varieties of nilpotent lie algebras of dimension less than six the set of all nilpotent ide algebra multiplications on a fixed vector space of dimension n over a field k forms an affine variety n isomorphism classes of these lie algebras are orbits in i under the change of basis action of gl n k by examining the orbit structure of n for small n we hope to gain a new perspective on the classification of nilpotent lie algebras in this paper we determine completely the orbit structure of the variety n when n 6 5 and k c we also show that the homomorphic image of a nilpotent lie algebra is in the closure of the orbit of that lie algebra taking the direct product with an abelian lie algebra to make the dimension right some of the orbit structure of n for n 6 and other geometric properties of n were described by vergne in vl v2 v3 recently carles has used vergne s results to study the variety of all lie algebras cl our motivation for studying the classification of nilpotent lie algebras is to gain information about the classification of nilpotent groups the correspondence between nilpotent lie algebras and nilpotent groups is described in go

a distortion theorem for quadrature domains for harmonic functions abstract we prove that any finitely connected domain in the plane can be distorted so that it becomes graviequivalent to a signed measure with arbitrarily small support precisely if d c is a bounded finitely connected domain with analytic boundary then for any a d and r 0 e 0 with b a r e d there exists a univalent function g in d with g z z z d and a signed measure with support in b a r such that for every integrable harmonic function h in g d we have h dx dy h d

analytic content and the isoperimetric inequality in higher dimensions abstract this paper establishes a conjecture of gustafsson and khavinson which relates the analytic content of a smoothly bounded domain in r n to the classical isoperimetric inequality the proof is based on a novel combination of partial balayage with optimal transport theory

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

existence of weak backward solutions to a generalized hele shaw flow moving boundary problem in the original hele shaw problem n 2 and u 6 the dirac measure at the origin a couple of essentially equivalent weak formulations of this problem have been given in 4 8 91 and in 3 f or a variant of the above problem in the case n 2 it has been proved in 4 81 that a classical solution this notion been made precise in two different ways in 4 81 also is a weak solution and in 4 8 91 and 3 that a weak solution exists and is unique for the time interval 0 0 for arbitrary given initial domains do it has also been proved 4 81 that at least under some extra hypotheses which probably are fulfilled for almost all f 0 in nonpathological cases ad consists of analytic curves when t 0 if d t 2 0 is a weak solution all the above results except the last one generalize without much trouble to arbitrary n b 2 as noted also in 9 section 31 some variant of the last result probably also holds for n 2 with analytic curve replaced by real analytic hypersurface no results seem to have been published on the existence or uniqueness of weak solutions for time intervals of the kind t 0 t 0 with do d given i e backward solutions the above mentioned results on the analyticity of dd show or indicate that a backward solution

nilpotent evolution algebras over arbitrary fields abstract the paper is devoted to the study of annihilator extensions of evolution algebras and suggests an approach to classify finite dimensional nilpotent evolution algebras subsequently nilpotent evolution algebras of dimension up to four are classified

laplacian flow of homogeneous g2 structures and its solitons we use the bracket flow algebraic soliton approach to study the laplacian flow of g2 structures and its solitons in the homogeneous case we prove that any homogeneous laplacian soliton is equivalent to a semi algebraic soliton that is a g invariant g2 structure on a homogeneous space g k that flows by pull back of automorphisms of g up to scaling algebraic solitons are geometrically characterized among laplacian solitons as those with a diagonal evolution unlike the ricci flow case where any homogeneous ricci soliton is isometric to an algebraic soliton we have found as an application of the above characterization an example of a left invariant closed semi algebraic soliton on a nilpotent lie group which is not equivalent to any algebraic soliton the normalized bracket flow evolution of such a soliton is periodic in the context of solvable lie groups with a codimension one abelian normal subgroup we obtain long time existence for any closed laplacian flow solution furthermore the norm of the torsion is strictly decreasing and converges to zero we also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding laplacian solitons

on the moment map for the variety of lie algebras we consider the moment map m pvn iu n for the action of gl n on vn 2 cn cn the critical points of the functional fn m 2 pvn r are studied in order to understand the stratification of ln pvn defined by the negative gradient flow of fn where ln is the algebraic subset of all lie algebras we obtain a description of the critical points which lie in ln in terms of those which are nilpotent as well as the minima and maxima of fn ln r a characterization of the critical points modulo isomorphism as the union of categorical quotients of suitable actions is considered and some applications to the study of ln are given

restriction operators balayage and doubly orthogonal systems of analytic functions abstract systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by walsh and szego and in full generality by bergman in principle these are very interesting objects allowing application to analytic continuation that is not restricted as weierstrassian continuation via power series either by circular geometry or considerations of locality however few explicit examples are known and in general one does not know even gross qualitative features of such systems the main contribution of the present paper is to prove qualitative results in a quite general situation it is by now very well known that the phenomenon of double orthogonality is not restricted to bergman spaces of analytic functions nor even indeed has it any intrinsic relation to analyticity its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a hilbert space of functions on some set to a subset thereof provided this restriction is injective and compact however in this paper only hilbert spaces of analytic functions are considered especially bergman spaces in the case of the hardy spaces fisher and micchelli discovered remarkable qualitative features of doubly orthogonal systems and we have shown how based on the classical potential theoretic notion of balayage and its modern generalizations one can deduce analogous results in the bergman space set up but with restrictions imposed on the geometry of the considered domains and measures these were not needed in the fisher micchelli analysis but are necessary here as shown by examples from a more constructive point of view we study the bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible as an inversion followed by a logarithmic derivative in terms of the polynomial equation of the boundary of the inner domain

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

finding einstein solvmanifolds by a variational method we use a variational approach to prove that any nilpotent lie algebra having a codimension one abelian ideal and anyone of dimension leq 5 admits a rank one solvable extension which can be endowed with an einstein left invariant riemannian metric a curve of 8 dimensional einstein solvmanifolds is also given

homogeneous einstein metrics and simplicial complexes on compact homogeneous spaces we investigate the hilbert action restricted to the space of homogeneous metrics of volume one based on a detailed understanding of the high energy levels of this action we assign to a compact homogeneous space a simplicial complex whose non contractibility is a sufficient condition for the existence of homogeneous einstein metrics

einstein solvmanifolds existence and non existence questions the aim of this paper is to study the problem of which solvable lie groups admit an einstein left invariant metric the space mathcal n of all nilpotent lie brackets on mathbb r n parametrizes a set of n 1 dimensional rank one solvmanifolds s mu mu in mathcal n containing the set of all those which are einstein in that dimension the moment map for the natural gl n action on mathcal n evaluated at mu in mathcal n encodes geometric information on s and suggests the use of strong results from geometric invariant theory for instance the functional on mathcal n whose critical points are precisely the einstein s s is the square norm of this moment map we use a gl n invariant stratification for the space mathcal n and show that there is a strong interplay between the strata and the einstein condition on the solvmanifolds s as an application we obtain criteria to decide whether a given nilpotent lie algebra can be the nilradical of a rank one einstein solvmanifold or not we find several examples of mathbb n graded even 2 step nilpotent lie algebras which are not a classification in the 7 dimensional 6 step case and an existence result for certain 2 step algebras associated to graphs are also given

immortal homogeneous ricci flows we show that for an immortal homogeneous ricci flow solution any sequence of parabolic blow downs subconverges to a homogeneous expanding ricci soliton this is established by constructing a new lyapunov function based on curvature estimates which come from real geometric invariant theory

on the classification of rigid lie algebras after having given the classification of solvable rigid lie algebras of low dimensions we study the general case concerning rigid lie algebras whose nilradical is filiform and present their classification

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

noncompact homogeneous einstein manifolds attached to graded lie algebras in this paper we study the nilradicals of parabolic subalgebras of semisimple lie algebras and the natural one dimensional solvable extensions of them we investigate the structures curvatures and einstein conditions of the associated nilmanifolds and solvmanifolds we show that our solvmanifold is einstein if the nilradical is two step new examples of einstein solvmanifolds with three step and four step nilradicals are also given

inequalities defining orbit spaces the orbit space of a representation of a compact lie group has a natural semialgebraic structure we describe explicit ways of finding the inequalities defining this structure and we give some applications

properties of some balayage operators with applications to quadrature domains and moving boundary problems in this paper we define and derive some properties of a fairly wide class of balayage operators we also apply our results to obtain geometric information about quadrature domains and solutions of certain moving boundary problems e g for hele shaw flows actually it was our interest in these applications that was the starting point for the present investigations and the balayage operators were developed rather as a tool our balayage operators are denoted f fp fp r and they depend on an open set r c i and a measure p in r actually we always assume that p has a density also denoted p in l r what f does is replace a given measure p by the nearest one in the energy norm v f p which satisfies v 5 p in r this presupposes that fi has finite energy if this is not the case there is another definition which is the one we will actually use among all distributions u in irn satisfying u i 17 the newtonian potential of p in irn and au i p in r there is a largest one u vp then f p ai by definition the operator of interest in the above mentioned applications is f with r irn p 1 the definition of f y when u has finite energy can be formulated as an elliptic variational inequality or linear complementarity problem and this is one of the standard tools when dealing with free boundary problems of the above kind see l 4 however we think that the balayage point of view is quite natural in our context the reason for considering the balayage operators for more general r and p than r rn p 1 is firstly that we need them in the proof of one of our theorems theorem 4 1 secondly and perhaps more important is that we think that the more general operators have quite a lot of intrinsic interest in particular as they turn out to contain classical balayage of positive measures as a special case namely with r bounded and p 0 recently we also learnt that discrete or numerical versions of these operators have been developed and applied in geophysics during the last three decades by dimiter zidarov who calls the balayage process partial gravi equivalent mass scattering see 5 and also for example 6 the paper is organized as follows in section 2 we define and establish the basic properties of our operators at the end we also explain their relation to quadrature domains and certain moving boundary problems section 3 is devoted to one single theorem theorem 3 1 and its corollaries these results concern the f with r lrn p 1 and give rather good geometric

applications of variational inequalities to the existence theorem on quadrature domains in this paper we shall study quadrature domains for the class of subharmonic functions by using the theory of variational inequalities we shall give a new proof of the existence and uniqueness theorem as an application we deal with hele shaw flows with a free boundary and show that their two weak solutions one of which was defined by the author using quadrature domains and the other was defined by gustafsson 3 using variational inequalities are identical with each other introduction in a previous paper 7 the author has defined the quadrature domains of positive measures for the class of subharmonic functions and studied their applications to complex function theory let v be a finite positive measure on the two dimensional euclidean space r2 let sl1 q be the class of subharmonic functions in an open set q which are integrable with respect to the two dimensional lebesgue measure m a nonempty open set q is called a quadrature domain of v for class sl1 if qi v is concentrated in q namely v qc 0 where qc denotes the complement of q qii jus s dv 1 a e in a connected open set w with finite area f 0 a e in wc and jf dm m w then q fm sl1 0 and there exists a minimum domain w in q fm sl1 namely q e q jm sl1 if and only if w c q and m q w o received by the editors october 5 1981 and in revised form february 23 1982 1980 mathematics subject classification primary 3 1aos 3 1bos secondary 35a15

selected topics on quadrature domains this is a selection of facts old and recent about quadrature domains the text written in the form of a survey is addressed to non experts and covers a variety of phenomena related to quadratur

cohomology operations for lie algebras if l is a lie algebra over r and z its centre the natural inclusion z l extends to a representation i z end h l r of the exterior algebra of z in the cohomology of l we begin a study of this representation by examining its poincare duality properties its associated higher cohomology operations and its relevance to the toral rank conjecture in particular by using harmonic forms we show that the higher operations presented by goresky kottwitz and macpherson 1998 form a subalgebra of end h l r and that they can be assembled to yield an explicit hirsch brown model for the borel construction associated to 0 z l l z 0

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

willmore orbits for isometric lie actions in this work we study the willmore submanifolds in a closed connected riemannian manifold which are orbits for the isometric action of a compact connected lie group we call them homogeneous willmore submanifolds or willmore orbits the criteria for these special willmore submanifolds is much easier than the general theory which requires a complicated euler lagrange equation our main theorem claims when the orbit type stratification for the group action satisfies certain conditions then we can find a willmore orbit in each stratified subset some classical examples of special importance like willmore torus veronese surface etc can be interpreted as willmore orbits and easily verified with our method our theorems provide a large number of new examples for willmore submanifolds as well as estimates for their numbers which are sharp in some classical cases

on the structure of partial balayage abstract we define partial balayage construct it with classical potential theoretic methods and establish the so called structure formula in full generality finally we give a stochastic construction of partial balayage

on commutative polarizations let l be a finite dimensional lie algebra over a field k of characteristic zero and let u l be its enveloping algebra with quotient division ring d l let p be a commutative lie subalgebra of l in o2 the necessary and sufficient condition on p was given in order for d p to be a maximal commutative subfield of d l in particular this condition is satisfied if p is a commutative polarization cp with respect to any regular f l and the converse holds if l is ad algebraic the purpose of this paper is to study lie algebras admitting these cp s and to demonstrate their widespread occurrence first we have the following characterisation if l is completely solvable p is a cp of l if and only if there exists a descending chain of lie subalgebras

a step towards the alekseevskii conjecture we refine existing structure results for non compact homogeneous einstein manifolds and provide a reduction in the classification problem of such spaces using this work we verify the generalized alekseevskii conjecture for a large class of homogeneous spaces

the ricci flow on solvmanifolds of real type abstract we show that for any solvable lie group of real type any homogeneous ricci flow solution converges in cheeger gromov topology to a unique non flat solvsoliton which is independent of the initial left invariant metric as an application we obtain results on the isometry groups of non flat solvsoliton metrics and einstein solvmanifolds

7 dimensional nilpotent lie algebras all 7 dimensional nilpotent lie algebras over c are determined by elementary methods a multiplication table is given for each isomorphism class distinguishing features are given proving that the algebras are pairwise nonisomorphic moduli are given for the infinite families which are indexed by the value of a complex parameter

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

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

examples of anosov diffeomorphisms we prove that if n is any graded rational lie algebra then the simply connected nilpotent lie group n n with lie algebra n r n r has a lattice such that the corresponding nilmanifold n n admits an anosov diffeomorphism this gives a procedure to construct easily several explicit examples of nilmanifolds admitting an anosov diffeomorphism and shows that a reasonable classification up to homeomorphism or even up to commensurability of such nilmanifolds would not be possible

parabolic subgroups of semisimple lie groups and einstein solvmanifolds in this paper we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple lie algebras these solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type we show that the ricci curvatures of our solvmanifolds coincide with the restrictions of the ricci curvatures of the ambient symmetric spaces consequently all of our solvmanifolds are einstein which provide a large number of new examples of noncompact homogeneous einstein manifolds we also show that our solvmanifolds are minimal but not totally geodesic submanifolds of symmetric spaces

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

counting rank two local systems with at most one unipotent monodromy the number of rank two overline bbb q ell local systems or overline bbb q ell smooth sheaves on x u otimes bbb f q bbb f where x is a smooth projective absolutely irreducible curve over bbb f q bbb f an algebraic closure of bbb f q and u is a closed point of x with principal unipotent monodromy at u and fixed by rm gal bbb f bbb f q is computed it is expressed as the trace of the frobenius on the virtual overline bbb q ell smooth sheaf found in the author s work with deligne on the moduli stack of curves with etale divisors of degree m ge 1 this completes the work with deligne in rank two this number is the same as that of representations of the fundamental group pi 1 x u otimes bbb f q bbb f invariant under the frobenius rm fr q with principal unipotent monodromy at u or cuspidal representations of rm gl 2 over the function field f bbb f q x of x over bbb f q with steinberg component twisted by an unramified character at u and unramified elsewhere trivial at the fixed id ele alpha of degree 1 this number is computed in theorem 4 1 using the trace formula evaluated at f u prod v not u chi k v with an iwahori component f u chi i u i u hence also the pseudo coefficient chi i u i u 2 chi k u of the steinberg representation twisted by any unramified character at u theorem 2 1 records the trace formula for rm gl 2 over the function field f the proof of the trace formula of theorem 2 1 recently appeared elsewhere theorem 3 1 computes following drinfeld the number of overline bbb q ell local systems or overline bbb q ell smooth sheaves on x otimes bbb f q bbb f fixed by rm fr q namely overline bbb q ell representations of the absolute fundamental group pi x otimes bbb f q bbb f invariant under the frobenius by counting the nowhere ramified cuspidal representations of rm gl 2 trivial at a fixed id ele alpha of degree 1 this number is expressed as the trace of the frobenius of a virtual overline bbb q ell smooth sheaf on a moduli stack this number is obtained on evaluating the trace formula at the characteristic function prod v chi k v of the maximal compact subgroup with volume normalized by k v 1 section 5 based on a letter of p deligne to the author dated august 8 2012 computes the number of such objects with any unipotent monodromy principal or trivial in our rank two case surprisingly this number depends only on x and rm deg s and not on the degrees of the points in s 1

on solvable lie groups of negative ricci curvature we study solvable lie groups which admit a left invariant metric of strictly negative ricci curvature we obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions

computing invariants and semi invariants by means of frobenius lie algebras abstract let u be the enveloping algebra of a finite dimensional lie algebra over a field k of characteristic zero z u its center and sz u its semi center a sufficient condition is given in order for sz u to be a polynomial algebra over k surprisingly this condition holds for many lie algebras especially among those for which the radical is nilpotent in which case sz u z u in particular it allows the explicit description of z u for more than half of all complex indecomposable nilpotent lie algebras of dimension at most 7

topological representation zeta functions of unipotent groups inspired by work surrounding igusa s local zeta function we introduce topological representation zeta functions of unipotent algebraic groups over number fields these group theoretic invariants capture common features of established p adic representation zeta functions associated with pro p groups derived from unipotent groups we investigate fundamental properties of the topological zeta functions considered here we also develop a method for computing them under non degeneracy assumptions as an application among other things we obtain a complete classification of topological representation zeta functions of unipotent algebraic groups of dimension at most six

solvable lie algebras and graphs abstract we define a solvable extension of the graph 2 step nilpotent lie algebras of 5 by adding elements corresponding to the 3 cliques of the graph we study some of their basic properties and we prove that two such lie algebras are isomorphic if and only if their graphs are isomorphic we also briefly discuss some metric properties providing examples of homogeneous spaces with nonpositive curvature operator and solvsolitons

existence and non existence of homogeneous einstein metrics on demontre un theoreme d existence general pour des metriques d einstein homogenes et on presente des espaces homogenes compacts simplement connexes qui ne portent pas une metrique d einstein homogene

the poisson center and polynomial maximal poisson commutative subalgebras especially for nilpotent lie algebras of dimension at most seven abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we collect some general results on the poisson center of s g including some simple criteria regarding its polynomiality and also on certain poisson commutative subalgebras of s g these facts are then used to complete our previous work on the subject ooms 2009 o5 5 i e to give an explicit description for the poisson center of all indecomposable nilpotent lie algebras of dimension at most seven among other things we also provide a polynomial maximal poisson commutative subalgebra of s g enjoying additional properties as a by product we show that a conjecture by milovanov is valid in this situation these results easily carry over to the enveloping algebra u g

the classification of 5 dimensional p nilpotent restricted lie algebras over perfect fields i abstract we first adapt a method due to skjelbred sund to classify p nilpotent restricted lie algebras it turns out that any p nilpotent restricted lie algebra of dimension n can be constructed as a central extension of a p nilpotent restricted lie algebra of dimension n 1 we apply these techniques to classify all p nilpotent restricted lie algebras of dimension 5 over a perfect field of characteristic p 5

the alekseevskii conjecture in low dimensions the long standing alekseevskii conjecture states that a connected homogeneous einstein space g k of negative scalar curvature must be diffeomorphic to mathbb r n this was known to be true only in dimensions up to 5 and in dimension 6 for non semisimple g in this work we prove that this is also the case in dimensions up to 10 when g is not semisimple for arbitrary g besides 5 possible exceptions we show that the conjecture holds up to dimension 8

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

methods for parametrizing varieties of lie algebras abstract a real n dimensional anticommutative nonassociative algebra is represented by an element of 2 r n r n for each 2 r n r n there is a unique subset i j k 1 i j n 1 k n so that the structure constant i j k with i j is nonzero if and only if i j k the set of all 2 r n r n with subset is denoted s r these sets stratify 2 r n r n we describe how to smoothly parametrize the lie algebras in s r up to isomorphism for satisfying certain frequently seen hypotheses

counting local systems with principal unipotent local monodromy let x1 be a curve of genus g projective and smooth over fq let s1 x1 be a reduced divisor consisting of n1 closed points of x1 let x s be obtained from x1 s1 by extension of scalars to an algebraic closure f of fq fix a primel not dividingq the pullback by the frobenius endomorphism fr ofx induces a permutation fr of the set of isomorphism classes of rank n irreducible ql local systems on x s it maps to itself the subset of those classes for which the local monodromy at each s2 s is unipotent with a single jordan block let t x1 s1 n m be the number of xed points of fr m acting on this subset under the assumption that n1 2 we show that t x1 s1 n m is given by a formula reminiscent of a lefschetz xed point formula the function m 7 t x1 s1 n m is of the form p ni m for suitable integers ni and eigenvalues i we use laorgue

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

construction of nice nilpotent lie groups abstract we illustrate an algorithm to classify nice nilpotent lie algebras of dimension n up to a suitable notion of equivalence applying the algorithm we obtain complete listings for n 9 on every nilpotent lie algebra of dimension 7 we determine the number of inequivalent nice bases which can be 0 1 or 2 we show that any nilpotent lie algebra of dimension n has at most countably many inequivalent nice bases

variation of complex structures and variation of lie algebras given a family of complex projective hypersurfaces in cp the torelli problem studied by p griffiths and his school asks whether the period map is injective on that family i e whether the family of complex hypersurfaces can be distinguished by means of their hodge structures a complex projective hypersurface in cp can be viewed as a complex hypersurface with isolated singularity in c t let v z c 1 f z o be a complex hypersurface with isolated singularity at the origin the moduli algebra of v 0 is a v

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified

a characterization of bounded symmetric domains by curvature this paper will prove that a bounded homogeneous domain is symmetric if and only if in the bergman metric all sectional curvatures are nonpositive introduction it is well known that a bounded symmetric domain has nonpositive sectional curvature in the bergman metric this paper is devoted to the converse namely that a bounded homogeneous domain equivalently homogeneous siegel domain which has nonpositive sectional curvature in the bergman metric must be a symmetric domain the proof uses the techniques of normal j algebras 13 as well as results of vinberg 16 17 the first author 1 3 and indirectly dorfmeister 5 6 the body of the paper is in two sections in the first we derive certain relations between the dimensions of the root spaces in an irreducible normal j algebra which in the presence of the additional curvature assumption show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root in the second section we show that the first dimension result implies that the cone of the corresponding siegel domain is self dual while the two dimension results together imply that the domain is quasi symmetric the proof is then finished because 3 a quasi symmetric domain with nonpositive sectional curvature in the bergman metric is known to be symmetric we remark that some of the subsidiary results mentioned above are not essentially new however for consistency of presentation it is necessary to have them in the language of normal jalgebras and for that reason they are included there is also overlap here with work of zelow lundquist 21 1 throughout this paper s j will denote a normal j algebra with admissible form w this means that s is a finite dimensional real split solvable lie algebra with almost complex structure j s s such that x y j jx y j x jy jx jy and w is a linear form on s such that the bilinear form o jx y is symmetric positive definite and j invariant note that the assumption that s is real split was incorrectly omitted in 2 let n s s and let a be the orthogonal complement of n in s by the basic structure theorem of pyatetskii shapiro 13 a is a commutative received by the editors july 13 1981 1980 mathematics subiect classification primary 32mi0 53c30 1983 american mathematical society 0002 9947 82 000 0759 09 00