complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

supersymmetry and singular perturbations abstract first it is shown in an abstract framework that supersymmetric quantum theory may give rise to very singular perturbations which make perturbation theories invalid then the abstract results are applied to a model of 1 dimensional supersymmetric quantum mechanics witten s model to obtain some families of singular hamiltonians some concrete examples which reveal their singularness are discussed in detail

quadratic maps and bockstein closed group extensions let be a central extension of the form where and are elementary abelian groups associated to there is a quadratic map given by the power map which uniquely determines the extension this quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of we say that is bockstein closed if is an ideal closed under the bockstein operator we find a direct condition on the quadratic map that characterizes when the extension is bockstein closed using this characterization we show for example that quadratic maps induced from the fundamental quadratic map given by yield bockstein closed extensions on the other hand it is well known that an extension is bockstein closed if and only if it lifts to an extension for some lattice in this situation one may write for a binding matrix with entries in we find a direct way to calculate the module structure of in terms of using this we study extensions where the lattice is diagonalizable triangulable and find interesting equivalent conditions to these properties

the algebra of grand unified theories the standard model is the best tested and most widely accepted theory of elementary particles we have today it may seem complicated and arbitrary but it has hidden patterns that are revealed by the relationship between three grand unied theories theories that unify forces and particles by extending the standard model symmetry group u 1 su 2 su 3 to a larger group these three are georgi and glashow s su 5 theory georgi s theory based on the group spin 10 and the pati salam model based on the group su 2 su 2 su 4 in this expository account for mathematicians we explain only the portion of these theories that involves nite dimensional group representations this allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve

an optimal nonlinear extension of banach stone theorem abstract we prove that if k and s are locally compact hausdorff spaces and there exists a bijective coarse m l quasi isometry t between the banach spaces of real continuous functions c 0 k and c 0 s with m 2 then k and s are homeomorphic this nonlinear extension of banach stone theorem 1933 1937 is in some sense optimal and improves some results of amir 1965 cambern 1967 jarosz 1989 dutrieux and kalton 2005 and gorak 2011 in the lipschitz case that is when l 0 we also improve the estimations of the distance of the map t from the isometries between the spaces c 0 k and c 0 s obtained by gorak when k and s are compact spaces or not as a consequence we get a linear sharp refinement of the amir cambern theorem

a general class of infinite dimensional dirac operators and path integral representation of their index abstract a general class of infinite dimensional dirac operators is defined in an abstract boson fermion fock space and some of their properties are studied in particular a path integral representation of their index is established which gives a topologically invariant integer valued functional on a space of functionals with values in a hilbert space further it is shown that in concrete realizations the supersymmetric quantum theories associated with the dirac operators yield the wess zumino models in supersymmetric quantum field theory

lie algebras and higher torsion in p groups abstract the primary aim of this paper is to study exceptional torsion in the integral cohomology of a family of p groups associated to p adic lie algebras a spectral sequence e r g is defined for any lie algebra g which models the bockstein spectral sequence of the corresponding group in characteristic p this spectral sequence is then studied for complex semisimple lie algebras like sl n c and the results there are transferred to the corresponding p group via the intermediary arithmetic lie algebra defined over z the results obtained this way for a fixed lie algebra scheme like sl n hold in a range in the corresponding bockstein spectral sequence for all but finitely many primes depending on the chosen range over c it is shown that e 1 g h g u g h bg where u g is the filtered dual of the universal enveloping algebra of g equipped with the dual adjoint action and bg is the free loop space of the classifying space of an associated compact connected real form lie group g to g when passing to characteristic p in the corresponding bockstein spectral sequence a char 0 to char p phase transition is observed for example it is shown that the algebra e 1 sl 2 f p requires at least 17 generators unlike its characteristic zero counterpart which only requires two

a criterion for the boundedness from below with a class of symmetric operators and its applications abstract a class of symmetric operators h acting in l2 m with an abstract measure space is considered and a criterion for the boundedness from below of h is presented the criterion gives also a method to estimate the lower bound of h as a special case of the criterion an affirmative answer is given to the question if an eigenfunction of a self adjoint operator in l2 m is strictly positive or nodeless then is the corresponding eigenvalue the lowest point of the spectrum the class includes schrodinger operators on r n with r n m rnand being the lebesgue measure it is illustrated how the method may give a simple proof of the boundedness from below for some schrodinger operators further it is shown 1 that in a special case there exists an operator in l2 m essentially isospectral to h and 2 that the existence of a nodeless eigenfunction of h implies the existence of a supersymmetric quantum mechanics in which h is unitarily equivalent to the restriction of the supersymmetric hamiltonian to a subspace where is the lower bound of h

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

bockstein closed 2 group extensions and cohomology of quadratic maps abstract a central extension of the form e 0 v g w 0 where v and w are elementary abelian 2 groups is called bockstein closed if the components q i h w f 2 of the extension class of e generate an ideal which is closed under the bockstein operator in this paper we study the cohomology ring of g when e is a bockstein closed 2 power exact extension the mod 2 cohomology ring of g has a simple form and it is easy to calculate the main result of the paper is the calculation of the bocksteins of the generators of the mod 2 cohomology ring using an eilenberg moore spectral sequence we also find an interpretation of the second page of the bockstein spectral sequence in terms of a new cohomology theory that we define for bockstein closed quadratic maps q w v associated to the extensions e of the above form

supersymmetric quantum mechanics defined as sesquilinear forms a formulation of supersymmetric quantum mechanics in terms of sesquilinear forms is presented otherwise occurring rather complicated domain questions are solved in a simple and natural way the authors formulate the axioms and describe a number of models supersymmetry transformations are implemented as superunitary transformations in a space defined over a grassmann algebra generalised bogoliubov transformations acting in that space are also formulated

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

meromorphic n 2 wess zumino supersymmetric quantum mechanics the ordinary holomorphic n 2 wess zumino model in supersymmetric quantum mechanics is extended to the case where the superpotential v z is a meromorphic function on c the extended model is analyzed in a mathematically rigorous way self adjoint extensions and the essential self adjointness of the supercharges are discussed the supersymmetric hamiltonian defined by one of the self adjoint extensions of the supercharges has no fermionic zero energy states vanishing theorem it is proven that if v z has only one pole at z 0 in c then the supercharges are essentially self adjoint on c 0 r2 0 c4 the special case where v z z p p n c 0 is analyzed in detail to prove the following facts i the number of the bosonic zero energy ground state s is equal to p 1 ii the supercharges are not fredholm

cohomology of uniformly powerful p groups in this paper we will study the cohomology of a family of p groups associated to fp lie algebras more precisely we study a category bgrp of p groups which will be equivalent to the category of fp bracket algebras lie algebras minus the jacobi identity we then show that for a group g in this category its fp cohomology is that of an elementary abelian p group if and only if it is associated to a lie algebra we then proceed to study the exponent of h g z in the case that g is as sociated to a lie algebra l to do this we use the bockstein spectral sequence and derive a formula that gives b 2 in terms of the lie algebra cohomologies of l we then expand some of these results to a wider category of p groups in particular we calculate the cohomology of the p groups n k which are defined to be the kernel of the mod p reduction gln z p k 1 z mod gln fp

a supersymmetric generalization of von neumann s theorem a formulation of supersymmetric quantum mechanics is given and superunitary and generalized canonical transformations are defined acting in a module next it is assumed that there are operators that give an irreducible representation of the canonical commutation and anticommutation relations respectively and it is proved that two such representations are connected by a uniquely determined superunitary transformation under suitable domain assumptions this extends the well known uniqueness theorem of von neumann to canonical anti commutation relations using anticommuting parameters

coarse version of the banach stone theorem abstract we show that if there exists a lipschitz homeomorphism t between the nets in the banach spaces c x and c y of continuous real valued functions on compact spaces x and y then the spaces x and y are homeomorphic provided l t l t 1 6 5 by l t and l t 1 we denote the lipschitz constants of the maps t and t 1 this improves the classical result of jarosz and the recent result of dutrieux and kalton where the constant obtained is 17 16 we also estimate the distance of the map t from the isometry of the spaces c x and c y

cohomology of metacyclic groups group n by a finite cyclic group k using homological perturbation theory we introduce the beginning of a free resolution of the integers z over the group ring zg of g in such a way that the resolution reflects the structure of g as an extension of n by k and we use this resolution to compute the additive structure of the integral cohomology of g in many cases we proceed by first establishing a number of special cases thereafter constructing suitable cohomology classes thereby obtaining a lower bound then computing characteristic classes introduced in an earlier paper and finally exploiting these classes obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension e the calculation is then completed by comparing the two bounds

an optimal linear approximation for a class of nonlinear operators between uniform algebras abstract assume that a b are uniform algebras on compact hausdorff spaces x and y respectively let t a b be a map nonlinear in general satisfying t a 1 b 1 and t 1 1 we show that if there exist constants 1 such that 1 f g 1 t f t g 1 f g 1 for all f a and g a 1 then there exists a homeomorphism b a between the silov boundaries of a and b such that 1 f y t f y f y for all f a and for all y b in particular a and b are homeomorphic moreover we give an example which shows that the multiple in the above inequality is best possible

linear and quadratic ranges in representation stability abstract we prove two general results concerning spectral sequences of fi modules these results can be used to significantly improve stable ranges in a large portion of the stability theorems for fi modules currently in the literature we work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range previously the best stable ranges known in these examples were exponential up to an additive constant our work on congruence subgroups verifies a conjecture of djament

path integral representation of the index of kahler dirac operators on an infinite dimensional manifold abstract operators of kahler dirac type are defined in an abstract infinite dimensional boson fermion fock space and a path integral representation of their index is established as preliminaries to this end some trace formulas associated with gibbs states are derived in both an abstract boson and fermion fock space this is done by introducing euclidean bose and fermi fields at finite temperature in each case in connection with supersymmetric quantum field theories the result gives a path integral formula of the so called witten index in a model with cutoffs

groups of automorphisms of the canonical commutation and anticommutation relations observables of supersymmetric quantum mechanics are coded by taking the antisymmetric tensor product with anticommuting parameters next the authors define superunitary transformations which mix bosonic and fermionic degrees of freedom in order to construct automorphisms of the canonical anti commutation relations c a cr conversely every automorphism of the c a cr is implemented by an essentially unique superunitary transformation

primitives and central detection numbers in group cohomology abstract fix a prime p given a finite group g let h g denote its mod p cohomology in the early 1990s henn lannes and schwartz introduced two invariants d 0 g and d 1 g of h g viewed as a module over the mod p steenrod algebra they showed that in a precise sense h g is respectively detected and determined by h d c g v for d d 0 g and d d 1 g with v running through the elementary abelian p subgroups of g the main goal of this paper is to study how to calculate these invariants we find that a critical role is played by the image of the restriction of h g to h c where c is the maximal central elementary abelian p subgroup of g a measure of this is the top degree e g of the finite dimensional hopf algebra h c h g f p a number that tends to be quite easy to calculate our results are complete when g has a p sylow subgroup p in which every element of order p is central using the benson carlson duality we show that in this case d 0 g d 0 p e p and a similar exact formula holds for d 1 as a bonus we learn that h e g p contains nontrivial essential cohomology reproving and sharpening a theorem of adem and karagueuzian in general we are able to show that d 0 g max e c g v v g if certain cases of benson s regularity conjecture hold in particular this inequality holds for all groups such that the difference between the p rank of g and the depth of h g is at most 2 when we look at examples with p 2 we learn that d 0 g 14 for all groups with 2 sylow subgroup of order up to 64 with equality realized when g su 3 4 en route we study two objects of independent interest if c is any central elementary abelian p subgroup of g then h g is an h c comodule and we prove that the subalgebra of h c primitives is always noetherian of krull dimension equal to the p rank of g minus the p rank of c if the depth of h g equals the rank of z g we show that the depth essential cohomology of g is nonzero reproving and extending a theorem of green and cohen macauley in a certain sense and prove related structural results