a limiting result for the structure of collisions between many independent diffusions we study a system of a large number n of independent diffusions undergoing collisions in a roughly constant mean free path regime we derive a propagation of chaos result when n goes to infinity concerning the limit behavior of the trajectory of a particle its first time of collision with all the other particles and the trajectory of the colliding particle

the classification of affine su 3 modular invariant partition functions a complete classification of thephysical modular invariant partition functions for the wznw models is known for very few affine algebras and levels the most significant being all levels ofsu 2 and level 1 of all simple algebras in this paper we solve the classification problem forsu 3 modular invariant partition functions all levels our approach will also be applicable to other affine lie algebras and we include some preliminary work in that direction including a sketch of a new proof forsu 2

arithmetic differential equations on gln i differential cocycles abstract the theory of differential equations has an arithmetic analogue 8 in which derivatives are replaced by fermat quotients one can then ask what is the arithmetic analogue of a linear differential equation the study of usual linear differential equations is the same as the study of the differential cocycle from gl n into its lie algebra given by the logarithmic derivative 14 equivalently by the maurer cartan connection 17 however we prove here that there are no such cocycles in the context of arithmetic differential equations in sequels of this paper 10 11 we will remedy the situation by introducing arithmetic analogues of lie algebras and a skew version of differential cocycles this will lead to a theory of linear arithmetic differential equations

on the kms condition for chiral conformal field theories using the kubo martin schwinger kms condition and exchange algebras we discuss how to compute correlation functions for the conformal exchange field theories in particular we derive the verlinde algebra from modular invariance of the conformal field theories we also show how the verlinde matrices can be related to the kms matrices

integrable and conformal boundary conditions for z k parafermions on a cylinder we study integrable and conformal boundary conditions for s 2 zk parafermions on a cylinder these conformal field theories are realized as the continuum scaling limit of critical a d e lattice models with negative spectral parameter the conformal boundary conditions labelled by a m g z2k are identified with associated integrable lattice boundary conditions labelled by r a ag 2 g where g is the coxeter number of the a d e graph g we obtain analytically the boundary free energies present general expressions for the parafermion cylinder partition functions and confirm these results by numerical calculations

automorphism modular invariants of current algebras we consider those two dimensional rational conformal field theories rcfts whose chiral algebras when maximally extended are isomorphic to the current algebra formed from some untwisted affine lie algebra at fixed level in this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the kac peterson modular matrices we classify all such partition functions when the underlying finite dimensional lie algebra is simple this gives all possible spectra for this class of rcfts while accomplishing this we also find the primary fields with second smallest quantum dimension

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

modular transformations of su n affine characters and their commutant we describe the algebra of matrices commuting with the action of the modular group on characters ofsu n k integrable representations using methods of finite quantum mechanics we find a canonical basis for this commutant over and prove the existence of an equivalent basis over with integral matrix elements a final section is devoted to the case ofsu 3

towards a classification of su 2 su 2 modular invariant partition functions the complete classification of wess zumino novikov witten modular invariant partition functions is known for very few affine algebras and levels the most significant being all levels of a1 and a2 and level 1 of all simple algebras here the classification problem is addressed for the nicest high rank semisimple affine algebras a1 1 r among other things all automorphism invariants are found explicitly for all levels k k1 kr and the classification for a 1 1 a 1 1 is completed for all levels k1 k2 the classification problem for a1 1 r is also solved for any levels ki with the property that for i j each gcd ki 2 kj 2 3 in addition some physical invariants are found which seem to be new together with some recent work by stanev the classification for all a1 1 k r could now be within sight

z nz conformal field theories in this paper we give a brief introduction to conformal field theory cft following the presentation of g segal we explain how to reconstruct part of a cft from its fusion rules the possible choices of s matrices are indexed by some automorphisms of the fusion algebra we illustrate this procedure by computing the modular properties of the possible genus one characters when the fusion algebra is the representation algebra of a finite group we also classify the modular invariant partition functions of these theories we recover as special cases the a n 1 wzw theories and the rational gaussian model

the basic geometry of witt vectors ii spaces this is an account of the algebraic geometry of witt vectors and arithmetic jet spaces the usual p typical witt vectors of p adic schemes of finite type are already reasonably well understood the main point here is to generalize this theory in two ways we allow not just p typical witt vectors but those taken with respect to any set of primes in any ring of integers in any global field for example this includes the big witt vectors we also allow not just p adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field we give similar generalizations of buium s formal arithmetic jet functor which is dual to the witt functor we also give concrete geometric descriptions of witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors

weak limits of stopped diffusions on considere le probleme d homogeneisation suivant soit le mouvement brownien dans r d d 3 tue sur la surface de plusieurs petits corps absorbants on etudie la limite lorsque le nombre de corps approche l infini et que la dimension des corps approche 0

modular invariants for affine su 3 theories at prime heights a proof is given for the existence of two and only two modular invariant partition functions in affine widehat su 3 k theories at heightsn k 3 which are prime numbers arithmetic properties of the ring of algebraic integers which is related tosu 3 weights are extensively used

duality and modular invariance in rational conformal field theories we investigate the polynomial equations which should be satisfied by the duality data for a rational conformal field theory we show that by these duality data we can construct some vector spaces which are isomorphic to the spaces of conformal blocks one can construct explicitly the inner product for the former if one deals with a unitary theory these vector spaces endowed with an inner product are the algebraic reminiscences of the hilbert spaces in a chern simons theory as by products we show that the polynomial equations involving the modular transformations for the one point blocks on the torus are not independent we discuss the solution of structure constants for a physical theory making some assumption we obtain a neat solution and this solution in turn implies that the quantum groups of the left sector and of the right sector must be the same although the chiral algebras need not be the same some examples are given finally we discuss the reconstruction of the quantum group in a rational conformal theory

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

random homogenization and singular perturbations in perforated domains the paper considers the singularly perturbed dirichlet problem u u f in a randomly perforated domain which is obtained from a bounded open set in r n after removing many holes of size q the perforated domain is described in terms of an ergodic dynamical system acting on a probability space imposing certain conditions on the domain the behaviour of u when 0 in lebesgue spaces l n is studied test functions together with the birkhoff ergodic theorem are the main tools of analysis the poisson distribution of holes of size p with the intensity r is then considered the above results apply in some cases other cases are treated by the wiener sausage approach

modular invariant partition functions and method of shift vector using the shift vector method a large class of self dual lattices of dimension l l is obtained which has a one to one correspondence with modular invariants of free bosonic theory compactified on a coroot lattice of a rank l lie group then a large number of modular invariants of affine lie algebras are derived explicitly two applications of this method are a direct derivation of the d series of su n and a new proof for the a d e classification of the su 2 k partition functions

automorphisms of the affinesu 3 fusion rules we classify the automorphisms of the chiral level k affinesu 3 fusion rules for any value ofk by looking for all permutations that commute with the modular matricess andt this can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed whenk is divisible by 3 the automorphism group z2 is generated by the charge conjugationc ifk is not divisible by 3 the automorphism group z2 z2 is generated byc and the altschuler lacki zaugg automorphism although the combinatorial analysis can become more involved the techniques used here forsu 3 can be applied to other algebras

the branching rules of conformal embeddings after recalling the main properties of a conformal embedding of lie algebras g id p which is defined by the equality of the sugawara central charges on both sides we launch a systematic study of their branching rules the bulk of the paper is devoted to the proof of a general formula in the case suim su m n su m at the end we give some applications to the con struction of modular invariant partition functions

differential subgroups of simple algebraic groups over p adic fields a theorem of cassidy states that any zariski closed differential algebraic subgroup of a simple linear algebraic group defined over a differential field is either the whole group or is conjugate to the subgroup of constant matrices an arithmetic analogue of this theorem is proved in which usual derivations on fields are replaced by certain nonlinear operators on p adic rings called p derivations