clustering based on gaussian processes in this letter we develop a gaussian process model for clustering the variances of predictive values in gaussian processes learned from a training data are shown to comprise an estimate of the support of a probability density function the constructed variance function is then applied to construct a set of contours that enclose the data points which correspond to cluster boundaries to perform clustering tasks of the data points an associated dynamical system is built and its topological invariant property is investigated the experimental results show that the proposed method works successfully for clustering problems with arbitrary shapes

frsvc towards making support vector clustering consume less abstract in spite of with great advantage of discovering arbitrary shapes of clusters support vector clustering svc is frustrated by large scale data especially on resource limited platform it is due to pricey storage and computation consumptions from solving dual problem and labeling clusters upon the pre computed kernel matrix and sampling point pairs respectively towards on it we first present a dual coordinate descent method to reformulate the solver that leads to a flexible training phase carried out on any runtime platform with without sufficient memory then a novel labeling phase who does connectivity analysis between two nearest neighboring decomposed convex hulls referring to clusters is proposed in which a new designed strategy namely sample once connected checking first tries to reduces the scope of sampling analysis by integrating them together a faster and reformulated svc frsvc is created with less consumption achieved according to comparative analysis of time and space complexities furthermore experimental results confirm a significant improvement on flexibility of selective efficiency without losing accuracy with which a balance can be easily reached on the basis of resources a platform equipped

almost independence and irreducibility in simple finite and algebraic groups abstract we study intersections of the form g 1 c 1 g 2 c 2 where c i are conjugacy classes of arbitrary finite simple groups and g i are group elements we show that generically g 1 c 1 g 2 c 2 c 1 c 2 g which means that the events g 1 c 1 g 2 c 2 are almost independent in g we also discuss the dimension and the irreducibility of such intersections in simple algebraic groups and expose the anomaly of sl 2 this work is motivated by recent questions of hrushovski

counting arithmetic lattices and surfaces we give estimates on the number al h x of conjugacy classes of arithmetic lattices of covolume at most x in a simple lie group h in particular we obtain a first concrete estimate on the number of arithmetic 3 manifolds of volume at most x our main result is for the classical case h psl 2 where we show that limx log al h x x log x 1 2 the proofs use several different techniques geometric bounding the number of generators of as a function of its covolume number theoretic bounding the number of maximal such and sharp estimates on the character values of the symmetric groups to bound the subgroup growth of

learning representative exemplars using one class gaussian process regression abstract an exemplar is an observation that represents a group of similar observations exemplars from data are examined to divide entire heterogeneous data into several homogeneous subgroups wherein each subgroup is represented by an exemplar with its inherent sparsity an exemplar based learning model provides a parsimonious model to represent or cluster large scale data a novel exemplar learning method using one class gaussian process gp regression is proposed in this study the proposed method constructs data distribution support from one class gp regression using automatic relevance determination prior and heterogeneous gp noise exemplars that correspond to the basis vectors of the constructed support function are then automatically located during the training process the proposed method is applied to various data sets to examine its operability characteristics of data representation and cluster analysis the exemplars of some real data generated by the proposed method are also reported

a hierarchy of coupled burgers systems possessing a hereditary structure a hierarchy of typical integrable coupled burgers systems is proposed by introducing a special spectral problem involving q dependent variables u0 u1 uq 1 these systems possess a hereditary structure and may reduce to a hierarchy of burgers equations under the reduction ui 0 0 or i or q 2 it is shown that their flows and lax operators commute mutually but each system is not hamiltonian

mixing and generation in simple groups let g be a finite simple group we show that a random walk on g with respect to the conjugacy class xg of a random element x g has mixing time 2 in particular it follows that xg 2 covers almost all of g which could be regarded as a probabilistic version of a longstanding conjecture of thompson we also show that if w is a non trivial word then almost every pair of values of w in g generates g

triangle generation of finite exceptional groups of low rank given a triple p 1 p 2 p 3 of primes the object of this paper is the study of the space hom t p1 t p2 t p3 g of homomorphisms from the triangle group t p1 t p2 t p3 to a finite simple exceptional group g of lie type b 2 2 2 g 2 g 2 or d 3 4 with a few exceptions we give precise asymptotic estimates for the size of hom t p1 t p2 t p3 g and determine the limiting probability that a randomly chosen homomorphism from t p1 t p2 t p3 to g is surjective as g infinity c 2010 elsevier inc all rights reserved

svdd based pattern denoising the support vector data description svdd is one of the best known one class support vector learning methods in which one tries the strategy of using balls defined on the feature space in order to distinguish a set of normal data from all other possible abnormal objects the major concern of this letter is to extend the main idea of svdd to pattern denoising combining the geodesic projection to the spherical decision boundary resulting from the svdd together with solving the preimage problem we propose a new method for pattern denoising we first solve svdd for the training data and then for each noisy test pattern obtain its denoised feature by moving its feature vector along the geodesic on the manifold to the nearest decision boundary of the svdd ball finally we find the location of the denoised pattern by obtaining the pre image of the denoised feature the applicability of the proposed method is illustrated by a number of toy and real world data sets

fuchsian groups finite simple groups and representation varieties let be a fuchsian group of genus at least 2 at least 3 if is non oriented we study the spaces of homomorphisms from to finite simple groups g and derive a number of applications concerning random generation and representation varieties precise asymptotic estimates for hom g are given implying in particular that as the rank of g tends to infinity this is of the form g 1 o 1 where is the measure of we then prove that a randomly chosen homomorphism from to g is surjective with probability tending to 1 as g combining our results with lang weil estimates from algebraic geometry we obtain the dimensions of the representation varieties text hom gamma bar g where bar g is gl n k or a simple algebraic group over k an algebraically closed field of arbitrary characteristic a key ingredient of our approach is character theory involving the study of the zeta function g s 1 s where the sum is over all irreducible complex characters of g

support vector clustering we present a novel clustering method using the approach of support vector machines data points are mapped by means of a gaussian kernel to a high dimensional feature space where we search for the minimal enclosing sphere this sphere when mapped back to data space can separate into several components each enclosing a separate cluster of points we present a simple algorithm for identifying these clusters the width of the gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters the structure of a dataset is explored by varying the two parameters maintaining a minimal number of support vectors to assure smooth cluster boundaries we demonstrate the performance of our algorithm on several datasets

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

diameters of finite simple groups sharp bounds and applications let g be a finite simple group and let s be a normal subset of g we determine the diameter of the cayley graph r g s associated with g and s up to a multiplicative constant many applications follow for example we deduce that there is a constant c such that every element of g is a product of c involutions and we generalize this to elements of arbitrary order we also show that for any word w w xl xd there is a constant c c w such that for any simple group g on which w does not vanish every element of g is a product of c values of w from this we deduce that every verbal subgroup of a semisimple profinite group is closed other applications concern covering numbers expanders and random walks on finite simple groups

invariable generation of infinite groups abstract a subset s of a group g invariably generates g if g s g s s s for each choice of g s g s s in this paper we study invariable generation of infinite groups with emphasis on linear groups our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable we also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

finite covers of random 3 manifolds a 3 manifold is haken if it contains a topologically essential surface the virtual haken conjecture posits that every irreducible 3 manifold with infinite fundamental group has a finite cover which is haken in this paper we study random 3 manifolds and their finite covers in an attempt to shed light on this difficult question in particular we consider random heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface for this model of random 3 manifold we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds our results contrast with the analogous probabilities for groups coming from random balanced presentations giving quantitative theorems to the effect that 3 manifold groups have many more finite quotients than random groups the next natural question is whether these covers have positive betti number for abelian covers of a fixed type over 3 manifolds of heegaard genus 2 we show that the probability of positive betti number is 0

the pre image problem in kernel methods in this paper we address the problem of finding the pre image of a feature vector in the feature space induced by a kernel this is of central importance in some kernel applications such as on using kernel principal component analysis pca for image denoising unlike the traditional method in which relies on nonlinear optimization our proposed method directly finds the location of the pre image based on distance constraints in the feature space it is noniterative involves only linear algebra and does not suffer from numerical instability or local minimum problems evaluations on performing kernel pca and kernel clustering on the usps data set show much improved performance

character ratios representation varieties and random generation of finite groups of lie type abstract we use character theory of finite groups of lie type to establish new results on representation varieties of fuchsian groups and also on probabilistic generation of groups of lie type

surjective word maps and burnside s p aq b theorem we prove surjectivity of certain word maps on finite non abelian simple groups more precisely we prove the following if n is a product of two prime powers then the word map x y mapsto x ny n is surjective on every finite non abelian simple group if n is an odd integer then the word map x y z mapsto x ny nz n is surjective on every finite quasisimple group these generalize classical theorems of burnside and feit thompson we also prove asymptotic results about the surjectivity of the word map x y mapsto x ny n that depend on the number of prime factors of the integer n

some identities and asymptotics for characters of the symmetric group w this paper is motivated by the problem arising from quantum gravity 9 x 2 of counting combinatorial types of triangulations s of a riemann surface x with given degrees of vertices let us color 2 simplexes of the barycentric subdivision s9 in black and white so that adjacent simplexes are of different color and consider a mapping p x a p onto the riemann sphere p which send white simplexes onto the northern hemisphere black simplexes onto the southern hemisphere and centers k simplexes of s into an equatorial point y k s 0 1 2 then deg p s 3n n is k the number of 2 simplexes in s and p is unramified outside y y y it 0 1 2 is easy to see that points of x over y and y have ramification index 2 1 2 respectively 3 while rmaification indices of points over y are just 0 degrees of vertices of s so the problem of counting triangulations reduces to the problem of counting ramified coverings p x a y of riemann surfaces with given ramification indices in section 2 we recall a connection between ramified coverings p x a y of a riemann surface y of genus g and irreducible characters x of the y symmetric group s n s deg p the starting point is the following forn w x w x mula essentially owing to hurwitz 6 see also 10 4 2

dynamic pattern denoising method using multi basin system with kernels in this paper we propose a novel pattern denoising method that utilizes the topological property of a support that describes the distribution of normal patterns to denoise noisy patterns the method first trains a support function which captures the domain of normal patterns and then construct a so called multi basin system associated with the trained support function by moving noisy patterns along the trajectories of the multi basin system noise is removed while the pattern recovers its normality the denoised pattern is obtained when the noisy pattern arrives at the attracting manifold generated by a set of normal patterns and this is the most similar normal pattern with the noisy pattern in the topological sense through simulations on some toy dataset and real image datasets we show that the proposed framework effectively removes the noise while preserving the information contained in the noisy pattern

subgroup graph methods for presentations of finitely generated groups and the connectivity of associated simplicial complexes abstract in this article we generalize the theory of subgroup graphs of subgroups of free groups developed by i kapovich and a myasnikov based on a work by j stallings to finite index subgroups of finitely generated groups given a presentation of a finitely generated group g x r and a finite connected x regular graph which fulfills the relators r we associate to a finite index subgroup h of g conversely the schreier coset graph of h with respect to x and g is such a graph firstly we study and prove various properties of h in relation to the graph secondly we prove that for many finitely generated infinite groups the order and nerve complexes that we associate to g are contractible in particular this is the case for free and free abelian groups fuchsian groups of genus g 2 infinite right angled coxeter groups artin and pure braid groups infinite virtually cyclic groups baumslag solitar groups as well as the free product of at least two of these and all finite index subgroups of these groups

counting and effective rigidity in algebra and geometry the purpose of this article is to produce effective versions of some rigidity results in algebra and geometry on the geometric side we focus on the spectrum of primitive geodesic lengths resp complex lengths for arithmetic hyperbolic 2 manifolds resp 3 manifolds by work of reid this spectrum determines the commensurability class of the 2 manifold resp 3 manifold we establish effective versions of these rigidity results by ensuring that for two incommensurable arithmetic manifolds of bounded volume the length sets resp the complex length sets must disagree for a length that can be explicitly bounded as a function of volume we also prove an effective version of a similar rigidity result established by the second author with reid on a surface analog of the length spectrum for hyperbolic 3 manifolds these effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras to prove these effective rigidity results we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest

alternating groups as quotients of two generator groups abstract graham higman has proved that for sufficiently large n the group a n is a homomorphic image of the triangle group 2 3 l where l 7 later his result was generalized by proving it for all l 7 in this paper we have shown that for sufficiently large n the group a n is a homomorphic image of 2 k l where l 5 k 3 and k 6 is even

constructing sparse kernel machines using attractors in this brief a novel method that constructs a sparse kernel machine is proposed the proposed method generates attractors as sparse solutions from a built in kernel machine via a dynamical system framework by readjusting the corresponding coefficients and bias terms a sparse kernel machine that approximates a conventional kernel machine is constructed the simulation results show that the constructed sparse kernel machine improves the efficiency of testing phase while maintaining comparable test error

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

alternating quotients of fuchsian groups abstract it is shown that any finitely generated non elementary fuchsian group has among its homomorphic images all but finitely many of the alternating groups a n this settles in the affirmative a long standing conjecture of graham higman

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

fast support based clustering method for large scale problems in many support vector based clustering algorithms a key computational bottleneck is the cluster labeling time of each data point which restricts the scalability of the method in this paper we review a general framework of support vector based clustering using dynamical system and propose a novel method to speed up labeling time which is log linear to the size of data we also give theoretical background of the proposed method various large scale benchmark results are provided to show the effectiveness and efficiency of the proposed method

a novel kernel method for clustering kernel methods are algorithms that by replacing the inner product with an appropriate positive definite function implicitly perform a nonlinear mapping of the input data into a high dimensional feature space in this paper we present a kernel method for clustering inspired by the classical k means algorithm in which each cluster is iteratively refined using a one class support vector machine our method which can be easily implemented compares favorably with respect to popular clustering algorithms like k means neural gas and self organizing maps on a synthetic data set and three uci real data benchmarks iris data wisconsin breast cancer database spam database

counting primes groups and manifolds let lambda sl 2 z be the modular group and let c n lambda be the number of congruence subgroups of lambda of index at most n we prove that lim n infinity log c n lambda log n 2 log log n 3 2 sqrt 2 4 the proof is based on the bombieri vinogradov riemann hypothesis on the average and on the solution of a new type of extremal problem in combinatorial number theory similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple lie groups if g is such a lie group and gamma is an irreducible lattice of g it turns out that the subgroup growth of gamma is independent of the lattice and depends only on the lie type of the direct factors of g it can be calculated easily from the root system the most general case of this result relies on the generalized riemann hypothesis but many special cases are unconditional the proofs use techniques from number theory algebraic groups finite group theory and combinatorics

generalized zeta function representation of groups and 2 dimensional topological yang mills theory the example of gl 2 q and pgl 2 q we recall the relation between zeta function representation of groups and two dimensional topological yang mills theory through mednikh formula we prove various generalisations of mednikh formulas and define generalization of zeta function representations of groups we compute some of these functions in the case of the finite group gl 2 x1d53d q and pgl 2 x1d53d q we recall the table characters of these groups for any q compute the frobenius schur indicator of their irreducible representations and give the explicit structure of their fusion rings

dynamic characterization of cluster structures for robust and inductive support vector clustering a topological and dynamical characterization of the cluster structures described by the support vector clustering is developed it is shown that each cluster can be decomposed into its constituent basin level cells and can be naturally extended to an enlarged clustered domain which serves as a basis for inductive clustering a simplified weighted graph preserving the topological structure of the clusters is also constructed and is employed to develop a robust and inductive clustering algorithm simulation results are given to illustrate the robustness and effectiveness of the proposed method

fuchsian groups coverings of riemann surfaces subgroup growth random quotients and random walks abstract fuchsian groups acting as isometries of the hyperbolic plane occur naturally in geometry combinatorial group theory and other contexts we use character theoretic and probabilistic methods to study the spaces of homomorphisms from fuchsian groups to symmetric groups we obtain a wide variety of applications ranging from counting branched coverings of riemann surfaces to subgroup growth and random finite quotients of fuchsian groups as well as random walks on symmetric groups in particular we show that in some sense almost all homomorphisms from a fuchsian group to alternating groups an are surjective and this implies higman s conjecture that every fuchsian group surjects onto all large enough alternating groups as a very special case we obtain a random hurwitz generation of an namely random generation by two elements of orders 2 and 3 whose product has order 7 we also establish the analogue of higman s conjecture for symmetric groups we apply these results to branched coverings of riemann surfaces showing that under some assumptions on the ramification types their monodromy group is almost always sn or an another application concerns subgroup growth we show that a fuchsian group has n o 1 index n subgroups where is the measure of and derive similar estimates for so called eisenstein numbers of coverings of riemann surfaces a final application concerns random walks on alternating and symmetric groups we give necessary and sufficient conditions for a collection of almost homogeneous conjugacy classes in an to have product equal to an almost uniformly pointwise our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups

estimating the support of a high dimensional distribution suppose you are given some data set drawn from an underlying probability distribution p and you want to estimate a simple subset s of input space such that the probability that a test point drawn from p lies outside of s equals some a priori specified value between 0 and 1 we propose a method to approach this problem by trying to estimate a function f that is positive on s and negative on the complement the functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data it is regularized by controlling the length of the weight vector in an associated feature space the expansion coefficients are found by solving a quadratic programming problem which we do by carrying out sequential optimization over pairs of input patterns we also provide a theoretical analysis of the statistical performance of our algorithm the algorithm is a natural extension of the support vector algorithm to the case of unlabeled data

word maps and waring type problems waring s problem asks whether every natural number is a sum of g k fcth powers where g is a suitable function this was solved affirmatively by hubert in 1909 optimizing g k has been a central problem in additive number theory ever since see na for more detail and background recently there has been considerable interest in group theoretic analogues of this phenomenon where the aim is to present group elements as short products of certain special elements these special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group the motivation for this is sometimes topological let g be a topologically finitely generated pro p group using expressions of elements of the commutator subgroup g1 as bounded products of commutators serre showed that g is closed and deduced that every finite index subgroup of g is open a recent deep result of nikolov and segal nsi ns2 shows that this holds for every finitely generated profinite group again the core of the proof is presenting group elements as short products of values of certain words

commutator maps measure preservation and t systems let g be a finite simple group we show that the commutator map g g g is almost equidistributed as g 1 this somewhat surprising result has many applications it shows that a for a subset x g we have 1 x g 2 x g o 1 namely is almost measure preserving from this we deduce that almost all elements g 2 g can be expressed as commutators g x y where x y generate g this enables us to solve some open problems regarding t systems and the product replacement algorithm pra graph we show that the number of t systems in g with two generators tends to infinity as g 1 this settles a conjecture of guralnick and pak a similar result follows for the number of connected components of the pra graph of g with two generators some of our results apply for more general finite groups and more general word maps our methods are based on representation theory combining classical character theory with recent results on character degrees and values in finite simple groups in particular the so called witten zeta function g s p 2irr g 1 s plays a key role in the proofs

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

probabilistic generative ranking method based on multi support vector domain description as the volume of database grows retrieval and ordering of information according to relevance has become an important and challenging task ranking problem has recently been considered and formulated as a machine learning problem among the various learning to rank methods the ranking support vector machines svms have been widely applied in various applications because of its state of the art performance in this paper we propose a novel ranking method based on a probabilistic generative model approach the proposed method utilizes multi support vector domain description multi svdd and constructs pseudo conditional probabilities for data pairs thus enabling the construction of an efficient posterior probability function of relevance judgment of data pairs results of experiments on both synthetic and real large scale datasets show that the proposed method can efficiently learn ranking functions better than ranking svms

domain described support vector classifier for multi classification problems in this paper a novel classifier for multi classification problems is proposed the proposed classifier based on the bayesian optimal decision theory tries to model the decision boundaries via the posterior probability distributions constructed from support vector domain description rather than to model them via the optimal hyperplanes constructed from two class support vector machines experimental results show that the proposed method is more accurate and efficient for multi classification problems

a coupled akns kaup newell soliton hierarchy a coupled akns kaup newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from hamiltonian pairs zero curvature representations and tri hamiltonian structures are established for all coupled akns kaup newell systems in the hierarchy therefore all systems have infinitely many commuting symmetries and conservation laws two reductions of the systems lead to the akns hierarchy and the kaup newell hierarchy and thus those two soliton hierarchies also possess tri hamiltonian structures

classical groups probabilistic methods and the 2 3 generation problem we study the probability that randomly chosen elements of prescribed type in a finite simple classical group g generate g in particular we prove a conjecture of kantor and lubotzky in this area the probabilistic approach is then used to determine the finite simple classical quotients of the modular group psl2 z up to finitely many exceptions

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities