self dual codes and modules for finite groups in characteristic two using representation theoretical methods we investigate self dual group codes and their extensions in characteristic 2 we prove that the existence of a self dual extended group code heavily depends on a particular structure of the group algebra kg which can be checked by an easy to handle criteria in elementary number theory surprisingly in the binary case such a code is doubly even if the converse of gleason s theorem holds true i e the length of the code is divisible by 8 furthermore we give a short representation theoretical proof of an earlier result of sloane and thompson which states that a binary self dual group code is never doubly even if the sylow 2 subgroups of g are cyclic it turns out that exactly in the case of a cyclic or klein four group as sylow 2 subgroup doubly even group codes do not exist

existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters abstract the existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved together with a partial regularity result

singularities of j holomorphic curves let v j be an almost complex manifold recall that a map from a riemann surface to v is j holomorphic if its derivative is complex linear a j holomorphic curve or j curve for short is the image of such a map nowhere locally constant the study of singularities of j holomorphic curves began with the statement of m gromov g about the positivity of intersections in the case when v is of dimension 4 the homological intersection of two compact j curves is a sum of positive local contributions and a similar property holds for the self intersection of one curve adjunction formula this property is crucial for applications of the theory to symplectic topology in dimension 4 and contact topology in dimension 3 gromov mcduff eliashberg hofer the positivity of intersections was proved by d mcduff mcd1 see also chapter vi in al then in mcd2 she proved that in any dimension a germ of j holomorphic map c 0 v v0 is topologically equivalent to a standard holomorphic germ from c 0 to cn 0 this was then strengthened to a c 1equivalence by m micallef and b white mw as a special case of a much more general result on singularities of generalized minimal surfaces they proved also that it applies to reducible germs

a stability property of symplectic packing we prove that for any closed symplectic 4 manifold m with h 2 m q there exists a number n 0 such that for every n n 0 m admits full symplectic packing by n equal balls we also indicate how to compute this n 0 our approach is based on donaldson s symplectic submanifold theorem and on tools from the framework of taubes theory of gromov invariants

the supremum of some canonical processes 1 introduction consider a random variable r v h such that eh2 l distributed like h to each t tk k 1 of 2 one can associate a r v xi ek l tkhk this defines a stochastic process xt t e2 by stochastic process we simply mean a collection of r v this process could be called the canonical process associated to h a long range and rather ambitious program would be to understand given h for which subsets t of 2 the process xt tct is bounded let us first discuss the case where h is centered gaussian the special importance of that case stems from the fact that for the question of boundedness discussed here gaussian processes of the type xt tct are the most general gaussian processes this fact is closely related to the rotational invariance of gaussian measures for a stochastic process xt tct where the set t is possibly uncountable we set

a note on self dual group codes we classify group algebras over galois rings containing self dual ideals i e ideals c which satisfy c c sup spl perp with respect to the natural nondegenerate bilinear form given on group algebras

on the boundedness of bernoulli processes we present a positive solution to the so called bernoulli conjecture concerning the characterization of sample boundedness of bernoulli processes we also discuss some applications and related open problems

symmetric semisimple modules of group algebras over finite fields and self dual permutation codes abstract a module of a finite group over a finite field with a symmetric non degenerate bilinear form which is invariant by the group action is called a symmetric module in this paper a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained and some applications to the self dual permutation codes are shown

periodic approximations of irrational pseudo rotations using pseudoholomorphic curves we prove that every c 1 smooth area preserving dieomorphism of the closed 2 disk having not more than one periodic point is the uniform limit of periodic c 1 smooth dieomorphisms in particular every smooth irrational pseudo rotation can be c 0 approximated by integrable systems this partially answers a long standing question of a katok regarding zero entropy hamiltonian systems in low dimensions our approach uses pseudoholomorphic curve techniques from symplectic geometry

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

hyperbolic modules and cyclic subgroups abstract let g be a finite group of odd order f a finite field of odd characteristic p and b a finite dimensional symplectic f g module we show that b is f g hyperbolic i e it contains a self perpendicular f g submodule iff it is f n hyperbolic for every cyclic subgroup n of g

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

majorizing measures without measures we give a reformulation of majorizing measures that does not involve measures but rather special sequences of partitions this formulation is more convenient to perform chaining with different distances

a general regularity theory for stable codimension 1 integral varifolds we give a necessary and sufficient geometric structural condition which we call the a structural hypothesis for a stable codimension 1 integral varifold on a smooth riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities the a structural hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded c 1 a hypersurfaces with boundary meeting only along their common boundary we establish that whenever a stable integral n varifold on a smooth n 1 dimensional riemannian manifold satisfies the a structural hypothesis for some a 0 1 2 its singular set is empty if n 6 discrete if n 7 and has hausdorff dimension n 7 if n 8 in view of well known examples this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses we also establish compactness of mass bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a structural hypothesis for some a 0 1 2 the a structural hypothesis on an n varifold for any a 0 1 2 is readily implied by either of the following two hypotheses i the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary ii the singular set of the varifold has vanishing n 1 dimensional hausdorff measure thus our theory subsumes the well known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions an optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

weighted homogeneous singularities and rational homology disk smoothings we classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings the nonexistence of rational homology disk smoothings is shown by symplectic geometric methods while the existence is verified via smoothings of negative weight

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

a dynamical characterization of universally tight lens spaces we give necessary and sufficient conditions for a closed connected co orientable contact 3 manifold m to be a standard lens space based on assumptions on the reeb flow associated to a defining contact form our methods also provide rational global surfaces of section for non degenerate reeb flows on l p q std with prescribed binding orbits

metabolic modules of finite group algebras over finite fields of characteristic two abstract let g be a finite group and let f be a finite field of characteristic 2 we introduce f special subgroups and f special elements of g in the case where f contains a p th primitive root of unity for each odd prime p dividing the order of g e g it is the case once f is a splitting field for all subgroups of g the f special elements of g coincide with real elements of odd order we prove that a symmetric fg module v is metabolic if and only if the restriction v d of v to every f special subgroup d of g is metabolic and also if and only if the characteristic polynomial on v defined by every f special element of g is a square of a polynomial over f some immediate applications to characters self dual codes and witt groups are given

on tensor induction for representations of finite groups abstract it is well known that if d is an irreducible complex representation of a finite group g then every direct summand of the restriction of d to a subgroup h must have degree at least as large as the degree of d divided by the index g h moreover d is induced from h if and only if the restriction does have a direct summand whose dimension is equal to this quotient this paper explores the possibility of an analogous result for tensor induction under the additional assumption that d is faithful quasi primitive and not a tensor product of projective representations of degree greater than 1 and that the fitting subgroup f g is not in the centre z g the main question is this if the restriction has a projective tensor factor whose degree is the g h th root of the degree of d does it follow that d is tensor induced from h among other results examples are given to show that the answer can be negative when the index is 2 an affirmative answer is proved for normal subgroups of odd index and also for arbitrary subgroups of odd prime index as might be expected the key lies in the study of f g z g as a symplectic module over a finite prime field in particular in exploring the connection between ordinary induction and form induction of such modules

phase transition for potentials of high dimensional wells for a potential function f w r k rc that attains its global minimum value at two disjoint compact connected submanifolds n in r k we discuss the asymptotics as 0 of minimizers u of the singular perturbed functional e u d r jruj 2 c 1 2 f u dx under suitable dirichlet boundary data g w r k in the expansion of e u with respect to 1 we identify the first order term by the area of the sharp interface between the two phases an area minimizing hypersurfacea and the energyc f of minimal connecting orbits between n c and n and the zerothorder term by the energy of minimizing harmonic maps into n both under the dirichlet boundary condition on and a very interesting partially constrained boundary condition on the sharp interfacea 2011 wiley periodicals inc

topology of symplectomorphism groups of s 2 s 2 in dimension 4 due to non existence of adequate tools very little is known about the topology of groups of diffeomorphisms for example it is unknown if the group of compactly supported diffeomorphisms of r is connected the situation is much better if one wants to study groups of symplectomorphisms this is due to the existence of powerful tools going by the name of pseudo holomorphic curve techniques and introduced in symplectic geometry by m gromov in his seminal paper of 1985 5 gromov proved in that paper among several other remarkable results the contractibility of the group of compactly supported symplectomorphisms of r with its standard symplectic form dx dy dx dy gromov also studied the following example letm be the symplectic manifold s s 1 0 0 where 0 r and 0 is a standard area form on s with total area equal to 1 denote by g the group of symplectomorphisms of m that act as the identity on h2 s s z gromov proved in 5 that g0 is homotopy equivalent to its subgroup of standard isometries so 3 so 3 he also showed why that would not be true for g with 0 and in 1987 d mcduff 10 constructed explicitly an element of infinite order in h1 g 0 in this paper we will give a more detailed description of g for 0 1 in particular we will prove the following on leave from instituto superior tecnico lisbon portugal partially supported by jnict programas ciencia and praxis fundacao luso americana para o desenvolvimento and a sloan doctoral dissertation fellowship current address school of mathematics institute for advanced study princeton nj 08540 usa e mail abreu math ias edu

gromov s compactness theorem for pseudo holomorphic curves we give a complete proof for gromov s compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary

a theorem on majorizing measures let t d be a metric space and r 1r an increasing convex function with 0 0 we prove that if m is a probability measure m on r which is majorizing with respect to d that is supper d t 0 1 1 m b x e de then e sup x s x t 32 s t t for each separable stochastic process x t t t which satisfies e x s x t d s t 1 for all s t t s t this is a strengthening of one of the main results from talagrand ann probab 18 1990 1 491 and its proof is significantly simpler

the structure of rational and ruled symplectic 4 manifolds this paper investigates the structure of compact symplectic 4 manifolds v w which contain a symplectically embedded copy c of s2 with nonnegative self intersection number such a pair v c w is called minimal if in addition the open manifold v c contains no exceptional curves i e symplectically embedded 2 spheres with self intersection 1 we show that every such pair v c w covers a minimal pair v c c which may be obtained from v by blowing down a finite number of disjoint exceptional curves in v c further the family of manifold pairs v c w under consideration is closed under blowing up and down we next give a complete list of the possible minimal pairs we show that v is symplectomorphic either to p2 with its standard form or to an s2 bundle over a compact surface with a symplectic structure which is uniquely determined by its cohomology class moreover this symplectomorphism may be chosen so that it takes c either to 2 a complex line or quadric in cp2 or in the case when v is a bundle to a fiber or section of the bundle department of mathematics state university of new york at stony brook stony brook new york 11794 3651 e mail address dusa math sunysb edu this content downloaded from 207 46 13 145 on wed 27 apr 2016 05 10 07 utc all use subject to http about jstor org terms

on the radial behavior of minimal surfaces and the uniqueness of their tangent cones suppose v is an m 1 dimensional minimal surface in r containing 0 as an isolated singular point what can one say about the structure of v near o this question initiated the present investigation and in certain special cases has a satisfactory answer 1 if one of the tangent cones to v at 0 is of the form 0 m where m is an m dimensional minimal submanifold of s and if also for each jacobi normal vectorfield z of m in sn 1 there is a one parameter family of minimal surfaces in s 1 having velocity z at m then 0 m is the unique tangent cone to v at 0 with v converging to 0 m for small radii r with rate rx it 0 2 in case m is the cartesian product of two standard spheres of appropriate radii then each jacobi vectorfield on m arises from isometric motions of sn 1 and the jacobi vectorfield hypothesis in 1 is satisfied this is shown in chapter 6 the uniqueness of tangent cones to one dimensional stationary varifolds with positive densities was shown in aa and the uniqueness of tangent cones to certain two dimensional area minimizing singular surfaces in r3 was shown by j taylor in tj1i tj2 tj3 also b white has shown such uniqueness for area minimizing hypersurfaces mod 4 in space dimen

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

filling by holomorphic curves in symplectic 4 manifolds we develop a general framework for embedded immersed jholomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4 dimensional symplectic manifolds in particular a deformation theory and an intersection theory for j holomorphic curves with boundary are developed bishop s local filling theorem is extended to almost complex manifolds existence and uniqueness of global fillings are given complete proofs then they are extended to the situation with nontrivial jholomorphic spheres culminating in the construction of singular fillings

a characterization of the tight 3 sphere ii recall that every closed oriented connected 3 manifold m admits a contact form we shall give conditions on a periodic orbit p0 of the reeb vector field defined by that allows the construction of an open book decomposition of m p0 into embedded planes asymptotic to p0 such that p0 is the binding orbit of the decomposition it follows that m is the tight 3 sphere as a by product a new dynamical criterion for the tightness of a contact structure is derived in the proof the results extend earlier results 1999 john wiley sons inc