equivalence relations for homology cylinders and the core of the casson invariant let r be a compact oriented surface of genus g with one boundary component homology cylinders over r form a monoid ic into which the torelli group i of r embeds by the mapping cylinder construction two homology cylinders m and m are said to be y k equivalent if m is obtained from m by twisting an arbitrary surface s in m with a homeomorphim belonging to the k th term of the lower central series of the torelli group of s the j k equivalence relation on ic is defined in a similar way using the k th term of the johnson filtration in this paper we characterize the y 3 equivalence with three classical invariants 1 the action on the third nilpotent quotient of the fundamental group of r 2 the quadratic part of the relative alexander polynomial and 3 a by product of the casson invariant similarly we show that the j 3 equivalence is classified by 1 and 2 we also prove that the core of the casson invariant originally defined by morita on the second term of the johnson filtration of i has a unique extension to the corresponding submonoid of ic that is preserved by y 3 equivalence and the mapping class group action

finiteness properties of the johnson subgroups abstract the main goal of this note is to provide evidence that the first rational homology of the johnson subgroup k g 1 of the mapping class group of a genus g surface with one marked point is finite dimensional building on work of dimca papadima 4 we use symplectic representation theory to show that for all g 3 the completion of h 1 k g 1 q with respect to the augmentation ideal in the rational group algebra of z 2 g is finite dimensional we also show that the terms of the johnson filtration of the mapping class group have infinite dimensional rational homology in some degrees in almost all genera generalizing a result of akita

structure of symplectic invariant lie subalgebras of symplectic derivation lie algebras abstract we study the structure of the symplectic invariant part h g 1 sp of the lie algebra h g 1 consisting of symplectic derivations of the free lie algebra generated by the rational homology group of a closed oriented surface g of genus g first we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree 20 in this framework we give a general constraint which is imposed on the sp invariant component of the bracket of two elements in h g 1 second we clarify the relations among h g 1 and the other two related lie algebras h g and h g which correspond to the cases of a closed surface g with and without base point g in particular based on a theorem of labute we formulate a method of determining these differences and describe them explicitly up to degree 20 third by giving a general method of constructing elements of h g 1 sp we reveal a considerable difference between two particular submodules of it one is the sp invariant part of a certain ideal j g 1 and the other is that of the johnson image finally we combine these results to determine the structure of h g 1 completely up to degree 6 including the unstable cases where the genus 1 case has an independent meaning in particular we see a glimpse of the galois obstructions explicitly from our point of view

the cokernel of the johnson homomorphisms of the automorphism group of a free metabelian group in this paper we determine the cokernel of the k th johnson homomorphisms of the automorphism group of a free metabelian group for k 2 and n 4 as a corollary we obtain a lower bound on the rank of the graded quotient of the johnson filtration of the automorphism group of a free group furthermore by using the second johnson homomorphism we determine the image of the cup product map in the rational second cohomology group of the ia automorphism group of a free metabelian group and show that it is isomorphic to that of the ia automorphism group of a free group which is already determined by pettet finally by considering the kernel of the magnus representations of the automorphism group of a free group and a free metabelian group we show that there are non trivial rational second cohomology classes of the ia automorphism group of a free metabelian group which are not in the image of the cup product map

real space renormalisation group method for the dilute gaussian model and its relation to the random walk problems near the percolation threshold an isomorphism between the gaussian model in the magnetic field and the time dependent conduction diffusion problem permits the author to develop a real space renormalisation group method for the diluted version of this model by which all single cluster results of the phenomenological scaling theory of gefen aharony and alexander 1983 are correctly reproduced an extension of this model to the case of strong bias then provides some restrictions on the validity of their results the result for the density of states which the author reproduces in the form predicted by alexander and orbach 1982 is subject to the same kind of restrictions

weight decompositions on etale fundamental groups let x 0 be a smooth or proper variety defined over a finite field k and let x be the base extension of x 0 to the algebraic closure bar k the geometric etale fundamental group pi 1 x bar x of x is a normal subgroup of the weil group so conjugation gives it a weil action for l not dividing the characteristic of k we consider the pro bbb q l algebraic completion of pi 1 x bar x as a nonabelian weil representation lafforgue s theorem and deligne s weil ii theorems imply that this affine group scheme is mixed in the sense that its structure sheaf is a mixed weil representation when x is smooth weight restrictions apply affecting the possibilities for the structure of this group this gives new examples of groups which cannot arise as etale fundamental groups of smooth varieties

rational points of universal curves in positive characteristics rational points of universal curves in positive characteristics by tatsunari watanabe department of mathematics duke university

multiple matrix gaussian graphs estimation matrix valued data where the sampling unit is a matrix consisting of rows and columns of measurements are emerging in numerous scientific and business applications matrix gaussian graphical model is a useful tool to characterize the conditional dependence structure of rows and columns in this article we employ nonconvex penalization to tackle the estimation of multiple graphs from matrix valued data under a matrix normal distribution we propose a highly efficient nonconvex optimization algorithm that can scale up for graphs with hundreds of nodes we establish the asymptotic properties of the estimator which requires less stringent conditions and has a sharper probability error bound than existing results we demonstrate the efficacy of our proposed method through both simulations and real functional magnetic resonance imaging analyses

arithmetic group symmetry and finiteness properties of torelli groups we examine groups whose resonance varieties characteristic varieties and sigma invariants have a natural arithmetic group symmetry and we explore implications on various niteness properties of subgroups we compute resonance varieties characteristic varieties and alexander polynomials of torelli groups and we show that all subgroups containing the johnson kernel have nite rst betti number when the genus is at least 4 we also prove that in this range the i adic completion of the alexander invariant is nite dimensional and the kahler property for the torelli group implies the nite generation of the johnson kernel

the structure of the torelli group i a finite set of generators for j this is the first of three papers concerning the so called torelli group let m mg be a compact orientable surface of genus g having n boundary components and let 9 xg be its mapping class group that is the group of orientation preserving diffeomorphisms of m which are 1 on the boundary am modulo isotopies which fix 3m pointwise this group is also known to the complex analysts as the teichmuller group or modular group if n 0 or 1 let further 4 jg be the subgroup of d1 which acts trivially on h1 m z the topologists have no traditional name for a but the analysts tell me it was known classically and is called the torelli group several interesting problems and conjectures exist concerning f the principal one can be found in kirby s problem list k and asks if gg is finitely generated in this first paper we shall answer the question affirmatively for both gg 0 and fgg when g 3 and shall give a fairly simple set of generators two other conjectures were made by the author the first involves the subgroup j of f which is generated by twists on nulhomologous simple closed curves ji produces a surjective homomorphism t fgg 1 a3h1 m z which kills c and it is conjectured there that in fact 5y ker t the proof of this is the content of the second paper in the third paper we use the results of the second to compute the abelianization f f explicitly thereby verifying another conjecture in ji the first reasonably simple but infinite set of generators for fg0 was produced by powell in p his generators consist of two types a twists on bounding simple closed curves b opposite twists on a bounding pair of disjoint homologous simple closed curves each of which are nonbounding using his result the author showed in j2 that the maps of the second type which we call bp maps for bounding pair are actually sufficient to generate both kg 0 and fg 1 for g 3 and in fact that we need only those whose two curves bound a genus one subsurface of m note that for g 2 all bp maps are trivial and hence the result fails in this case in the finite set of generators produced in this paper only

stochastic quasi likelihood for case control point pattern data we propose a novel stochastic quasi likelihood estimation procedure for case control point processes quasi likelihood for point processes depends on a certain optimal weight function and for the new method the weight function is stochastic since it depends on the control point pattern the new procedure also provides a computationally efficient implementation of quasi likelihood for univariate point processes in which case a synthetic control point process is simulated by the user under mild conditions the proposed approach yields consistent and asymptotically normal parameter estimators we further show that the estimators are optimal in the sense that the associated godambe information is maximal within a wide class of estimating functions for case control point processes the effectiveness of the proposed method is further illustrated using extensive simulation studies and two data examples

residual p properties of mapping class groups and surface groups let m p be the mapping class group of a punctured oriented surface e p where p may be empty and let t p p be the kernel of the action of m p on h 1 p f p we prove that t p p is residually p in particular this shows that m p is virtually residually p for a group g we denote by ip g the kernel of the natural action of out g on h 1 g f p in order to achieve our theorem we prove that under certain conditions g is conjugacy p separable and has property a the group ip g is residually p the fact that free groups and surface groups have property a is due to grossman the fact that free groups are conjugacy p separable is due to lyndon and schupp the fact that surface groups are conjugacy p separable is from a technical point of view the main result of the paper

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

vanishing resonance and representations of lie algebras we explore a relationship between the classical representation theory of a complex semisimple lie algebra g and the resonance varieties r v k v attached to irreducible g modules v and submodules k v v in the process we give a precise roots and weights criterion insuring the vanishing of these varieties or equivalently the finiteness of certain modules w v k over the symmetric algebra on v in the case when g sl2 c our approach sheds new light on the modules studied by weyman and eisenbud in the context of green s conjecture on free resolutions of canonical curves in the case when g sln c or sp2g c our approach yields a unified proof of two vanishing results for the resonance varieties of the outer torelli groups of surface groups results which arose in recent work by dimca hain and the authors on homological finiteness in the johnson filtration of mapping class groups and automorphism groups of free groups

potts model dirac propagator and conformational statistics of semiflexible polymers a new discretized version of the dirac propagator ind space and one time dimensions is obtained with the help of the 2d state one dimensional potts model the euclidean version of this propagator describes all conformational properties of semiflexible polymers it also describes all properties of fully directed self avoiding walks the case of semiflexible copolymers composed of a random sequence of fully flexible and semirigid monomer units is also considered as a by product some new results for disordered one dimensional ising and potts models are obtained in the case of the potts model the nontrivial extension of the results to higher dimensions is discussed briefly

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

onsager s reaction field for the potts model from the path integral a new path integral formulation for theq state potts model is proposed this formulation reproduces known results for the ising model q 2 and naturally extends these results for arbitraryq the mean field results for both the ising and the potts models are obtained as a leading saddle point contribution to the corresponding functional integrals while the systematic computation of corrections to the saddle point contribution produces the onsager reaction field terms which forq 2 coincide with results already known for the ising model

studies in the history of probability and statistics l karl pearson and the rule of three karl pearson s role in the transformation that took the 19th century statistics of laplace and gauss into the modern era of 20th century multivariate analysis is examined from a new point of view by viewing pearson s work in the context of a motto he adopted from charles darwin a philosophical theme is identified in pearson s statistical work and his three major achievements are briefly described copyright 2012 oxford university press