on extensions of free nilpotent lie algebras of type 2 in this paper we study the structure of lie algebras which have free t nilpotent lie algebras n2 t of type 2 as nilradical and give a detailed construction for them we prove that the dimension of any lie algebra g of this class is dimn2 t k if g is solvable k 2 otherwise the levi subalgebra of g is sl2 k the split simple 3 dimensional lie algebra of 2 2 matrices of trace zero and then k 4 as an application of the main results we get the classification over algebraically closed fields of lie algebras with nilradical n2 1 n2 2 and n2 3

n dimensional superintegrable systems from symplectic realizations of lie coalgebras the construction of n dimensional nd integrable systems from coalgebras is reviewed in the case of poisson coalgebras a necessary condition for the integrability of the nd coalgebra hamiltonian coming from a given coalgebra is obtained in terms of the dimension of the symplectic realization and the number of nonlinear casimir functions from this viewpoint the full set of three four and five dimensional lie poisson coalgebras is analysed and many new families of multiparametric nd integrable systems coming from the cases that fulfil the integrability condition are obtained including the explicit form of the integrals of the motion the superintegrability of these hamiltonians is also emphasized and the generalization of the whole construction to the quantum mechanical case is straightforward

low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

classification of solvable lie algebras with a given nilradical by means of solvable extensions of its subalgebras abstract we construct all solvable lie algebras with a specific n dimensional nilradical n n 3 which contains the previously studied filiform n 2 dimensional nilpotent algebra n n 2 1 as a subalgebra but not as an ideal rather surprisingly it turns out that the classification of such solvable algebras can be deduced from the classification of solvable algebras with the nilradical n n 2 1 also the sets of invariants of coadjoint representation of n n 3 and its solvable extensions are deduced from this reduction in several cases they have polynomial bases i e the invariants of the respective solvable algebra can be chosen to be casimir invariants in its enveloping algebra

solvable lie algebras of dimension six six dimensional solvable lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes this completes mubarakzyanov s classification of the real six dimensional solvable lie algebras

the wente problem for a singular elliptic equation the jacobian det u of some vector field u belonging to w 1 n rn rn plays a particular role in various partial differential equation arising from calculus of variations mechanics and geometry although det u is only integrable but due to its special structure this quantity has some suitable regularity properties recently in 8 coifman et al have shown that det u belongs to the hardy space h1 rn a strict subspace of l1 rn in this paper we shall focus on the case n 2 let be a smooth and bounded domain in r2 given 0 a non negative measurable function and let u a b and be functions defined on consider the following problem div det u ax1bx2 ax2bx1 in 0 on 1

the cone of type a level 1 conformal blocks divisors abstract we prove that the type a level 1 conformal blocks divisors on m 0 n span a finitely generated full dimensional subcone of the nef cone each such divisor induces a morphism from m 0 n and we identify its image as a git quotient parameterizing configurations of points supported on a flat limit of veronese curves we show how scaling git linearizations gives geometric meaning to certain identities among conformal blocks divisor classes this also gives modular interpretations in the form of git constructions to the images of the hyperelliptic and cyclic trigonal loci in m g under an extended torelli map

structure of real lie algebras abstract restrictions are presented on the radical structure of an indecomposable lie algebra these restrictions are consequences of both the levi theorem and the representation theory of semisimple lie algebras as an application all real lie algebras of dimension 9 that admit a nontrivial levi decomposition are found these algebras are expressed in terms of known solvable and simple algebras of smaller dimensions and in terms of representations of simple algebras

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

the moduli spaces of hyperelliptic curves and binary forms hg b2g 2 lethg andb2g 2 denote the moduli spaces of stable hyperelliptic curves of genus g and semistable binary forms of degree 2g 2 respectively these are certainly different compactifications of hg b2g 2 in fact the boundary hg hg is a divisor whereas b2g 2 is a one point compactification of the moduli space of stable binary forms moreover hg and b2g 2 are constructed as quotients via different group actions hg is defined as the closure of hg in mg the moduli space of stable curves of genus g and this is constructed using the group pgl 6g 5 whereas b2g 2 is constructed classically using the group sl2 c it is the aim of this note to work out the relation between the spaceshg and b2g 2 the main result is the following theorem which generalizes theorem 5 6 of 1 where we give a proof in the special case g 2

on levi extensions of nilpotent lie algebras abstract levi s theorem decomposes any arbitrary lie algebra over a field of characteristic zero as a direct sum of a semisimple lie algebra named levi factor and its solvable radical given a solvable lie algebra r a semisimple lie algebra s is said to be a levi extension of r in case a lie structure can be defined on the vector space s r the assertion is equivalent to s der r where der r is the derivation algebra of r for some representation of s onto r our goal in this paper is to present some general structure results on nilpotent lie algebras admitting levi extensions based on free nilpotent lie algebras and modules of semisimple lie algebras in low nilpotent index a complete classification will be given the results are based on linear algebra methods and leads to computational algorithms

tautological classes on the moduli spaces of stable maps to p r via torus actions we present a localization proof of the fact that the cohomology of the moduli spaces of genus zero stable maps to projective spaces is entirely tautological in addition we obtain a description of a bialynicki birula decomposition of the stack of stable maps in the context of gathmann s moduli spaces of maps with prescribed contact order to a fixed hyperplane we show that the decomposition is filterable

configurations braids and homotopy groups the main results of this article are certain connections between braid groups and the homotopy groups of the 2 sphere the connections are given in terms of brunnian braids over the disk and over the 2 sphere the techniques arise from the natural structure of simplicial and structures on fundamental groups of configuration spaces

a class of solvable lie algebras and their casimir invariants a nilpotent lie algebra with an n 1 dimensional abelian ideal is studied all indecomposable solvable lie algebras with as their nilradical are obtained their dimension is at most n 2 the generalized casimir invariants of and of its solvable extensions are calculated for n 4 these algebras figure in the petrov classification of einstein spaces for larger values of n they can be used in a more general classification of riemannian manifolds

the modular variety of hyperelliptic curves of genus three the modular variety of nonsingular and complete hyperelliptic curves with level two structure of genus 3 is a 5 dimensional quasi projective variety which admits several standard compactifications the first one realizes this variety as a subvariety of the siegel modular variety of level two and genus three it has 36 irreducible isomorphic components one of the purposes of this paper will be to describe the equations of one of these components two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with 8 branch points there are two important compactifications of this configuration space the first one y uses the semistable degenerated point configurations in p 1 8 this variety also can be identified with a baily borel compactified ball quotient y b 1 i we will describe these results in some detail and obtain new proofs including some finer results for them the other compactification uses the fact that families of marked projective lines can degenerate to stable marked curves of genus 0 we use the standard notation m 0 8 for this compactification we have a diagram the horizontal arrow is only birational but not everywhere regular in this paper we find another realization of this triangle which uses the fact that there are graded algebras closely related to algebras of modular forms a b such that x proj a y proj b

all solvable extensions of a class of nilpotent lie algebras of dimension n and degree of nilpotency n 1 we construct all solvable lie algebras with a specific n dimensional nilradical of degree of nilpotency n 1 and with an n 2 dimensional maximal abelian ideal we find that for given n such a solvable algebra is unique up to isomorphisms using the method of moving frames we construct a basis for the casimir invariants of the nilradical we also construct a basis for the generalized casimir invariants of its solvable extension consisting entirely of rational functions of the chosen invariants of the nilradical

on quantum cohomology rings for hypersurfaces in cp n 1 using the torus action method we construct a one variable polynomial representation of quantum cohomology ring for degree k hypersurface in cpn 1 the results interpolate the well known result of cpn 2 model and the one of calabi yau hypersuface in cpn 1 we find in the k n 2 case the principal relation of this ring has a very simple form compatible with toric compactification of moduli space of holomorphic maps from cp1 to cpn 1

non semisimple lie algebras with levi factor fraktur so 3 fraktur sfraktur l 2 bbb r and their invariants we analyse the number of functionally independent generalized casimir invariants for non semisimple lie algebras r with levi factors isomorphic to 3 and 2 in dependence of the pair r formed by a representation r of and a solvable lie algebra we show that for any dimension n 6 there exist lie algebras r with non trivial levi decomposition such that r 0

solvable lie algebras with naturally graded nilradicals and their invariants the indecomposable solvable lie algebras with graded nilradical of maximal nilindex and a heisenberg subalgebra of codimension one are analysed and their generalized casimir invariants are calculated it is shown that rank one solvable algebras have a contact form which implies the existence of an associated dynamical system moreover due to the structure of the quadratic casimir operator of the nilradical these algebras contain a maximal non abelian quasi classical lie algebra of dimension 2n 1 indicating that gauge theories with ghosts are possible on these subalgebras

a new look at the moduli space of stable hyperelliptic curves we carry out the log minimal model program for the moduli space bar h g of stable hyperelliptic curves and show that certain log canonical models of bar h g are isomorphic to the proper transform of bar h g in the corresponding log canonical models of bar m g for g 3 we retrieve the compact moduli space bar b 8 of binary forms as a log canonical model and obtain a decomposition of the natural map bar h 3 to bar b 8 into successive divisorial contractions of the boundary divisors as a byproduct we also obtain an isomorphism of bar b 8 with the git quotient of the chow variety of bicanonically embedded hyperelliptic curves of genus three

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

solvable lie algebras with an mathbb n graded nilradical of maximal nilpotency degree and their invariants the class of solvable lie algebras with an graded nilradical of maximal nilpotency index is classified it is shown that such solvable extensions are unique up to isomorphism the generalized casimir invariants for the graded nilradicals and their associated solvable extensions are computed by the method of moving frames

classification of solvable real rigid lie algebras with a nilradical of dimension n 6 the complete classification of real solvable rigid lie algebras possessing a nilradical of dimension at most six is given eleven new isomorphism classes of indecomposable algebras are obtained it is further shown that the resulting solvable lie algebras have a vanishing second chevalley cohomology group thus correspond to algebraically rigid lie algebras

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

on the structure of maximal solvable extensions and of levi extensions of nilpotent lie algebras we establish an improved upper estimate on the dimension of any solvable lie algebra with its nilradical isomorphic to a given nilpotent lie algebra next we consider levi decomposable algebras with a given nilradical and investigate restrictions on possible levi factors originating from the structure of characteristic ideals of we present a new perspective on turkowski s classification of levi decomposable algebras up to dimension 9

simultaneous semi stable reduction for curves with ade singularities a key tool in the study of algebraic surfaces and their moduli is brieskorn s simultaneous resolution for families of algebraic surfaces with simple du val or ade singularities in this paper we show that a similar statement holds for families of curves with at worst simple ade singularities for a family x b of ade curves we give an explicit and natural resolution of the rational map b 99k mg moreover we discuss a lifting of this map to the moduli stack mg i e a simultaneous semi stable reduction for the family x b in particular we note that in contrast to what might be expected from the case of surfaces the natural weyl cover of b is not a sufficient base change for a lifting of the map b 99k mg to mg

best constant involving the l norm of evolutionary wente s problem associated to wave operator abstract in this paper we consider the solution of evolutionary wente s problem with the wave operator in r r 2 we study in particular the best constant involving the l norm of these solutions

the geometry of eight points in projective space representation theory lie theory and dualities this paper deals with the geometry of the space git quotient m8 of 8 points in p1 and the gale quotient n0 8 of the git quotient of 8 points in p3

solvable lie algebras with triangular nilradicals all finite dimensional indecomposable solvable lie algebras having the triangular algebra t n as their nilradical are constructed the number of non nilpotent elements f in satisfies and the dimension of the lie algebra is

solvable lie algebras with heisenberg ideals all finite dimensional indecomposable solvable lie algebras l n f having the heisenberg algebra h n as the nilradical are constructed the number of non nilpotent elements f that can be added to h n satisfies f or n 1 the casimir and generalized casimir operators of the algebras l n f are obtained