enclosed convex hypersurfaces with maximal affine area in this paper we study the regularity of closed convex surfaces which achieve maximal affine area among all the closed convex surfaces enclosed in a given domain in the euclidean 3 space we prove the c 1 regularity for general domains and c 1 1 regularity if the domain is uniformly convex

the bernstein problem for affine maximal hypersurfaces in this paper we prove the validity of the chern conjecture in affine geometry 18 namely that an affine maximal graph of a smooth locally uniformly convex function on two dimensional euclidean space r 2 must be a paraboloid more generally we shall consider the n dimensional case r n showing that the corresponding result holds in higher dimensions provided that a uniform strict convexity condition holds we also extend the notion of affine maximal to non smooth convex graphs and produce a counterexample showing that the bernstein result does not hold in this generality for dimension n 10

groups with just one character degree divisible by a given prime the ito michler theorem asserts that if no irreducible character of a finite group g has degree divisible by some given prime p then a sylow p subgroup of g is both normal and abelian in this paper we relax the hypothesis and we assume that there is at exactly one multiple of p that occurs as the degree of an irreducible character of g we show that in this situation a sylow p subgroup of g is almost normal in g and it is almost abelian

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

affine complete locally convex hypersurfaces an open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in euclidean n 1 space is euclidean complete for n 2 in this paper we give the affirmative answer as an application it follows that an affine complete affine maximal surface in r 3 must be an elliptic paraboloid

lie point symmetries and stochastic differential equations we discuss lie point symmetries of stochastic ordinary differential equations and the interrelations between these and analogous symmetries of the associated fokker planck equation for the probability measure

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

compactness results for immersions of prescribed gaussian curvature i analytic aspects abstract we extend recent results of guan and spruck proving existence results for constant gaussian curvature hypersurfaces in hadamard manifolds

the existence of hypersurfaces of constant gauss curvature with prescribed boundary in this paper we are concerned with the problem of finding hypersurfaces of constant gauss kronecker curvature k hypersurfaces in r n 2 with prescribed boundary given a disjoint collection 1 m of closed smooth embedded n 1 dimensional submanifolds of r decide whether there exist immersed k hypersurfaces m in r with m locally this problem reduces to questions concerning monge ampere type equations and we seek solutions for which the resulting equation is elliptic this means that we must confine ourselves to the class of locally strictly convex hypersurfaces i e those whose principal curvatures are all positive such hypersurfaces locally lie on one side of their tangent planes at any point but need not do so globally as they have non empty boundary finding hypersurfaces with prescribed curvature and boundary has been a major challenge in geometric analysis because of the highly nonlinear nature of the problem and the lack of variational methods beginning around 1980 some success was achieved due to breakthroughs in the theory of monge ampere equations and general fully nonlinear equations but only for hypersurfaces which are globally graphs of functions over domains with geometric restrictions e g strictly convex domains this was the case even for k hypersurfaces where the only general existence results were consequences of the existence theory for monge ampere equations see 6 20 23 and was restricted to strictly convex domains this means that the resulting surfaces must be simply connected graphs a very strong restriction geometrically the first idea that a more general result was possible came in the paper of hoffmanrosenberg spruck 18 and subsequently such a general result was developed in 12 and 11 in these papers the authors proved an essentially optimal existence theorem for monge ampere equations in domains of arbitrary geometry and thus the limit of

existence of convex hypersurfaces with prescribed gauss kronecker curvature let f x be a given positive function in rn 1 in this paper we consider the existence of convex closed hypersurfaces x so that its gausskronecker curvature at x x is equal to f x this problem has variational structure and the existence of stable solutions has been discussed by tso j diff geom 34 1991 389 410 using the mountain pass lemma and the gauss curvature flow we prove the existence of unstable solutions to the problem

locally convex hypersurfaces of constant curvature with boundary for some constant k where m 1 n denotes the principal curvatures of m important examples include the classical plateau problem for minimal or constant mean curvature surfaces and the corresponding problem for gauss curvature which was treated recently by the authors 12 and independently by trudinger wang 24 in this paper as in our previous work 12 we are concerned with locally convex hypersurfaces accordingly the function f is assumed to be defined in the convex cone n r each component i 0 in r and satisfy the fundamental structure conditions

special lagrangian curvature we define the notion of special lagrangian curvature showing how it may be interpreted as an alternative higher dimensional generalisation of two dimensional gaussian curvature we obtain first a local rigidity result for this curvature when the ambient manifold has negative sectional curvature we then show how this curvature relates to the canonical special legendrian structure of spherical subbundles of the tangent bundle of the ambient manifold this allows us to establish a strong compactness result in the case where the special lagrangian angle equals n 1 2 we obtain compactness modulo a unique mode of degeneration where a sequence of hypersurfaces wraps ever tighter round a geodesic

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

topology of riemannian submanifolds with prescribed boundary we prove that a smooth compact submanifold of codimension 2 immersed in r n 3 bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces this settles a question of guan and spruck related to a problem of yau analogous results for complete fillings of arbitrary riemannian submanifolds are obtained as well on the other hand we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular our proofs employ among other methods a relative version of nash s isometric embedding theorem and the theory of alexandrov spaces with curvature bounded below including the compactness and stability theorems of gromov and perelman

many projectively unique polytopes we construct an infinite family of 4 polytopes whose realization spaces have dimension smaller or equal to 96 this in particular settles a problem going back to legendre and steinitz whether and how the dimension of the realization space of a polytope is determined bounded by its f vector from this we derive an infinite family of combinatorially distinct 69 dimensional polytopes whose realization is unique up to projective transformation this answers a problem posed by perles and shephard in the sixties moreover our methods naturally lead to several interesting classes of projectively unique polytopes among them projectively unique polytopes inscribed to the sphere the proofs rely on a novel construction technique for polytopes based on solving cauchy problems for discrete conjugate nets in s d a new alexandrov van heijenoort theorem for manifolds with boundary and a generalization of lawrence s extension technique for point configurations

noether s theorem in symmetric stochastic calculus of variations quite recently a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics which is an extension of nelson s stochastic mechanics in this article a noether s theorem is formulated within this calculus of variations conservation laws of momentum angular momentum and energy are proved which are related with the same laws in quantum mechanics

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

mathbf h principles for hypersurfaces with prescribed principle curvatures and directions we prove that any compact orientable hypersurface with boundary immersed resp embedded in euclidean space is regularly homotopic resp isotopic to a hypersurface with principal directions which may have any prescribed homotopy type and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant further we construct regular homotopies resp isotopies which control the principal curvatures and directions of hypersurfaces in a variety of ways these results which we prove by holonomic approximation establish some h principles in the sense of gromov and generalize theorems of gluck and pan on embedding and knotting of positively curved surfaces in 3 space

rank 3 permutation modules of the finite classical groups abstract the cross characteristic permutation modules for the actions of the finite classical groups on singular 1 spaces of their natural modules are studied the composition factors and submodule lattices are determined

on locally convex hypersurfaces with boundary in this paper we give some geometric characterizations of locally convex hypersurfaces in particular we prove that for a given locally convex hypersurface m with boundary there exists r 0 depending only on the diameter of m and the principal curvatures of m on its boundary such that the r neighbourhood of any given point on m is convex as an application we prove an existence theorem for a plateau problem for locally convex hypersurfaces of constant gauss curvature

cross characteristic representations of even characteristic symplectic groups we classify the small irreducible representations of sp 2n q with q even in odd characteristic this improves even the known results for complex representations the smallest representation for this group is much larger than in the case when q is odd this makes the problem much more difficult

primes dividing the degrees of the real characters let g be a finite group and let irr g denote the set of all complex irreducible characters of g the ito michler theorem asserts that if a prime p does not divide the degree of any in irr g then a sylow p subgroup p of g is normal in g we prove a real valued version of this theorem where instead of irr g we only consider the subset irrrv g consisting of all real valued irreducible characters of g we also prove that the character degree graph associated to irrrv g has at most 3 connected components similar results for the set of real conjugacy classes of g have also been obtained

nonsolvable groups satisfying the prime power hypothesis let g be a finite group with the degree set cd g cd g of its complex irreducible characters we call that g satisfies the prime power hypothesis if for distinct degrees 1 1 cd g 1 1 cd g the greatest common divisor gcd 1 1 gcd 1 1 is a prime power in this paper we show that cd g 18 cd g 18 if g is a nonsolvable group satisfying the prime power hypothesis

some a priori bounds for solutions of the constant gauss curvature equation abstract in this work we give a priori height and gradient estimates for solutions of the prescribed constant gauss curvature equation in euclidean space we shall consider convex radial graphs with positive constant mean curvature the estimates are established by considering in such a graph the riemannian metric given by the second fundamental form of the immersion

normal forms for stochastic differential equations we address the following problem from the intersection of dynamical systems and stochastic analysis two sde dx t j 0 m f j x t dw t j and dx t j 0 m g j x t dw t j in d with smooth coefficients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random diffeomorphism coordinate transformation h with h 0 0 and dh 0 id which conjugates the corresponding local flows

the affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions