restrictions of representations of a surface group to a pair of free subgroups let be the fundamental group of a compact orientable genus m surface and let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero define two free rank m subgroups of by a a 1 a m and b b 1 b m where a 1 a m b 1 b m j 1 m a j b j is the standard presentation of we consider representations of of a and of b into g restriction of representations induces a morphism from c g the variety of closed conjugacy classes of representations of to c a g c b g we prove that if m is greater than the semisimple rank of g then this morphism is dominant and almost all fibers are finite

reductive subgroups of reductive groups in nonzero characteristic abstract let g be a possibly nonconnected reductive linear algebraic group over an algebraically closed field k and let n n the group g acts on gn by simultaneous conjugation let h be a reductive subgroup of g we prove that if k has nonzero characteristic then the natural map of quotient varieties hn h gn g is a finite morphism we use methods introduced by vinberg who proved the same result in characteristic zero as an application we show that if is a finite group then the character variety c g of closed conjugacy classes of representations from to g is finite

separability and complete reducibility of subgroups of the weyl group of a simple algebraic group of type e7 abstract let g be a connected reductive algebraic group defined over an algebraically closed field k the aim of this paper is to present a method to find triples g m h with the following three properties property 1 g is simple and k has characteristic 2 property 2 h and m are closed reductive subgroups of g such that h m g and g m is a reductive pair property 3 h is g completely reducible but not m completely reducible we exhibit our method by presenting a new example of such a triple in g e 7 then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction

complete reducibility and separability let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 a subgroup of g is said to be separable in g if its global and infinitesimal centralizers have the same dimension we study the interaction between the notion of separability and serre s concept of g complete reducibility for subgroups of g the separability hypothesis appears in many general theorems concerning g complete reducibility we demonstrate that many of these results fail without this hypothesis on the other hand we prove that if g is a connected reductive group and p is very good for g then any subgroup of g is separable we deduce that under these hypotheses on g a subgroup h of g is g completely reducible provided the lie algebra of g is semisimple as an h module recently guralnick has proved that if h is a reductive subgroup of g and c is a conjugacy class of g then the intersection of c and g is a finite union of h conjugacy classes for generic p when certain extra hypotheses hold including separability this follows from a well known tangent space argument due to richardson but in general it rests on lusztig s deep result that a connected reductive group has only finitely many unipotent conjugacy classes we show that the analogue of guralnick s result is false if one considers conjugacy classes of n tuples of elements from h for n 1

a normal subgroup of a strongly reductive subgroup is strongly reductive abstract let h be a strongly reductive subgroup of a reductive linear algebraic group g over an algebraically closed field k we prove that any closed normal subgroup n of h is also a strongly reductive subgroup of g if g gl n k then this is a consequence of clifford s theorem from representation theory

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

l and l 2 semi global stabilisation of continuous time periodic linear systems with bounded controls this article is concerned with l and l 2 semi global stabilisation of continuous time periodic linear systems with bounded controls two problems namely l semi global stabilisation with controls having bounded magnitude and l 2 semi global stabilisation with controls having bounded energy are solved based on solutions to a class of periodic lyapunov differential equations pldes resulting from the problem of minimal energy control with guaranteed convergence rate under the assumption that the open loop system is asymptotically null controllable with constrained controls periodic feedback are established to solve the concerned problems the proposed plde based approaches possess the advantage that the resulting controllers are easy to implement since the designers need only to solve a linear differential equation a numerical example is worked out to illustrate the effectiveness of the proposed approach

a generalised pid type control scheme with simple tuning for the global regulation of robot manipulators with constrained inputs in this paper a globally stabilising pid type control scheme with a generalised saturating structure for robot manipulators under input constraints is proposed it gives rise to various families of bounded pid type controllers whose implementation is released from the exact knowledge of the system parameters and model structure compared to previous approaches of the kind the proposed scheme is not only characterised by its generalised structure but also by its very simple tuning criterion the simplest hitherto obtained in the considered analytical framework experimental results on a 3 degree of freedom direct drive manipulator corroborate the efficiency of the proposed approach

the modified equation approach to flux corrected abstract for convective problems the modified equation can be considered as the actual partial differential equation solved by a given numerical scheme using finite differences such an expression characterizes the dissipative and dispersive properties of the scheme adjusting the parameters of flux corrected transport fct algorithms to cancel the successive truncation terms in the modified equation can be used in place of fourier analysis when the velocity is no longer constant and uniform this technique is used to propose a time centered fct algorithm in which diffusion antidiffusion coefficients are velocity gradient dependent and which has reduced diffusion and noise level

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

a description of the brauer severi scheme of trace rings of generic matrices abstract here we use invariant theory to describe the brauer severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants using this approach we first prove that the brauer severi scheme of a trace ring is isomorphic to proj q for a graded ring q whose generators we describe in the first section this description also has a relevant interpretation over base fields of arbitrary characteristic in the second section of this paper we show that the brauer severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as brauer severi schemes of certain algebras

on a question of kulshammer for homomorphisms of algebraic groups abstract let g be a linear algebraic group over an algebraically closed field of characteristic p 0 we show that if h 1 and h 2 are connected subgroups of g such that h 1 and h 2 have a common maximal unipotent subgroup and h 1 r u h 1 and h 2 r u h 2 are semisimple then h 1 and h 2 are g conjugate moreover we show that if h is a semisimple linear algebraic group with maximal unipotent subgroup u then for any algebraic group homomorphism u g there are only finitely many g conjugacy classes of algebraic group homomorphisms h g such that u is g conjugate to this answers an analogue for connected algebraic groups of a question of b kulshammer in kulshammer s original question h is replaced by a finite group and u by a sylow p subgroup of h the answer is then known to be no in general we obtain some results in the general case when h is non connected and has positive dimension along the way we prove existence and conjugacy results for maximal unipotent subgroups of non connected linear algebraic groups when g is reductive we formulate kulshammer s question and related conjugacy problems in terms of the nonabelian 1 cohomology of unipotent radicals of parabolic subgroups of g and we give some applications of this cohomological approach in particular we analyse the case when g is a semisimple group of rank 2

hat g local systems on smooth projective curves are potentially automorphic dfg grants fg 1920 and spp 1489 erc grant 290766 aamot nsf grant dms 1404769 nsf grant dms 1161671 humboldt research award clay mathematics institute erc grant no 714405 gmlp

laplace operator and polynomial invariants abstract let a be a finite dimensional simple algebra not necessarily associative over the field of complex numbers c and let g denote the automorphism group aut a suppose that a has a symmetric nondegenerate associative g invariant bilinear form x y and a compact real form i e a subalgebra b over r of dimension dim r b dim c a where a is equal to the span of b over c and the restriction of x y to b is positive definite we describe generators of the algebra of polynomial g invariants of a system of several vectors from a in terms of x y and laplace operators in particular we give generators of the algebra of polynomial invariants of the adjoint representation of a simple linear algebraic group of any exceptional type e 6 as a consequence we get the first main theorem on matrix invariants invariants of minimal representation of g 2 and f 4

maximal representations of uniform complex hyperbolic lattices let rho be a maximal representation of a uniform lattice gamma subset rm su n 1 n geq 2 in a classical lie group of hermitian type h we prove that necessarily h rm su p q with p geq qn and there exists a holomorphic or antiholomorphic rho equivariant map from complex hyperbolic space to the symmetric space associated to rm su p q this map is moreover a totally geodesic homothetic embedding in particular up to a representation in a compact subgroup of rm su p q the representation rho extends to a representation of rm su n 1 in rm su p q

high order zip differencing of convective terms the zip flux form for differencing the term wv x where w is a convected quantity and v is a convective velocity is observed to be equivalent to differencing the alternative expression wvx wxv using centered second order finite differences the advantage of this form is that one class of nonlinear computational instabilities is eliminated based on this observation the extension of the zip flux concept to arbitrarily high order accuracy is given computational examples show the advantage both of the zip flux concept itself and of its higher order forms within the context of flux corrected transport fct algorithms

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

involutions and higher order automorphisms of higgs bundle moduli spaces this research was partially supported by the spanish mineco under the icmat severo ochoa grant no sev 2011 0087 and grant no mtm2013 43963 p and by the european commission marie curie irses moduli programme pirses ga 2013 612534

complete reducibility and commuting subgroups let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 we study j p serre s notion of g complete reducibility for subgroups of g specifically for a subgroup h and a normal subgroup n of h we look at the relationship between g complete reducibility of n and of h and show that these properties are equivalent if h n is linearly reductive generalizing a result of serre we also study the case when h mn with m a g completely reducible subgroup of g which normalizes n in our principal result we show that if g is connected n and m are connected commuting g completely reducible subgroups of g and p is good for g then h mn is also g completely reducible

on the method of modified equations vi asymptotic analysis of and asymptotic successive corrections techniques for two point boundary value problems in ode s the modified equation technique is extended to two point boundary value problems and a second order accurate implicit centered finite difference scheme for non homogeneous second order ordinary differential equations with linear boundary conditions is analyzed the first second and third modified equations or equivalent second equivalent and simply modified equations respectively for this scheme and its boundary conditions are presented it is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results for a linear problem multiple scales and summed up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme asymptotic successive corrections techniques are also applied to the three kinds of modified equations to obtain higher order schemes higher order boundary conditions are easily treated in the asymptotic successive corrections technique although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries the methods introduced in this paper are applied to homogeneous and non homogeneous second order linear and non linear ordinary differential equations and yield very accurate results

flux corrected transport this paper describes a class of explicit eulerian finite difference algorithms for solving the continuity equation which are built around a technique called flux correction these flux corrected transport fct algorithms are of indeterminate order but yield realistic accurate results in addition to the mass conserving property of most conventional algorithms the fct algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well this first paper concentrates on a simple one dimensional version of fct utilizing shasta a new transport algorithm for the continuity equation which is described in detail

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

further advancements on the output feedback global continuous control for the finite time and exponential stabilisation of bounded input mechanical systems desired conservative force compensation and experiments global saturating proportional saturating derivative sp sd type continuous control for the finite time or local exponential stabilisation of mechanical systems with bounded inputs is achieved a

complete reducibility and conjugacy classes of tuples in algebraic groups and lie algebras let h be a reductive subgroup of a reductive group g over an algebraically closed field k we consider the action of h on g n the n fold cartesian product of g with itself by simultaneous conjugation we give a purely algebraic characterization of the closed h orbits in g n generalizing work of richardson which treats the case h g this characterization turns out to be a natural generalization of serre s notion of g complete reducibility this concept appears to be new even in characteristic zero we discuss how to extend some key results on g complete reducibility in this framework we also consider some rationality questions

on global output feedback regulation of euler lagrange systems with bounded inputs we identify a class of euler lagrange systems with bounded inputs that can be globally asymptotically stabilized via dynamic output feedback in particular we prove that if the system is fully actuated and the forces due to the potential field can be dominated by the constrained control signals then global output feedback regulation is still possible incorporating some suitable saturation functions in the controller

orbit closures and invariants let g be a reductive linear algebraic group h a reductive subgroup of g and x an affine g variety let x h denote the set of fixed points of h in x and n g h the normalizer of h in g in this paper we study the natural map of quotient varieties psi x h x h n g h rightarrow x g induced by the inclusion x h subseteq x we show that given g and h psi x h is a finite morphism for all affine g varieties x if and only if h is a g completely reducible subgroup of g in the sense defined by serre this was proved in characteristic 0 by luna in the 1970s we discuss some applications and give a criterion for psi x h to be an isomorphism we show how to extend some other results in luna s paper to positive characteristic and also prove the following theorem let h and k be reductive subgroups of g then the double coset hgk is closed for generic g in g if and only if h cap gkg 1 is reductive for generic g in g

brief semiglobal stability of saturated linear pid control for robot manipulators this paper addresses the position regulation problem of robot manipulators under control input constraints it is proven that the robot system under a saturated linear pid control is semiglobally asymptotically stable if the torque bounds are larger than gravitational torques and if the proportional gain is large enough and the integral gain is small enough the stability analysis makes use of singular perturbation tools and the results are illustrated via numerical simulations

compactification d espaces de representations de groupes de type fini let be a finitely generated group and g be a noncompact semisimple connected real lie group with finite center we consider the space fancyscript x gamma g of conjugacy classes of semisimple representations of into g which is the maximal hausdorff quotient of rm hom gamma g g we define the translation vector of g in g with value in a weyl chamber as a natural refinement of the translation length of g in the symmetric space associated with g we construct a compactification of fancyscript x gamma g induced by the marked translation vector spectrum generalizing thurston s compactification of the teichmuller space we show that the boundary points are projectivized marked translation vector spectra of actions of on affine buildings with no global fixed point an analoguous result holds for any reductive group g over a local field

monodromy of the sl n and gl n hitchin fibrations we compute the monodromy of the hitchin fibration for the moduli space of l twisted sl n mathbb c and gl n mathbb c higgs bundles for any n on a compact riemann surface of genus g 1 we require the line bundle l to either be the canonical bundle or satisfy deg l 2g 2 the monodromy group is generated by picard lefschetz transformations associated to vanishing cycles of singular spectral curves we construct such vanishing cycles explicitly and use this to show that the sl n mathbb c monodromy group is a skew symmetric vanishing lattice in the sense of janssen using the classification of vanishing lattices over mathbb z we completely determine the structure of the monodromy groups of the sl n mathbb c and gl n mathbb c hitchin fibrations as an application we determine the image of the restriction map from the cohomology of the moduli space of higgs bundles to the cohomology of a non singular fibre of the hitchin fibration

conjugacy classes of n tuples in lie algebras and algebraic groups let g be a reductive algebraic group over an algebraically closed field f of characteristic zero and let l g be the lie algebra of g then g acts on g by inner automorphisms and it acts on l g by the adjoint representation by taking the diagonal actions we get actions of g on the spaces of n tuples g n and l g in this paper we will prove a number of geometric properties of the orbits of g on these spaces of n tuples