homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

a homological representation formula of colored alexander invariants we give a formula of the colored alexander invariant in terms of the homological representation of the braid groups which we call truncated lawrence s representation this formula generalizes the famous burau representation formula of the alexander polynomial

semi martingale inequalities via the garsia rodemich rumsey lemma and applications to local times abstract with the help of the garsia rodemich rumsey lemma some improvements of the burkholder davis gundy inequalities for continuous martingales are obtained as well as certain estimates of the l p norm of the supremum of a the local times of a continuous semi martingale and b the difference of the local times of two continuous martingales

parameters for which the lawrence krammer representation is reducible we show that the representation introduced by lawrence and krammer to show the linearity of the braid group is generically irreducible however for some values of its two parameters when these are specialized to complex numbers it becomes reducible we construct a representation of degree n n 1 2 of the bmw algebra of type a n 1 as a representation of the braid group on n strands it is equivalent to the lawrence krammer representation where the two parameters of the bmw algebra are related to those appearing in the lawrence krammer representation we give the values of the parameters for which the representation is reducible and give the proper invariant subspaces in some cases we use this representation to show that for these special values of the parameters the bmw algebra of type a n 1 is not semisimple

homological representations of the hecke algebra in this paper a topological construction of representations of thean 1 series of hecke algebras associated with 2 row young diagrams will be given this construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection the fibres of the vector bundle are homology spaces of configuration spaces of points in c with a suitable twisted local coefficient system it is also shown that there is a close correspondence between this construction and the work of tsuchiya and kanie who constructed hecke algebra representations from the monodromy ofn point functions in a conformal field theory onp1 this work has significance in relation to the one variable jones polynomial which can be expressed in terms of characters of the iwahori hecke algebras associated with 2 row young diagrams it gives rise to a topological description of the jones polynomial which will be discussed elsewhere l2

noncommutative martingale deviation and poincare type inequalities with applications we prove a deviation inequality for noncommutative martingales by extending oliveira s argument for random matrices by integration we obtain a burkholder type inequality with satisfactory constant using continuous time we establish noncommutative poincare type inequalities for nice semigroups with a positive curvature condition these results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to bobkov and gotze in the commutative case to demonstrate our setting is general enough we give various examples including certain group von neumann algebras random matrices and classical diffusion processes among others

on twisted lawrence krammer representations abstract lawrence krammer representations are an important family of linear representations of artin tits groups of small type which are known under some assumptions on the parameters to be faithful when the type is spherical or more generally when they are restricted to the artin tits monoid and irreducible when the type is connected here we investigate an analogue of these representations introduced by digne in the spherical cases for every artin tits monoid that appears as the submonoid of fixed points of an artin tits monoid of small type under a group of graph automorphisms and for the corresponding artin tits group under the same assumptions on the parameters as in the small type cases we first show that these so called twisted lawrence krammer representations are faithful and we then prove by computing their formulas when the group of graph automorphisms is of order two or three their irreducibility in all the spherical and connected cases but one

bmw algebras of simply laced type abstract it is known that the recently discovered representations of the artin groups of type a n the braid groups can be constructed via bmw algebras we introduce similar algebras of type d n and e n which also lead to the newly found faithful representations of the artin groups of the corresponding types we establish finite dimensionality of these algebras moreover they have ideals i 1 and i 2 with i 2 i 1 such that the quotient with respect to i 1 is the hecke algebra and i 1 i 2 is a module for the corresponding artin group generalizing the lawrence krammer representation finally we give conjectures on the structure the dimension and parabolic subalgebras of the bmw algebra as well as on a generalization of deformations to brauer algebras for simply laced spherical type other than a n

the braid group b4 is linear we study a linear representation b n gl m z q 1 t 1 with m n n 1 2 we will show that for n 4 this representation is faithful we prove a relation with the new charney length function we formulate a conjecture implying that is faithful for all n

braid groups are linear in a previous work 11 the author considered a representation of the braid group p bn glm z ql1 t l m n n 1 2 and proved it to be faithful for n 4 bigelow 3 then proved the same representation to be faithful for all n by a beautiful topological argument the present paper gives a different proof of the faithfulness for all n we establish a relation between the charney length in the braid group and exponents of t a certain bn invariant subset of the module is constructed whose properties resemble those of convex cones we relate line segments in this set with the thurston normal form of a braid

noncommutative maximal ergodic theorems this paper is devoted to the study of various maximal ergodic theorems in noncommutative spaces in particular we prove the noncommutative analogue of the classical dunford schwartz maximal ergodic inequality for positive contractions on and the analogue of stein s maximal inequality for symmetric positive contractions we also obtain the corresponding individual ergodic theorems we apply these results to a family of natural examples which frequently appear in von neumann algebra theory and in quantum probability

the lawrence krammer bigelow representations of the braid groups via uq sl2 abstract we construct representations of the braid groups b n on n strands on free z q 1 s 1 modules w n l using generic verma modules for an integral version of u q sl 2 we prove that the w n 2 are isomorphic to the faithful lawrence krammer bigelow representations of b n after appropriate identification of parameters of laurent polynomial rings by constructing explicit integral bases and isomorphism we also prove that the b n representations w n l are irreducible over the fractional field q q s

poincare type inequalities on the discrete cube and in the car algebra we prove l p poincare inequalities with suitable dimension free constants for functions on the discrete cube 1 1 n as well known such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by bobkov and gotze we also get inequalities between the l p norms of left vert nabla f right vert and delta alpha f alpha 0 moreover l p spaces may be replaced by more general ones similar results hold true replacing functions on the cube by matrices in the algebra spanned by n fermions and the l p norm by the schatten norm c p

the universalr matrix burau representation and the melvin morton expansion of the colored jones polynomial abstract p melvin and h morton 9 studied the expansion of the colored jones polynomial of a knot in powers of q 1 and color they conjectured an upper bound on the power of color versus the power of 1 they also conjectured that the bounding line in their expansion generated the inverse alexander conway polynomial these conjectures were proved by d bar natan and s garoufalidis 1 we have conjectured 12 that other lines in the melvin morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the alexander conway polynomial here we prove this conjecture by using the r matrix formula for the colored jones polynomial and presenting the universal r matrix as a perturbed burau matrix

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

subconvex bounds on mathrm gl _3 via degeneration to frequency zero for a fixed cusp form pi on mathrm gl 3 mathbb z and a varying dirichlet character chi of prime conductor q we prove that the subconvex bound begin aligned l left pi otimes chi tfrac 1 2 right ll q 3 4 delta end aligned holds for any delta 1 36 this improves upon the earlier bounds delta 1 1612 and delta 1 308 obtained by munshi using his mathrm gl 2 variant of the delta method the method developed here is more direct we first express chi as the degenerate zero frequency contribution of a carefully chosen summation formula a la poisson after an elementary amplification step exploiting the multiplicativity of chi we then apply a sequence of standard manipulations reciprocity voronoi cauchy schwarz and the weil bound to bound the contributions of the nonzero frequencies and of the dual side of that formula

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

on lawrence krammer representations lawrence krammer representations or simply lk representations for short are linear representations of artin tits groups of small type over the ring z x 1 y 1 which are of particular interest since they are known to be faithful when the type is spherical and irreducible when the type is spherical and connected here we define and study lk representations over an arbitrary commutative ring r under some conditions on r and on the parameters of an lk representation which generalize the classical settings we recover the faithfulness property and give a new proof of irreducibility that works for every non necessarily spherical connected type we then study the equivalence of distinct lk representations of a given artin tits group for distinct choices of parameters finally we generalize to our setting the known classification of lk representations in the spherical cases and proceed with their classification in the affine cases

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

reading the dual garside length of braids from homological and quantum representations we show that lawrence s representation and linear representations of the braid groups from generic highest weight representations of u q mathfrak sl 2 detect the dual garside length of braids in a simple and natural way that is by expressing a representation as a matrix over a laurent polynomial ring using a certain natural basis the span of the variable is equal to a constant multiple of the dual garside length

conditioned square functions for noncommutative martingales we prove a weak type 1 1 inequality involving conditioned versions of square functions for martingales in noncommutative l p spaces associated with finite von neumann algebras as application we determine the optimal orders for the best constants in the noncommutative burkholder rosenthal inequalities from ann probab 31 2003 948 995 we also discuss bmo norms of sums of noncommuting order independent operators

conformal blocks of minimal models on a riemann surface we give explicit integral representations for conformal blocks of minimal models on arbitrary compact riemann surfaces

martingale transforms and lp norm estimates of riesz transforms on complete riemannian manifolds under the condition that the bakry emery ricci curvature is bounded from below we prove a probabilistic representation formula of the riesz transforms associated with a symmetric diffusion operator on a complete riemannian manifold using the burkholder sharp l p inequality for martingale transforms we obtain an explicit and dimension free upper bound of the l p norm of the riesz transforms on such complete riemannian manifolds for all 1 p in the euclidean and the gaussian cases our upper bound is asymptotically sharp when p 1 and when p

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

on krammer s representation of the braid group a connection is made between the krammer representation and the birman murakami wenzl algebra inspired by a dimension argument a basis is found for a certain irrep of the algebra and relations which generate the matrices are found following a rescaling and change of parameters the matrices are found to be identical to those of the krammer representation the two representations are thus the same proving the irreducibility of one and the faithfulness of the other

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that