the mathematics of microstructure and the design of new materials the pathological energy function e u u 2 for u 0 e 0 1 has no minimizer as u decreases to 0 the energy also decreases but there is no way to achieve the value 0 although examples like this might seem to be unimaginably far from scientific thought they are at the heart of a new approach 1 to understand the complex microstructure and macroscopic response of materials that undergo phase transformations the free energy of such materials typically has no minimizer and the observed microstructures complex fine scale patterns of domains of different atomic lattice structure as shown below in a micrograph of cualni by c chu and r d j fig 1 have their origin in the material s ultimately futile attempt to find the minimum energy state 2 the lack of a ground state prohibits prediction of the macroscopic response from microscopic data via the standard procedure determine the free energy find the minimizing state and evaluate its macroscopic properties emerging mathematical methods linked to profound work in the 1940s by l c young and recently surveyed in 3 nevertheless deliver well defined macroscopic quantities obtained via averaging over all low energy states one area where predictions obtained in this new way have played a role is the recent synthetization of a new magnetostrictive material 4 5 whose magnetostrictive strain is 50 times larger than that of giant magnetostrictive materials formerly those with the largest strain

windings of the 2d free rouse chain we study the long time dynamical properties of a chain of harmonically bound brownian particles this chain is allowed to wander everywhere in the plane we show that the scaling variables for the occupation times tj areas aj and winding angles j j 1 n labels the particles take the same general form as in the usual brownian motion we also compute the asymptotic joint laws p tj p aj p j and discuss the correlations occurring in those distributions

the random walk winding number problem convergence to a diffusion process with excluded area the winding number of a random walk in the plane is the net angle through which the walker encircles the origin the author discusses the moments and asymptotic form of the winding number distribution for walks of constant step size l the author shows that the root mean square winding number theta rms grows logarithmically with the number of steps n as n to infinity in contrast with the n1 2 growth of the radial distribution the corresponding diffusion process however has infinite theta rms and does not provide a good approximation for the random walk distribution instead a diffusion process considered recently by rudnick and hu 1987 where the area surrounding the origin has been removed out to a radius r provides the correct asymptotic distribution the author finds that the optimum radius for convergence of the finite step and diffusion distributions is precisely r e 2l

maps between classifying spaces of kac moody groups abstract we classify up to homotopy the self maps of the classifying space of any non affine kac moody group of rank two

classification of problematic subgroups of u n we classify p toral subgroups of u n that can have non contractible fixed points under the action of u n on the com plex ln of partitions of c n into mutually orthogonal subspaces

winding of the plane brownian motion on ecrit le mouvement brownien dans un format complexe x a 1b et on note son invariance conforme ce mouvement brownien x dans le plan perfore est releve a m 3 et on etudie si ce mouvement est recurseur

the classification of p compact groups for p odd a p compact group as defined by dwyer and wilkerson is a purely homotopically defined p local analog of a compact lie group it has long been the hope and later the conjecture that these objects should have a classification similar to the classification of compact lie groups in this paper we finish the proof of this conjecture for p an odd prime proving that there is a one to one correspondence between connected p compact groups and finite reflection groups over the p adic integers we do this by providing the last and rather intricate piece namely that the exceptional compact lie groups are uniquely determined as p compact groups by their weyl groups seen as finite reflection groups over the p adic integers our approach in fact gives a largely self contained proof of the entire classification theorem for p odd

normalizers of maximal tori normalizers and p normalizers of maximal tori in p compact groups can be characterized by the euler characteristic of the associated homogeneous spaces applied to centralizers of elementary abelian p groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus i e that normalizer commutes with centralizer

singular perturbation of an elastic energy with a singular weight abstract we study the singular perturbation of an elastic energy with a singular weight the minimization of this energy results in a multi scale pattern formation we derive an energy scaling law in terms of the perturbation parameter and prove that although one cannot expect periodicity of minimizers the energy of a minimizer is uniformly distributed across the sample finally following the approach developed by alberti and muller 2001 we prove that a sequence of minimizers of the perturbed energies converges to a young measure supported on piecewise linear periodic functions of slope 1 whose period depends on the location in the domain and the weights in the energy

filtrations of global equivariant k theory arone and lesh j reine angew math 604 73 136 2007 fund math 207 1 29 70 2010 constructed and studied spectrum level filtrations that interpolate between connective topological or algebraic k theory and the eilenberg maclane spectrum for the integers in this paper we consider global equivariant generalizations of these filtrations and another closely related class of filtrations the modified rank filtrations of the k theory spectra themselves we lift arone and lesh s description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups it turns out that these representation ring filtrations are considerably easier to express in a global equivariant context than over a fixed compact lie group furthermore they have formal similarities to the filtration on burnside rings induced by the symmetric products of spheres which was computed by schwede j am math soc 30 3 673 711 2017

quillen grassmannians as non modular homotopy fixed points the aim of this paper is to compute the homotopy fixed points of the homotopy action described by unstable adams operations on the classifying spaces of unitary groups the same technique can be applied to compute homotopy fixed points of the action of certain automorphisms on p compact groups called generalized grassmannians we use the description of quillen grassmannians to describe the set of homotopy representations of elementary abelian p groups into them

standard stratifications of ei categories and alperin s weight conjecture abstract we characterize the finite ei categories whose representations are standardly stratified with respect to the natural preorder on the simple representations the orbit category of a finite group with respect to any set of subgroups is always such a category taking the subgroups to be the p subgroups of the group we reformulate alperin s weight conjecture in terms of the standard and proper costandard representations of the orbit category we do this using the properties of the ringel dual construction and a theorem of dlab which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules we indicate that these results hold in the generality of an algebra whose simple modules are preordered rather than partially ordered

existence and uniqueness of linking systems chermak s proof via obstruction theory we present a version of a proof by andy chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems this proof differs from the one in ch2 in that it is based on the computation of derived functors of certain inverse limits this leads to a much shorter proof but one which is aimed mostly at researchers familiar with homological algebra

the isomorphism and isogeny theorems for reductive algebraic groups let g be a connected reductive algebraic group defined over an algebraically closed field k and let t be a maximal torus of g letx be the character group oftand r subset x the root system of grelative tot thusx is isomorphic to mathbb z r with r dim t the rank of g and for each alpha in r there exists a unipotent subgroup u alpha which is isomorphic to the additive group of k and is normalized by t according to the character alpha 1 1 if u alpha k to u alpha is an isomorphism then t u alpha left a right t 1 u alpha left alpha left t right a right for all t in t and a in k

the goodwillie tower of the identity functor and the unstable periodic homotopy of spheres we investigate goodwillie s taylor tower of the identity functor from spaces to spaces more specifically we reformulate johnson s description of the goodwillie derivatives of the identity and prove that in the case of an odd dimensional sphere the only layers in the tower that are not contractible are those indexed by powers of a prime moreover in the case of a sphere the tower is finite in v k periodic homotopy

for which pseudo reflection groups are thep adic polynomial invariants again a polynomial algebra abstract letwbe a finite group acting on a latticelover thep adic integers z p the analysis of the ring of invariants of the associatedw action on the algebra z p l of polynomial functions onlis a classical question of invariant theory ifpis coprime to the order ofw classical results show thatwis a pseudo reflection group if and only if the ring of invariants is again polynomial we analyze the situation for odd primes dividing the order ofwand in particular determine those pseudo reflection groups for which the ring of invariants z p l wis a polynomial algebra

discrete approximations for complex kac moody groups abstract we construct a map from the classifying space of a discrete kac moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological kac moody group and prove that it is a homology equivalence at primes q different from p this generalizes a classical result of quillen friedlander and mislin for lie groups as an application we construct unstable adams operations for general kac moody groups compatible with the frobenius homomorphism our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete kac moody groups

vector bundles over classifying spaces of compact lie groups the completion theorem of atiyah and segal as says that the complex k theory group k bg of the classifying space of any compact lie group g is isomorphic to r g the representation ring completed with respect to its augmentation ideal however the group k bg bg z bu does not directly contain information about vector bundles over the infinite dimensional complex bg itself the purpose of this paper is to compare the grothendieck group of vector bundles over bg which we denote k bg with both k bg and r g the main result is an algebraic description of k bg in terms of the representation rings of certain subgroups of g as one consequence we show that of the natural maps

fibrations of classifying spaces we investigate fibrations of the form z y x where two of the three spaces are classifying spaces of compact connected lie groups we obtain certain finiteness conditions on the third space which make it also a classifying space our results allow to express some of the basic notions in group theory in terms of homotopy theory i e in terms of classifying spaces as an application we prove that every retract of the classifying space of a compact connected lie group is again a classifying space

a filtration of spectra arising from families of subgroups of symmetric groups let ffn be a family of subgroups of enx which is closed under taking subgroups and conjugates such a family has a classifying space byfn and we showed in an earlier paper that a cormpatible choice of tfn for each n gives a simplicial monoid jjn btf which group completes to an infinite loop space in this paper we define a filtration of the associated spectrum whose filtration quotients given an extra condition on the families can be identified in terms of the classifying spaces of the families of subgroups that were chosen this gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum we calculate two examples the first is related to elementary abelian p groups and the second gives a new expression for the desuspension of sp so spm i so as a suspension spectrum

the asymptotic joint distribution of windings of planar brownian motion let z z t t 0 be a complex valued brownian motion starting at zo and consider the real valued winding processes 0i 02 0n where 0j i is the continuous total angle wound by z about zj up to time t and z z zn are n 1 distinct points in the complex plane c let c i c n jb b be n 2 mutually independent one dimensional processes where the first n are standard cauchy processes and the last 2 are standard brownian motions let r inf t l u local time at 0 of b up to time t wj cj u b n j l n

equivariant stable homotopy and sullivan s conjecture let g be a p group and let x be a g complex let eg denote a contractible space on which g acts freely by the homotopy fixed point set of x we mean the fixed point set f eg x g where f eg x denotes the function space of all maps eg x equipped with a g action by 9f 9f9 1 if we let denote the one point space with trivial g action we may also consider the g space f x it is canonically g homeomorphic to x the g map e g induces a map x g f x g in 19 d sullivan proposed the following

asymptotic laws for the winding angles of planar brownian motion using constrained path integrals we study the winding angle distribution of a two dimensional brownian motion around a given point by a careful analysis of the spectral properties of some schrodinger like hamiltonians we obtain a generalization of the messulam yor law various limiting cases are considered

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

strictly commutative complex orientation theory for a multiplicative cohomology theory e complex orientations are in bijective correspondence with multiplicative natural transformations to e from complex bordism cohomology mu if e is represented by a spectrum with a highly structured multiplication we give an iterative process for lifting an orientation mu rightarrow e to a map respecting this extra structure based on work of arone lesh the space of strictly commutative orientations is the limit of an inverse tower of spaces parametrizing partial lifts stage 1 corresponds to ordinary complex orientations and lifting from stage m 1 to stage m is governed by the existence of an orientation for a family of e modules over a fixed base space f m when e is p local we can say more we find that this tower only changes when m is a power of p and if e is e n local the tower is constant after stage p n moreover if the coefficient ring e is p torsion free the ability to lift from stage 1 to stage p is equivalent to a condition on the associated formal group law that was shown necessary by ando

twisted orbifold k theory we use equivariant methods to define and study the orbifold k theory of an orbifold x adapting techniques from equivariant k theory we construct a chern character and exhibit a multiplicative decomposition for k orb x in particular showing that it is additively isomorphic to the orbifold cohomology of x a number of examples are provided we then use the theory of projective representations to define the notion of twisted orbifold k theory in the presence of discrete torsion an explicit expression for this is obtained in the case of a global quotient

the z theorem for compact lie groups keywords categories of finite p subgroups z theorem compact lie group control of fusion p toral subgroups mod p cohomology reference ctg article 1991 001doi 10 1007 bf01445193 record created on 2008 12 16 modified on 2017 05 12

extensions of p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we study and classify extensions of p local finite groups and also compute the fundamental group of the classifying space of a p local finite group

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

loop structures on homotopy fibers of self maps of spheres let s 2 n 1 k denote the fiber of the degree k map on the sphere s 2 n 1 if k p r where p is an odd prime and n divides p 1 then s 2 n 1 k is known to be a loop space it is also known that s 3 2 r is a loop space for r 3 in this paper we study the possible loop structures on this family of spaces for all primes p in particular we show that s 3 4 is not a loop space our main result is that whenever s 2 n 1 p r is a loop space the loop structure is unique up to homotopy

dynamic stability of non minimizing phase mixtures we study the dynamics of pattern formation in the one dimensional partial differential equation u tt w u x x u xxt u 0 proposed recently by ball holmes pego and swart as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 u x 2 1 2 this problem possesses infinitely many equilibrium solutions corresponding to different microstructures or phase mixtures all locally energetically unstable in the natural norms however many are locally dynamically stable theorem 1 1 and the paper proves this theorem and investigates other properties of these equilibria

rational equivalences between classifying spaces the paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected lie groups with an application to genus sets of such spaces

a finite loop space not rationally equivalent to a compact lie group we construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact lie group hence providing a counterexample to a classical conjecture aided by machine calculation we verify that our counterexample is minimal i e that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact lie group extending the classical known bound of 5

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

filtered spectra arising from permutative categories given a special category c satisfying some mild hypotheses we construct a sequence of spectra interpolating between the spectrum associated to c and the eilenberg maclane spectrum hz examples of categories to which our construction applies are the category of finite sets the category of finite dimensional vector spaces and the category of finitely generated free modules over a reasonable ring in the case of finite sets our construction recovers the filtration of hz by symmetric powers of the sphere spectrum in the case of finite dimensional complex vector spaces we obtain an apparently new sequence of spectra am that interpolate between bu and hz we think of am as a bu analogue of spm s and describe far reaching formal similarities between the two sequences of spectra for instance in both cases the mth subquotient is contractible unless m is a power of a prime and in vk periodic homotopy the filtration has only k 2 nontrivial terms there is an intriguing relationship between the bu analogues of symmetric powers and weiss s orthogonal calculus parallel to the not yet completely understood relationship between the symmetric powers of spheres and the goodwillie calculus of homotopy functors we conjecture that the sequence am when rewritten in a suitable chain complex form gives rise to a minimal projective resolution of the connected cover of bu this conjecture is the bu analogue of a theorem of kuhn and priddy about the symmetric power filtration the calculus of functors provides substantial supporting evidence for the conjecture

a version of alperinʼs weight conjecture for finite category algebras let p be a prime and k an algebraically closed field of characteristic p we construct a functor c oc on the category of finite categories with the property that if g c is a finite group then oc is the orbit category of p subgroups of g this leads to an extension of alperin s weight conjecture to any finite category c stating that the number of isomorphism classes of simple kc modules should be equal to that of the weight algebra w koc of oc we show that the versions of alperin s weight conjecture for finite groups and for finite categories are in fact equivalent 2013 elsevier inc

representations of categories and their applications we define for each small category c a category algebra rc over a base ring r and study its representations when c is an ei category we develop a theory of vertices and sources for rc mod which parameterizes the indecomposable rc modules as a main application we use our theory to find formulas for computing higher inverse limits over c

extensions of linking systems with p group kernel we study extensions of p local finite groups where the kernel is a p group in particular we construct examples of saturated fusion systems mathcal f which do not come from finite groups but which have normal p subgroups a vartriangleleft mathcal f such that mathcal f a is the fusion system of a finite group one of the tools used to do this is the concept of a transporter system which is modelled on the transporter category of a finite group and is more general than a linking system

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

on the mod p cohomology of bpu p we study the mod p cohomology of the classifying space of the projective unitary group pu p we first prove that conjectures due to j f adams and kono and yagita 1993 about the structure of the mod p cohomology of the classifying space of connected compact lie groups hold in the case of pu p finally we prove that the classifying space of the projective unitary group pu p is determined by its mod p cohomology as an unstable algebra over the steenrod algebra for p 3 completing previous work by dwyer miller and wilkerson 1992 and broto and viruel 1998 for the cases p 2 3

asymptotic winding angle distribution for two dimensional levy flights the author studies the winding angle theta of planar levy flights around a given point say the origin o in particular the author calculates the variance theta 2 and show that after a long time t it is proportional to in t for brownian random walks theta 2 varies as in t 2 in the same conditions the author shows that the variable x theta in t is approximately distributed according to a gaussian law finally the authors compare results with computer simulations

minami webb splittings and some exotic p local finite groups we give a new proof of the minami webb formula for classifying spaces of finite groups by exploiting symonds s resolution of webb s conjecture the methods are applicable to obtain a stable decomposition of minami s type for the classifying spaces of the three exotic p local finite groups which were introduced by ruiz and viruel at the prime 7 we obtain a similar decomposition for the classifying spaces of a family of exotic p local finite groups which were constructed by broto levi and oliver

homology decompositions for p compact groups we construct a homotopy theoretic setup for homology decompositions of classifying spaces of p compact groups this setup is then used to obtain a subgroup decomposition for p compact groups which generalizes the subgroup decomposition with respect to p stubborn subgroups for a compact lie group constructed by jackowski mcclure and oliver

on the p compact groups corresponding to the p adic reflection groups g q r n there exists an infinite family of p compact groups whose weyl groups correspond to the finite p adic pseudoreflection groups g q r n of family 2a in the clark ewing list in this paper we study these p compact groups in particular we construct an analog of the classical whitney sum map a family of monomorphisms and a spherical fibration which produces an analog of the classical j homomorphism finally we also describe a faithful complexification homomorphism from these p compact groups to the p completion of unitary compact lie groups

homotopy equivalences of p completed classifying spaces of finite groups we study homotopy equivalences of p completions of classifying spaces of finite groups to each finite group g and each prime p we associate a finite category p c g with the following properties two p completed classifying spaces bg p and bg p have the same homotopy type if and only if the associated categories p c g and p c g are equivalent furthermore the topological monoid aut bg p of self equivalences is determined by the self equivalences of the associated category p c g

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

group actions on spheres with rank one isotropy let g be a rank two finite group and let h denote the family of all rank one p subgroups of g for which rankp g 2 we show that a rank two finite group g which satisfies certain explicit group theoretic conditions admits a finite g cw complex x s n with isotropy in h whose fixed sets are homotopy spheres our construction provides an infinite family of new non linear g cw complex examples

connected finite loop spaces with maximal tori finite loop spaces are a generalization of compact lie groups however they do not enjoy all of the nice properties of compact lie groups for example having a maximal torus is a quite distinguished property actually an old conjecture due to wilkerson says said that every connected finite loop space with a maximal torus is equivalent to a compact connected lie group we give some more evidence for this conjecture by showing that the associated action of the weyl group on the maximal torus always represents the weyl group as a crystallographic group we also develop the notion of normalizers of maximal tori for connected finite loop space and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal tori is equivalent to a compact connected lie group if it has the right normalizer of the maximal torus actually in the cases under consideration the information about the weyl group is sufficient to give the answer all this is done by first studying the analoguous local problems