary antipode furthermore we study levy processes whose marginal states are invariant under the adjoint action finally some related aspects of the potential theory as dirichlet form derivation and spectral triple are investigated we discuss how the above results apply to compact groups group c algebras of discrete groups free orthogonal quantum groups on and twisted suq 2 quantum group

gamma functions for function fields and drinfeld modules the purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field in particular we explore among other things their connection with the theory of drinfeld modules of rank one which is function field cyclotomic theory their interpolations functional equations the nature of their special values transcendence algebraicity connection with periods and various analogies most of the facts about the classical gamma function are recalled when necessary readers may refer to g k or artin s book on the classical gamma function now we describe more precisely the contents of this paper let fq be a finite field of characteristic p section 0 covers the background material on drinfeld modules it also describes the gamma function f for fq t with

diameter estimate for compact quasi einstein metrics in this paper we prove a gradient estimate for the potential function of a nontrivial compact quasi einstein metric with constant 0 based on this gradient estimate we derive the lower bound estimate for the diameter

congruences between derivatives of geometric l functions we prove a natural refinement of a theorem of lichtenbaum describing the leading terms of zeta functions of curves over finite fields in terms of weil etale cohomology we then use this result to prove the validity of chinburg s 3 conjecture for all abelian extensions of global function fields to prove natural refinements and generalisations of the refined stark conjectures formulated by amongst others gross tate rubin and popescu to prove a variety of explicit restrictions on the galois module structure of unit groups and divisor class groups and to describe explicitly the fitting ideals of certain weil etale cohomology groups in an appendix coauthored with k f lai and k s tan we also show that the main conjectures of geometric iwasawa theory can be proved without using either crystalline cohomology or drinfeld modules

values of certain l series in positive characteristic we introduce a class of deformations of the values of the goss zeta function we prove with the use of the theory of deformations of vectorial modular forms as well as with other techniques a formula for their value at 1 and some arithmetic properties of values at other positive integers our formulas involve anderson and thakur s function we discuss how our formulas may be used to investigate the existence of a kind of functional equation for the goss zeta function

noncommutative geometry and reality we introduce the notion of real structure in our spectral geometry this notion is motivated by atiyah s kr theory and by tomita s involution j it allows us to remove two unpleasant features of the connes lott description of the standard model namely the use of bivector potentials and the asymmetry in the poincare duality and in the unimodularity condition

shtukas and jacobi sums we show how the analogues of jacobi sums in the context of function fields introduced and studied in t1 t2 t3 can be obtained from shtukas introduced and studied in d2 d3 m we apply this to obtain some results on the prime factorization of analogues of gauss sums and to prove an analogue of the gross koblitz formula for general function field generalizing the results in t2 for this purpose we also introduce and interpolate a new analogue of gamma function

on special l values of t modules we show that taelman s conjecture on special l values of ander son t modules holds for a large class of t modules this class contains all mixed a finite and uniformizable t modules whose hodge pink weights are at least 1 as a consequence we deduce various log algebraicity identities for tensor powers of the carlitz module generalizing the work of anderson thakur

new monotonicity formulas for ricci curvature and applications i we prove three new monotonicity formulas for manifolds with a lower ricci curvature bound and show that they are connected to rate of convergence to tangent cones in fact we show that the derivative of each of these three monotone quantities is bounded from below in terms of the gromov hausdorff distance to the nearest cone the monotonicity formulas are related to the classical bishop gromov volume comparison theorem and perelman s celebrated monotonicity formula for the ricci flow we will explain the connection between all of these moreover we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the green function that we prove this is parallel to the fact that perelman s monotonicity is closely related to the sharp gradient estimate for the heat kernel of li yau in cm4 one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross sections of einstein manifolds finally there are obvious parallelisms between our monotonicity and the positive mass theorem of schoen yau and witten

generalized stark formulae over function fields we establish formulae of stark type for the stickelberger elements in the function field setting our result generalizes a work of hayes and a conjecture of gross it is used to deduce a p adic version of rubin stark conjecture and burns conjecture

deforming maps for lie group covariant creation and annihilation operators any deformation of a weyl or clifford algebra a can be realized through a deforming map i e a formal change of generators in a this is true in particular if a is covariant under a lie algebra g and its deformation is induced by some triangular deformation uhg of the hopf algebra ug we propose a systematic method to construct all the corresponding deforming maps together with the corresponding realizations of the action of uhg the method is then generalized and explicitly applied to the case that uhg is the quantum group uhsl 2 a preliminary study of the status of deforming maps at the representation level shows in particular that deformed fock representations induced by a compact uhg can be interpreted as standard undeformed fock representations describing particles with ordinary bose or fermi statistics

monotonicity formulas via the bakry emery curvature abstract we prove several monotone formulas for smooth metric measure spaces with nonnegative m bakry emery curvature first we define a f r and v f r for the weighted green function and establish the monotone formulas these formulas are parallel to the weighted volume comparison theorem on smooth metric measure spaces with nonnegative m bakry emery curvature second we define the weighted nash entropy fisher information and perelman s entropy then we get the monotone formulas along the weighted heat equation via the m bakry emery curvature

the distribution of the zeros of the goss zeta function for a mathbb f _2 x y y 2 y x 3 x 1 let f be a global function field over a finite constant field and infty a place of f the ring a of functions regular away from infty in f is a dedekind domain for such a goss defined a zeta function which is a continuous function from mathbb z p to the ring of entire power series with coefficients in the completion f infty of f at infty he asks what one can say about the distribution of the zeros of the entire function at any parameter of mathbb z p in the simplest case a is the polynomial ring in one variable over a finite field here the question was settled completely by j sheats after previous work by j diaz vargas b poonen and d wan for any parameter in mathbb z p the zeros of the power series have pairwise different valuations and they lie in f infty in the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings namely a mathbb f 2 x y y 2 y x 3 x 1 this a has class number 1 it is the affine coordinate ring of a supersingular elliptic curve and the place infty is mathbb f 2 rational the answer is slightly different from the above case of polynomial rings for arbitrary a such that infty is a rational place of f we describe a pattern in the distribution of zeros which we observed in some computational experiments finally we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic a of genus 2

noncommutative manifolds the instanton algebra and isospectral deformations we give new examples of noncommutative manifolds that are less standard than the nc torus or moyal deformations of n they arise naturally from basic considerations of noncommutative differential topology and have non trivial global features

the surface c c on jacobi varieties and 2nd order theta functions preprint enviat per a la seva publicacio en una revista cientifica acta mathematica volume 157 1986 1 22 https projecteuclid org euclid acta 1485890436

integrable systems and number theory in finite characteristic the purpose of this paper is to give an overview of applications of the concepts and techniques of the theory of integrable systems to number theory in finite characteristic the applications include explicit class field theory and langlands conjectures for function fields effect of the geometry of the theta divisor on factorization of analogs of gauss sums special values of function field gamma zeta and l functions analogs of theorems of weil and stickelberger control of the intersection of the jacobian torsion with the theta divisor the techniques are the krichever drinfeld dictionaries and the theory of solitons akhiezer baker and tau functions developed in this context of arithmetic geometry by anderson 2001 elsevier science b v all rights reserved

gauss sums for fq t the purpose of this paper is to define gauss sums taking values in function fields of one variable over a finite field and to prove analogues of various classical and recent results these results include stickelberger s theorem the hasse davenport theorem weil s theorem on jacobi sums as hecke characters and the gross koblitz theorem for comparison the reader may consult g k and references given there

the hirota equation over finite fields algebro geometric approach and multisoliton solutions we consider the hirota equation the discrete analogue of the generalized toda system over a finite field we present the general algebro geometric method of construction of solutions of the equation as an example we construct analogues of the multisoliton solutions for which the wavefunctions and the function can be found using rational functions within the class of multisoliton solutions we isolate generalized breather type solutions which have no direct counterparts in the complex field case

smooth dense subalgebras and fourier multipliers on compact quantum groups we define and study dense frechet subalgebras of compact quantum groups realised as smooth domains associated with a dirac type operator with compact resolvent further we construct spectral triples on compact matrix quantum groups in terms of clebsch gordon coefficients and the eigenvalues of the dirac operator mathcal d grotendieck s theory of topological tensor products immediately yields a schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo differential operators on them it is also shown that regular pseudo differential operators are closed under compositions as a by product we develop elements of the distribution theory and corresponding fourier analysis we give applications of our construction to obtain sufficient conditions for lp lq boundedness of coinvariant linear operators we provide necessary and sufficient conditions for algebraic differential calculi on hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra c infty mathcal d we check explicitly that these conditions hold true on the quantum su2q for both its 3 dimensional and 4 dimensional calculi

on lfp spectrum and τ quasi einstein metric abstract in this paper we study l f p spectrum estimates for some smooth potential function f on complete noncompact riemannian manifolds via the bakry emery curvature as its applications we give the upper bound estimate of the l f 2 spectrum for the complete noncompact quasi einstein metric we give an example to show the sharpness of our estimate we also get a lower bound estimate when 0 1 and 0

interpolation and φ moment inequalities of noncommutative martingales this paper is devoted to the study of moment inequalities for noncommutative martingales in particular we prove the noncommutative moment analogues of martingale transformations stein s inequalities khintchine s inequalities for rademacher s random variables and burkholder gundy s inequalities the key ingredient is a noncommutative version of marcinkiewicz type interpolation theorem for orlicz spaces which we establish in this paper

large scale behavior of kernels of schrodinger operators in this paper we study the asymptotic behavior of solutions to schrodinger equations on manifolds with nonnegative ricci curvature and euclidean volume growth as a first application we get a sharp asymptotic estimate on the green s function for the laplacian which sharpens results of li and yau next we establish a definite gap in the possible rates of decay of exterior harmonic functions on these manifolds when the manifold also has quadratic curvature decay we get supremum estimates on the c 2 norm of the green s function this has function theoretic implications for these manifolds we then give a generalization to manifolds with nonnegative ricci curvature outside a compact set finally we study schrodinger operators where the potential decays quadratically at infinity in this setting we obtain a geometric version at infinity of a classical theorem of bers

taylor coefficients of anderson thakur series and explicit formulae for each positive characteristic multiple zeta value defined by thakur in function field arithmetic world scientific publishing river edge 2004 the first and third authors in chang and mishiba invent math 2020 https doi org 10 1007 s00222 020 00988 1 constructed a t module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value the objective of this paper is to use the taylor coefficients of anderson thakur series and t motivic carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector

perelman s entropy formula for the witten laplacian on riemannian manifolds via bakry emery ricci curvature in this paper we study perelman s mathcal w entropy formula for the heat equation associated with the witten laplacian on complete riemannian manifolds via the bakry emery ricci curvature under the assumption that the m dimensional bakry emery ricci curvature is bounded from below we prove an analogue of perelman s and ni s entropy formula for the mathcal w entropy of the heat kernel of the witten laplacian on complete riemannian manifolds with some natural geometric conditions in particular we prove a monotonicity theorem and a rigidity theorem for the mathcal w entropy on complete riemannian manifolds with non negative m dimensional bakry emery ricci curvature moreover we give a probabilistic interpretation of the mathcal w entropy for the heat equation of the witten laplacian on complete riemannian manifolds and for the ricci flow on compact riemannian manifolds

the dirac operator on compact quantum groups for the q deformation gq 0 q 1 of any simply connected simple compact lie group g we construct an equivariant spectral triple which is an isospectral deformation of that defined by the dirac operator d on g our quantum dirac operator dq is a unitary twist of d considered as an element of ug cl g the commutator of dq with a regular function on gq consists of two parts one is a twist of a classical commutator and so is automatically bounded the second is expressed in terms of the commutator of the associator with an extension of d we show that in the case of the drinfeld associator the latter commutator is also bounded