probabilistic generation of finite simple groups ii abstract in earlier work it was shown that each nonabelian finite simple group g has a conjugacy class c such that whenever 1 x g the probability is greater than 1 10 that g x y for a random y c much stronger asymptotic results were also proved here we show that allowing equality the bound 1 10 can be replaced by 13 42 and excluding an explicitly listed set of simple groups the bound 2 3 holds we use these results to show that any nonabelian finite simple group g has a conjugacy class c such that if x 1 x 2 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y similarly aside from one infinite family and a small explicit finite set of simple groups g has a conjugacy class c such that if x 1 x 2 x 3 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y x 3 y we also prove analogous but weaker results for almost simple groups

generation and random generation from simple groups to maximal subgroups abstract let g be a finite group and let d g be the minimal number of generators for g it is well known that d g 2 for all non abelian finite simple groups we prove that d h 4 for any maximal subgroup h of a finite simple group and that this bound is best possible we also investigate the random generation of maximal subgroups of simple and almost simple groups by applying a recent theorem of jaikin zapirain and pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant we then apply our results to the study of permutation groups in particular we show that if g is a finite primitive permutation group with point stabilizer h then d g 1 d h d g 4

on the orders of primitive groups with restricted nonabelian composition factors we prove that given c0 there exists c such that the following holds if g is a primitive permutation group of degree n no composition factor of which is an alternating group of degree greater than c0 or a classical group of dimension greater than c0 then g nc in particular if the nonabelian composition factors of g have bounded order then g is polynomially bounded

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

mesh generation refinement using fractal concepts and iterated function systems a novel method of mesh generation is proposed which is based on the use of fractal concepts to derive contractive affine transformations the transformations are constructed in such a manner that the attractors of the resulting maps are a union of the points lines and surfaces in the domain in particular the mesh nodes may be generated recursively as a sequence of points which are obtained by applying the transformations to a coarse background mesh constructed from the given boundary data a delaunay triangulation or similar edge connection approach can then be performed on the resulting set of nodes in order to generate the mesh local refinement of an existing mesh can also be performed using the procedure the method is easily extended to three dimensions in which case the delaunay triangulation is replaced by an analogous 3 d tesselation

random generation of finite and profinite groups and group enumeration the following article appeared in annals of mathematics 173 2 2011 769 814 and may be found at http annals math princeton edu 2011 173 2 p04

explicit representations of maximal subgroups of the monster abstract most of the maximal subgroups of the monster are now known but in many cases they are hard to calculate in we produce explicit small representations of all the maximal subgroups which are not 2 local the representations we construct are available on the world wide web at http brauer maths qmul ac uk atlas

the structure of 2d semi simple field theories i classify the cohomological 2d field theories based on a semi simple complex frobenius algebra a they are controlled by a linear combination of classes and by an extension datum to the deligne mumford boundary their effect on the gromov witten potential is described by givental s fock space formulae this leads to the reconstruction of gromov witten ancestor invariants from the quantum cup product at a single semi simple point and the first chern class of the manifold confirming givental s higher genus reconstruction conjecture this in turn implies the virasoro conjecture for manifolds with semi simple quantum cohomology the classification uses the mumford conjecture proved by madsen and weiss european congress of mathematics pp 283 303 2005

on conjugacy classes of maximal subgroups of finite simple groups and a related zeta function we prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of lie type of bounded rank is bounded for exceptional groups this solves a longstanding open problem the proof uses among other tools some methods from geometric invariant theory using this result we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of lie type groups of fixed rank drawing conclusions regarding the behaviour of the corresponding zeta function g s m maxg g m s which appears in many probabilistic applications more specifically we are able to show that for simple groups g and for any fixed real number s 1 g s 0 as g this confirms a conjecture made in 27 we also apply these results to prove the conjecture made in 29 that the symmetric group sn has n o 1 conjugacy classes of primitive maximal subgroups 2000 mathematics classification numbers 20e28 20g15 20d06

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

numerical simulation of dendritic solidification with convection three dimensional flow a numerical method for the simulation of the effect of melt flow on the three dimensional growth of a dendrite is described the method is an extension of the technique for two dimensional flow described in al rawahi and tryggvason j comput phys 180 2002 471 and is based on the explicit tracking of connected marker points that describe the liquidsolid interface an explicit projection method is used to solve the energy and the navier stokes equations on a regular stationary grid and the solidified region is represented by setting the velocities in the solid phase to zero the latent heat released during solidification is calculated using the normal temperature gradient near the interface the method is validated by a comparison with an exact solution for a stefan problem and a grid refinement study the simulations show that the speed of a dendrite arm growing into the flow is increased due to an increase in the temperature gradient on the upstream side and the formation of side branches is promoted as in two dimensions the effect of the flow on the growth of dendrite arms growing in the downstream direction is smaller than in two dimensions due to a smaller wake

modeling the growth and interaction of multiple dendrites in solidification using a level set method abstract a level set method is presented to study the growth and interaction of multiple dendrites in solidification the method couples thermal and solute diffusion with propagation of multiple interfaces a single signed distance function is used to track the solid liquid interface with the aid of markers the value of which is the orientation angle for identification of different crystals the problem of evolving multiple crystal interfaces is reduced to two tasks 1 tracking one level set variable signed distance function and 2 determination of the marker for a newly solidified finite element nodal point tracking a single level set variable is implemented by solving the level set equation with interface velocity computed from an extended stefan equation using the marker information crystal orientation determination of the marker for a newly solidified finite element nodal point is implemented by using an algorithm modified from the fast marching technique both of these two steps are computationally efficient and the approach is suitable for incorporating effects of multiple crystals convergence and accuracy of this approach are demonstrated by using different grid spacings and comparing with results obtained from the multi phase level set method a parametric study is performed to investigate the effects of solidification speed and thermal gradient on the resulting solidification microstructure pattern numerical results of columnar to equiaxed transition cet qualitatively agree with an analytical estimation and are similar to previous numerical results obtained using a phase field method a convergence study is performed to determine the appropriate grid spacing for numerical simulation at lower surface tension cet occurs at a lower thermal gradient for a giving solidification speed secondary dendrite formation is more apparent with lower surface tension the differences and similarities between the three dimensional and two dimensional growth results are analyzed randomness in crystal orientation and required undercooling for nucleation are modeled and found to have a great effect on the microstructure pattern the efficiency of the present approach is finally demonstrated with an example that includes the growth of hundreds of crystals with consideration of randomness effects

adaptive mesh computation of polycrystalline pattern formation using a renormalization group reduction of the phase field crystal model we implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes our numerical approach uses the systematic renormalization group formulation of a phase field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field a quantity that is spatially uniform except near topological defects grain boundaries and other lattice imperfections our algorithm employs a hybrid formulation of the amplitude equations combining cartesian and polar decompositions of the complex amplitude we show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions

analyticity of the total ancestor potential in singularity theory abstract k saito s theory of primitive forms gives a natural semi simple frobenius manifold structure on the space of miniversal deformations of an isolated singularity on the other hand givental introduced the notion of a total ancestor potential for every semi simple point of a frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non semisimple points as well in this paper we prove givental s conjecture by using the eynard orantin recursion

numerical method for coupled fluid flow heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology abstract a numerical method is presented that can be used for both solid body stress analysis and fluid flow predictions independently as well as in a coupled manner the method uses an integral form of equations governing mass energy and momentum balance for an arbitrary control volume a detailed description is provided of a novel second order accurate spatial discretisation technique which can accommodate unstructured moving meshes with cells of arbitrary topology this is accompanied by a fully implicit temporal discretisation which makes the method stable for any time step size the resulting set of coupled non linear algebraic equations is solved by employing a segregated approach leading to a decoupled set of linear algebraic equations for each dependent variable with a sparse diagonally dominant coefficient matrix these equations are solved by an iterative conjugate gradient solver which retains the sparsity of the coefficient matrix thus achieving a very efficient use of computer resources the method has been tested on a number of fluid flow and stress analysis problems and numerical calculations show favourable agreement with analytical and or experimental results in order to demonstrate the full capabilities of the present method an example of numerically complex coupled fluid flow stress analysis and heat transfer calculations is presented

maximal subgroups in finite and profinite groups we prove that if a finitely generated profinite group g is not generated with positive probability by finitely many random elements then every finite group f is obtained as a quotient of an open subgroup of g the proof involves the study of maximal subgroups of profinite groups as well as techniques from finite permutation groups and finite chevalley groups confirming a conjecture from ann of math 137 1993 203 220 we then prove that a finite group g has at most igic maximal soluble subgroups and show that this result is rather useful in various enumeration problems

the 4 triangles longest side partition of triangles and linear refinement algorithms in this paper we study geometrical properties of the iterative 4 triangles longest side partition of triangles and of a 3 triangles partition as well as practical algorithms based on these partitions used both directly for the triangulation refinement problem and as a basis for point insertion strategies in delaunay refinement algorithms the 4 triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides by means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles while they remain obtuse in the following sense the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount at least equal to the smallest angle of each iteration this allows us to improve the known bound on the smallest angle without making use of previous results and to obtain a better a priori bound on the number of similarly distinct triangles as a function of the geometry of the initial triangle numerical evidence showing that the practical behavior of the 4 triangles partition is in complete agreement with this theory is included a 4 triangles refinement algorithm is also discussed and illustrated furthermore we show that the time cost of the algorithm is linear independently of the size of the triangulation

identification of the givental formula with the spectral curve topological recursion procedure we identify the givental formula for the ancestor formal gromov witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve as an application we prove a conjecture of norbury and scott on the reconstruction of the stationary sector of the gromov witten potential of mathbb c rm p 1 via a particular spectral curve

tau structure for the double ramification hierarchies in this paper we continue the study of the double ramification hierarchy of buryak commun math phys 336 3 1085 1107 2015 after showing that the dr hierarchy satisfies tau symmetry we define its partition function as the logarithm of the tau function of the string solution and show that it satisfies various properties string dilaton and divisor equations plus some important degree constraints we then formulate a stronger version of the conjecture from buryak 2015 for any semisimple cohomological field theory the dubrovin zhang and double ramification hierarchies are related by a normal i e preserving the tau structure dubrovin et al in adv math 293 382 435 2016 miura transformation which we completely identify in terms of the partition function of the cohft in fact using only the partition functions the conjecture can be formulated even in the non semisimple case where the dubrovin zhang hierarchy is not defined we then prove this conjecture for various cohfts trivial cohft hodge class gromov witten theory of mathbb cp 1 3 4 and 5 spin classes and in genus 1 for any semisimple cohft finally we prove that the higher genus part of the dr hierarchy is basically trivial for the gromov witten theory of smooth varieties with non positive first chern class and their analogue in fan jarvis ruan witten quantum singularity theory fan et al in ann math 178 1 1 106 2013

phase field model for three dimensional dendritic growth with fluid flow we study the effect of fluid flow on three dimensional 3d dendrite growth using a phase field model on an adaptive finite element grid in order to simulate 3d fluid flow we use an averaging method for the flow problem coupled to the phase field method and the semi implicit approximated projection method siapm we describe a parallel implementation for the algorithm using the charm fem framework and demonstrate its efficiency we introduce an improved method for extracting dendrite tip position and tip radius facilitating accurate comparison to theory we benchmark our results for 2d dendrite growth with solvability theory and previous results finding them to be in good agreement the physics of dendritic growth with fluid flow in three dimensions is very different from that in two dimensions and we discuss the origin of this behavior

towards a more efficient dynamic mesh adaptation methodology for continuum discretization in complex engineering problems a novel and efficient method of adaptive mesh generation for dynamically adaptive unstructured grids is proposed a locally refined triangulation is constructed on a coarse background mesh subdividing each triangle in the refinement region r into four congruent sub triangles iteratively by connecting edge midpoints until triangles of a prescribed lengthscale are obtained the unavoidable propagation outside the refinement region r is restricted to a single triangle in the coarse background mesh the triangles in the immediate vicinity of region r are broken down using the concept of iterated function systems widely used in fractal modeling by recursive generation of sub triangles with a gradation towards the region r triangles a quantitative assessment of the present algorithm proves its superiority over other comparable models reported in the literature the time cost of the algorithm is linear and the method can be easily extended to three dimensions

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

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