composition factors of monodromy groups the authors prove the 1990 conjecture of guralnick and thompson on composition factors of monodromy groups using riemann s existence theorem the conjecture translates into a problem on primitive permutation groups this group theoretic problem had been reduced to a question about actions of classical groups on subspaces of their natural modules the key ingredients in the present proof are the authors earlier fixed point ratio estimates for such actions and a result of scott on the generation of linear groups

generalization of selberg s frac 3 16 theorem and affine sieve an analogue of the well known frac 3 16 lower bound for the first eigenvalue of the laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non elementary subgroup l of sl 2 z the proof in the case that the hausdorff of the limit set of l is bigger than frac 1 2 is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting in the case that delta is less than frac 1 2 we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function these spectral gaps are then applied to sieving problems on orbits of such groups

spectral partitioning eigenvalue bounds and circle packings for graphs of bounded genus in this paper we address two long standing questions about finding good separators in graphs of bounded genus and degree 1 it is a classical result of gilbert hutchinson and tarjan j algorithms 5 1984 pp 391 407 that one can find asymptotically optimal separators on these graphs if given both the graph and an embedding of it onto a low genus surface does there exist a simple efficient algorithm to find these separators given only the graph and not the embedding 2 in practice spectral partitioning heuristics work extremely well on these graphs is there a theoretical reason why this should be the case we resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio o sqrt smash b g n and vertex bisectors of size o sqrt gn in these graphs both of which are optimal as our main technical lemma we prove an o g n bound on the second smallest eigenvalue of the laplacian of such graphs and show that this is tight thereby resolving a conjecture of spielman and teng while this lemma is essentially combinatorial in nature its proof comes from continuous mathematics drawing on the theory of circle packings and the geometry of compact riemann surfaces

stochastic representations for the wave equation on graphs and their scaling limits abstract this paper is devoted to an interacting particle system that provides probabilistic interpretation of the wave equation on graphs a feynman kac type formula is established connecting the expectation of the process with the wave equation on graphs non asymptotic l 2 estimates are presented it is then shown that the high density hydrodynamic limit of the system is given by the wave equation in euclidean space the sharpness of scaling limit result is demonstrated by a phase transition phenomenon

comparison of stochastic and deterministic models of a linear chemical reaction with diffusion particles placed in n cells on the unit interval give birth or die according to linear rates adjacent cells are coupled by diffusion with a rate proportional to n2 cell numbers are divided by a density parameter to represent concentrations and the resulting space time markov process is compared to a corresponding deterministic model the solution to a partial differential equation the models are viewed as hilbert space valued processes and compared by means of a law of large numbers and central limit theorem new and nearly optimal results are obtained by exploiting the ornstein uhlenbeck type structure of the stochastic model

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

uniform expansion bounds for cayley graphs of sl2 fp we prove that cayley graphs of sl2 fp are expanders with respect to the projection of any fixed elements in sl 2 z generating a non elementary subgroup and with respect to generators chosen at random in sl2 fp

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

on a weak variant of the geometric torsion conjecture a consequence of the geometric torsion conjecture for abelian varieties over function fields is the following let k be an algebraically closed field of characteristic 0 for any integers d g 0 there exists an integer n n k d g 1 such that for any function field l k with transcendence degree 1 and genus g and any d dimensional abelian variety a l containing no nontrivial k isotrivial abelian subvariety a l tors a n in this paper we deal with a weak variant of this statement where a l runs only over abelian varieties obtained from a fixed d dimensional abelian variety by base change more precisely let k k be a function field with transcendence degree 1 and a k an abelian variety containing no nontrivial k isotrivial abelian subvariety then we show that if k has genus 1 or if a k has semistable reduction over all but possibly one place then for any integer g 0 there exists an integer n n a g 1 such that for any finite extension l k with genus g a l tors a n previous works of the authors show that this holds without any restriction on k for the l primary torsion with l a fixed prime so it is enough to prove that there exists an integer n n a g 1 such that for any finite extension l k with genus g the prime divisors of a l tors are all n

mesoscopic approach to minority games in herd regime abstract we study minority games in efficient regime by incorporating the utility function and aggregating agents with similar strategies we develop an effective mesoscale notion of the state of the game using this approach the game can be represented as a markov process with substantially reduced number of states with explicitly computable probabilities for any payoff the finiteness of the number of states is proved interesting features of an extensive random variable called aggregated demand viz its strong inhomogeneity and presence of patterns in time can be easily interpreted using markov theory and quenched disorder approach we can explain important macroscopic characteristics of the game behavior of variance per capita and predictability of the aggregated demand we prove that in the case of linear payoff many attractors in the state space are possible

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

on random time and on the relation between wave and telegraph equations kac s conjecture relating the solution of wave and telegraph equations in higher dimensions through a poisson process driven random time is established through the concepts of stochastic calculus new expression is derived for the probability density function of the random time we demonstrate how the relationship between the solution of a lossy wave and that of a lossless wave equation can be exploited to derive some statistical identities relevance of the results presented to the study of pulse propagation in a dispersive medium characterized by a lorentz or drude model is discussed and new evolution equations for 2 d maxwell s equations are presented for the drude medium it is shown that the computational time required for updating the electric field using the stochastic technique is expected to go up as o t

stochastic process leading to wave equations in dimensions higher than one stochastic processes are proposed whose master equations coincide with classical wave telegraph and klein gordon equations similar to predecessors based on the goldstein kac telegraph process the model describes the motion of particles with constant speed and transitions between discreet allowed velocity directions a new ingredient is that transitions into a given velocity state depend on spatial derivatives of other states populations rather than on populations themselves this feature requires the sacrifice of the single particle character of the model but allows to imitate the huygens principle and to recover wave equations in arbitrary dimensions

congruence properties of zariski dense subgroups i this paper deals with the following general situation we are given an algebraic group g defined over a number field k and a subgroup f of the group g k of k rational points of g then what should it mean for f to be a large subgroup we might require f to be a lattice in g to be arithmetic to contain many elements of a specific kind to have a large closure in some natural topology et cetera there are many theorems proving implications between conditions of size of this kind we shall consider the case of g a k simple group usually q simple the zariski topology of g for which the closed sets are those defined by the vanishing of polynomial functions is very coarse on the other hand the various valuations v of k each give rise to a strong topology on g k if v is non archimedean and kv is the completion of k with respect to v then g kv is a non archimedean lie group and if is the ring of integers of kv the open subgroup g 5 j of g kv of integral points is defined for all but finitely many v the zariski closure f of a subgroup f of g k is a c algebraic subgroup of g while the strong closure fy is a lie subgroup of g kv for elementary reasons the dimension of tv is at most the dimension of f our results are in the opposite direction we show under suitable conditions on g that if f g then for almost all v we have that tv contains g ov to illustrate this with a specific example if ml 5 mk are matrices in sl2 q generating a group f which is zariskidense in sl2 then for all sufficiently large prime numbers p we have f p sl2 zp further this result is effective in the sense that we could in principle check the hypothesis for given mx mk and derive a bound on p beyond which the conclusion holds it is perhaps worth remarking that this special case which is the simplest application of our results may be proved with much less effort than the general theorem

a feynman kac type formula for the deterministic and stochastic wave equations and other p d e s we establish a probabilistic representation for a wide class of linear deterministic p d e s with potential term including the wave equation in spatial dimensions 1 to 3 our representation applies to the heat equation where it is related to the classical feynman kac formula as well as to the telegraph and beam equations if the potential is a random spatially homogeneous gaussian noise then this formula leads to an expression for the moments of the solution