deformed oscillator algebra approach of some quantum superintegrable lissajous systems on the sphere and of their rational extensions we extend the construction of 2d superintegrable hamiltonians with separation of variables in spherical coordinates using combinations of shift ladder and supercharge operators to models involving rational extensions of the two parameter lissajous systems on the sphere these new families of superintegrable systems with integrals of arbitrary order are connected with jacobi exceptional orthogonal polynomials of type i or ii and supersymmetric quantum mechanics moreover we present an algebraic derivation of the degenerate energy spectrum for the one and two parameter lissajous systems and the rationally extended models these results are based on finitely generated polynomial algebras casimir operators realizations as deformed oscillator algebras and finite dimensional unitary representations such results have only been established so far for 2d superintegrable systems separable in cartesian coordinates which are related to a class of polynomial algebras that display a simpler structure we also point out how the structure function of these deformed oscillator algebras is directly related with the generalized heisenberg algebras spanned by the nonpolynomial integrals

chow groups of projective varieties of very small degree see 12 5 and the references given there these facts together with various conjectures on the cohomology and chow groups of algebraic varieties suggest that the chow groups of x might satisfy chl x q chl pk q q for l 1 compare with remark 5 6 and corollary 5 7 this is explicitly formulated by v srinivas and k paranjape in 16 conjecture 1 8 the chain of reasoning goes roughly as follows suppose x is smooth one expects a good filtration 0 f j 1 f j f 0 ch x x q whose graded pieces f f l 1 are controlled by h2j l x x see 10 according to grothendieck s generalized conjecture 8 the groups h x should be generated by the image under the gysin morphism of the homology of a codimension subset together with the classes coming from p applying this to the diagonal in x x should then force the triviality of the chow groups in the desired range for zero cycles the conjecture follows from roitman s theorem see 17 and 18

all complete intersection varieties are fano visitors abstract we prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective fano variety this enables us to define a new invariant of smooth projective varieties and raise many interesting questions

on the relationship between monstrous moonshine and the uniqueness of the moonshine module we consider the relationship between the conjectured uniqueness of the moonshine module open image in new window and monstrous moonshine the genus zero property of the modular invariance group for each monster group thompson series we first discuss a family of possiblezn meromorphic orbifold constructions of open image in new window based on automorphisms of the leech lattice compactified bosonic string we reproduce the thompson series for all 51 non fricke classes of the monster groupm together with a new relationship between the centralisers of these classes and 51 corresponding conway group centralisers generalising a well known relationship for 5 such classes assuming that open image in new window is unique we consider meromorphic orbifoldings of open image in new window and show that monstrous moonshine holds if and onlyzr if the only meromorphic orbifoldings of open image in new window are open image in new window itself or the leech theory this constraint on the meromorphic orbifoldings of open image in new window therefore relates monstrous moonshine to the uniqueness of open image in new window in a new way

algebraic varieties with quasi projective universal cover we prove that the universal cover of a normal projective variety x is quasi projective if and only if a finite etale cover of x is a fiber bundle over an abelian variety with simply connected fiber

bps algebras genus zero and the heterotic monster in this note we expand on some technical issues raised in paquette et al 2016 commun number theory phys 10 433 526 by the authors as well as providing a friendly introduction to and summary of our previous work we construct a set of heterotic string compactifications to 0 1 dimensions intimately related to the monstrous moonshine module of frenkel lepowsky and meurman and orbifolds thereof using this model we review our physical interpretation of the genus zero property of monstrous moonshine furthermore we show that the space of second quantized bps states forms a module over the monstrous lie algebras m g some of the first and most prominent examples of generalized kac moody algebras constructed by borcherds and carnahan in particular we clarify the structure of the module present in the second quantized string theory we also sketch a proof of our methods in the language of vertex operator algebras for the interested mathematician

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

monstrous moonshine from orbifolds we consider the monster module of frenkel lepowsky and meurman as az2 orbifold of a bosonic string compactified by the leech lattice we show that the main conway and norton monstrous moonshine properties stating that the thompson series for each monster group conjugacy class has a modular invariance group of genus zero follow from an orbifold construction based on an orbifold group composed of monster group elements it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each thompson series it is also shown that the power map formula of conway and norton follows from the same vacuum structure finally we demonstrate the validity of the vacuum conjectures for sectors twisted by leech lattice automorphisms in many cases

on the proper moduli spaces of smoothable kahler einstein fano varieties in this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth kahler einstein fano manifolds inside an appropriate hilbert scheme in particular we prove that being k semistable is a zariski open condition and we establish the uniqueness of the gromov hausdorff limit for a punctured flat family of kahler einstein fano manifolds based on these we construct a proper scheme parameterizing the s equivalent classes of q gorenstein smoothable k semistable q fano varieties and we verify various necessary properties to guarantee that it is a good moduli space

super whittaker vector at c 3 2 the degenerate whittaker vector of the superconformal algebra can be represented in terms of jack superpolynomials however in this representation the norm of the whittaker vector involves a scalar product with respect to which the jack superpolynomials are not orthogonal in this note we point out that this defect can be cured at c 3 2 by means of a trick specific to the supersymmetric case at c 3 2 we thus end up with a closed form expression for the norm of the degenerate super whittaker vector granting the super version of the agt conjecture this closed form expression should be equal to the symmetric su 2 pure gauge instanton partition function the corresponding equality taking the form of a rather nontrivial combinatorial identity

on fano foliations abstract in this paper we address fano foliations on complex projective varieties these are foliations whose anti canonical class k is ample we focus our attention on a special class of fano foliations namely del pezzo foliations on complex projective manifolds we show that these foliations are algebraically integrable with one exceptional case when the ambient space is p n we also provide a classification of del pezzo foliations with mild singularities

beauty and the beast superconformal symmetry in a monster module frenkel lepowsky and meurman have constructed a representation of the largest sporadic simple finite group the fischer griess monster as the automorphism group of the operator product algebra of a conformal field theory with central chargec 24 in string terminology their construction corresponds to compactification on az2 asymmetric orbifold constructed from the torusr24 where is the leech lattice in this note we point out that their construction naturally embodies as well a larger algebraic structure namely a super virasoro algebra with central charge 16 with the supersymmetry generator constructed in terms of bosonic twist fields

rational curves dynkin diagrams and fano manifolds with nef tangent bundle a fano manifold x with nef tangent bundle is of flag type if it has the same kind of elementary contractions as a complete flag manifold in this paper we present a method to associate a dynkin diagram mathcal d x with any such x based on the numerical properties of its contractions we then show that mathcal d x is the dynkin diagram of a semisimple lie group as an application we prove that campana peternell conjecture holds when x is a flag type manifold whose dynkin diagram is a n i e we show that x is the variety of complete flags of linear subspaces in mathbb p n

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

on fano manifolds of large pseudoindex abstract we describe fano manifolds of large pseudoindex that are rationally connected with respect to some numerically independent families of rational curves

spinor bundles on quadrics we define some stable vector bundles on the complex quadric hypersurface qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on q4 gr l 3 we call them spinor bundles when n 2fc 1 there is one spinor bundle of rank 2k 1 when n 2k there are two spinor bundles of rank 2k 1 their behavior is slightly different according as n 0 mod 4 or n 2 mod 4 as an application we describe some moduli spaces of rank 3 vector bundles on q5 and qe introduction let qn be the smooth quadric hypersurface of the complex pro jective space pn 1 in this paper we define in a geometrical way some vector bundles on the quadric qn as the natural generalization of the universal bundle and the dual of the quotient bundle on q4 gr l 3 we call them spinor bundles on q4 this definition is equivalent to the usual one spinor bundles are homogeneous and stable according to the definition of mum ford takemoto we study their first properties using the geometrical description given and some standard techniques available in oss we also use a theorem of ramanan see um about the stability of homoge neous bundles induced by irreducible representations when n is odd there is only one spinor bundle while when n is even there are two nonisomorphic spinor bun dles when n is even the behavior of spinor bundles is slightly different according as n 0 mod4 or n 2 mod4 in ot2 we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles qn spin n 2 p cty st is a homogeneous manifold and the semisimple part of the lie algebra of p ai is o n at the level of lie algebras spinor bundles are defined from the spin and half spin representations of o n the paper is divided into three sections in 1 we give some preliminary results and we define the spinor bundles in 2 we study the first properties of spinor bundles in 3 as an application we describe some moduli spaces of rank 3 vector bundles on q5 and qq

p1 bundles admitting another smooth morphism of relative dimension one abstract we give the complete classification of p 1 bundles over projective manifolds of picard number one each of which admit another smooth morphism of relative dimension one

new ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two dimensional systems new ladder operators are constructed for a rational extension of the harmonic oscillator associated with type iii hermite exceptional orthogonal polynomials and characterized by an even integer m the eigenstates of the hamiltonian separate into m 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebra these ladder operators are used to construct a higher order integral of motion for two superintegrable two dimensional systems separable in cartesian coordinates the polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite dimensional unitary irreducible representations

degenerations of rationally connected varieties we prove that a degeneration of rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected

on fano manifolds with nef tangent bundles admitting 1 dimensional varieties of minimal rational tangents let x be a fano manifold of picard number 1 with numerically effective tangent bundle according to the principal case of a conjecture of campana peternell s x should be biholomorphic to a rational homogeneous manifold g p where g is a simple lie group and p g is a maximal parabolic subgroup in our opinion there is no overriding evidence for the campana peternell conjecture for the case of picard number 1 to be valid in its full generality as part of a general programme that the author has undertaken with jun muk hwang to study uniruled projective manifolds via their varieties of minimal rational tangents a new geometric approach is adopted in the current article in a special case consisting of a recovering the generic variety of minimal rational tangents c x and b recovering the structure of a rational homogeneous manifold from c x the author proves that when b 4 x 1 and the generic variety of minimal rational tangents is 1 dimensional x is biholomorphic to the projective plane p 2 the 3 dimensional hyperquadric q 3 or the 5 dimensional fano homogeneous contact manifold of type g 2 to be denoted by k g 2 the principal difficulty is part a of the scheme we prove that c x c pt x x is a rational curve of degrees 3 and show that d 1 resp 2 resp 3 corresponds precisely to the cases of x p 2 resp q 3 resp k g 2 let be the normalization of a choice of a chow component of minimal rational curves on x nefness of the tangent bundle implies that is smooth furthermore it implies that at any point x x the normalization x of the corresponding chow space of minimal rational curves marked at is smooth after proving that x is a rational curve our principal object of study is the universal family u of giving a double fibration p u u x which gives p 1 bundles there is a rank 2 holomorphic vector bundle v on whose projectivization is isomorphic to p u we prove that v is stable and deduce the inequality d 4 from the inequality c 2 1 v 4c 2 v resulting from stability and the existence theorem on hermitian einstein metrics the case of d 4 is ruled out by studying the structure of the curvature tensor of the hermitian einstein metric on v in the special case where c 2 1 v 4c 2 v

fano 5 folds with nef tangent bundles and picard numbers greater than one we prove that smooth fano 5 folds with nef tangent bundles and picard numbers greater than one are rational homogeneous manifolds

lengths of chains of minimal rational curves on fano manifolds abstract in this paper we consider a natural question how many minimal rational curves are needed to join two general points on a fano manifold x of picard number 1 in particular we study the minimal length of such chains in the cases where the dimension of x is at most 5 the coindex of x is at most 3 and x equips with a structure of a double cover as an application we give a better bound on the degree of fano 5 folds of picard number 1

families over a base with a birationally nef tangent bundle mathematicssubjectclassi cation 1991 14j10 14d99itisawell knownconsequenceofthetorellitheoremthatasmoothprojec tivefamilyofcurvesofgenusatleast2overaprojectiverationalorellipticcurveisisotrivial thatis the bresofthefamilyareisomorphic sincetheautomorphismgroupofacurveofgenusatleast2is nite thisalsoimpliesthatthefamilybecomestrivialaftera nitebasechange the above statement was generalized for smooth projective families ofminimalsurfacesofgeneraltypein migliorini95 andforsmoothprojec tivefamiliesofvarieties ofarbitrarydimension withamplecanonicalbundlein kov acs96 botharticlesstudiedfamiliesovercurves theaimofthisarticleistopresentafurthergeneralization namelyletthebaseofthefamilyhavearbitrarydimension 0 1theorem 4 2theorem letf x s beasmoothmorphismofpro jectivealgebraicvarietiessuchthatthecanonicalbundleofevery breoffisample assumethats isbirationaltoans

on the minimal length of extremal rays for fano four folds the minimum of intersection numbers of the anti canonical divisor with rational curves on a fano manifold is called pseudo index it is expected that the intersection number of anti canonical divisor attains to the minimum on an extremal ray i e there exists an extremal rational curve whose intersection number with the anti canonical divisor equals the pseudo index in this note we prove this for smooth fano four folds having birational contractions

projective manifolds whose tangent bundles are numerically effective in 1979 mori mo proved the so called hartshorne frankel conjecture every projective n dimensional manifold with ample tangent bundle is isomorphic to the complex projective space p a differential geometric analogon assuming the existence of a k ihler metric on x with positive holomorphic bisectional curvature is independently due to siu yau sy thus it seems natural to classify projective manifolds x whose tangent bundle tx satisfy a degenerate condition of ampleness numerical effectivity abbreviated by nef this means that the tautological quotient line bundle d 1 on f tx is numerically effective i e c 0

2 cycles on higher fano hypersurfaces let x d be a smooth hypersurface of degree d in mathbb p mathbb c n suppose that the fano variety mathrm f x d of lines of x d is smooth we prove that the griffiths group mathrm griff 1 mathrm f mathrm x d of mathrm f x d is trivial if the hypersurface mathrm x d is of 2 fano type as a result we give a positive answer to a question of professor voisin about the first griffiths groups of fano varieties in some cases base on this result we prove that mathrm ch 2 mathrm x d mathbb z for a smooth 3 fano hypersurface mathrm x d subseteq mathbb p n mathbb c with smooth fano variety of lines

kontsevich spaces of rational curves on fano hypersurfaces we investigate the spaces of rational curves on a general hypersurface in particular we show that for a general degree d hypersurface in pn with n d 2 the space m0 0 x e of degree e kontsevich stable maps from a rational curve to x is an irreducible local complete intersection stack of dimension e n d 1 n 4 this resolves all but one case of a conjecture of coskun harris and starr

g elliptic systems and the genus zero problem for m_ 24 toward the end of the 1970s j h conway and s p norton wrote a quite unique and influential paper 3 in which they produced incontrovertible evidence based partly on ideas of j mckay and j g thompson 13 that the fischer griess monster simple group m was intimately related to the modular function j x familiar from the theory of modular forms thus was so called moonshine brought into the world among other things it was conjectured that there is a natural infinite dimensional complex vector space v which carries a z grading into finite dimensional subspaces f v x e vx 0 v2 0 such that each vi is a cmmodule moreover the graded character chvt q j2n dimf tf of f should be the fourier expansion of j r namely q 196884 21 493 760 if the indeterminate q is interpreted as e for r in the upper half plane h there should in addition be an analogous interpretation for

potentially non klt locus and its applications we introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor the potentially non klt locus is a subset of x which is birationally transformed precisely into the non klt locus on a k x minimal model of x we prove basic properties of potentially non klt locus in comparison with those of classical non klt locus as applications we give a new characterization of varieties of fano type and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor

an extremal mathcal n 2 superconformal field theory here we provide an example of an extremal chiral mathcal n 2 superconformal field theory at c 24 the construction is based on a mathbb z 2 orbifold of the theory associated to the a 1 24 niemeier lattice the statespace is governed by representations of the sporadic group m 23

calculus of twisted vertex operators starting from an arbitrary isometry of an arbitrary even lattice twisted and shifted vertex operators are introduced under commutators these operators provide realizations of twisted affine lie algebras this construction generalizing a number of known ones is based on a self contained calculus

fano n folds with nef tangent bundle and picard number greater than n 5 we prove that fano n folds with nef tangent bundle and picard number greater than n 5 are rational homogeneous manifolds

on manifolds whose tangent bundle is big and 1 ample a line bundle over a complex projective variety is called big and 1 ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1 dimensional fibers a vector bundle is called big and 1 ample if the relative hyperplane line bundle over its projectivisation is big and 1 ample the main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more whose tangent bundle is big and 1 ample is equal either to a projective space or to a smooth quadric since big and 1 ample bundles are almost ample the present result is yet another extension of the celebrated mori paper projective manifolds with ample tangent bundles ann of math 110 1979 593 606 the proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective in the appendix we re prove these results

fricke lie algebras and the genus zero property in moonshine we give a new simpler proof that the canonical actions of finite groups on fricke type monstrous lie algebras yield genus zero functions in generalized monstrous moonshine using a borcherds kac moody lie algebra decomposition due to jurisich we describe a compatibility condition arising from the no ghost theorem in bosonic string theory that yields the genus zero property we give evidence for and against the conjecture that such a compatibility for symmetries of the monster lie algebra gives a characterization of the monster group

meromorphic c 24 conformal field theories modular invariant conformal field theories with just one primary field and central charge c 24 are considered it has been shown previously that if the chiral algebra of such a theory contains spin 1 currents it is either the leech lattice cft or it contains a kac moody sub algebra with total central charge 24 in this paper all meromorphic modular invariant combinations of the allowed kac moody combinations are obtained the result suggests the existence of 71 meromorphic c 24 theories including the 41 that were already known

varieties with small discriminant variety let x be a smooth complex projective variety let l be an ample and spanned line bundle on x v c h x l defining a morphism v x p n and let d x v be its discriminant locus the variety parameterizing the singular elements of v we present two bounds on the dimension of d x v and its main component relying on the geometry of v x p n classification results for triplets x l v reaching the bounds as well as significant examples are provided

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

nonabelian orbifolds and the boson fermion correspondence for a positive integerl divisible by 8 there is a bosonic holomorphic vertex operator algebra voa v gamma l associated to the spin lattice l for a broad class of finite groupsg of automorphisms of v gamma l we prove the existence and uniqueness of irreducibleg twisted v gamma l modules and establish the modular invariance of the partition functionsz g h for commuting elements ing in particular for any finite group there are infinitely many holomorphic voas admittingg for which these properties hold the proof is facilitated by a boson fermion correspondence which gives a voa isomorphism between v gamma l and a certain fermionic construction and which extends work of frenkel and others

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

an orbifold theory of genus zero associated to the sporadic groupm24 let v gamma l be the self dual or holomorphic bosonic conformal field theory associated with the spin lattice l of rankl divisible by 24 in earlier work of the authors we showed how it is possible to establish the existence and uniqueness of irreducibleg twisted sectors for v gamma l for certain automorphismsg of v gamma l and to establish the modular invariance of the space of partition functionsz g h corresponding to commuting pairsg h of elements in certain groupsg of automorphisms of v gamma l in the present work we show that if we takel 24 andg the sporadic simple groupm24 then the corresponding orbifold has thegenus zero property that is eachz g h is either identically zero or ahauptmodul i e it generates the field of functions on the subgroup ofsl2 which fixesz g h which then necessarily has genus zero

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

connection between quantum systems involving the fourth painleve transcendent and k step rational extensions of the harmonic oscillator related to hermite exceptional orthogonal polynomial the purpose of this communication is to point out the connection between a 1d quantum hamiltonian involving the fourth painleve transcendent piv obtained in the context of second order supersymmetric quantum mechanics and third order ladder operators with a hierarchy of families of quantum systems called k step rational extensions of the harmonic oscillator and related with multi indexed xm1 m2 mk hermite exceptional orthogonal polynomials of type iii the connection between these exactly solvable models is established at the level of the equivalence of the hamiltonians using rational solutions of the fourth painleve equation in terms of generalized hermite and okamoto polynomials we also relate the different ladder operators obtained by various combinations of supersymmetric constructions involving darboux crum and krein adler supercharges their zero modes and the corresponding energies these results will demonstrate and clarify the relation observed for a particular case in previous papers

families of superintegrable hamiltonians constructed from exceptional polynomials we introduce a family of exactly solvable two dimensional hamiltonians whose wave functions are given in terms of laguerre and exceptional jacobi polynomials the hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems additional higher order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

congruences for rational points on varieties over finite fields we prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper varieties with trivial chow group of 0 cycles we also obtain congruences for the number of rational points of singular varieties appearing as fibres of a proper family with smooth total and base space and such that the chow group of 0 cycles of the generic fibre is trivial in particular this leads to a vast generalization of the classical chevalley warning theorem the above results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of h esnault on the number of rational points of smooth proper varieties over finite fields with trivial chow group of 0 cycles

on the birational unboundedness of higher dimensional q fano varieties we show that the family of mathbb q factorial and log terminal mathbb q fano n folds with picard number one is birationally unbounded for n 6

mld s vs thresholds and flips minimal log discrepancies mld s are related not only to termination of log flips 22 and thus to the existence of log flips 11 but also to the ascending chain condition acc of some global invariants and invariants of singularities in the log minimal model program lmmp in this paper we draw clear links between several central conjectures in the lmmp more precisely our main result states that the lmmp the acc conjecture for mld s and the boundedness of canonical morifano varieties in dimension d imply the following the acc conjecture for a lc thresholds in particular for canonical and log canonical lc thresholds in dimension d the acc conjecture for lc thresholds in dimension d 1 termination of log flips in dimension d 1 for effective pairs and existence of pl flips in dimension d 2 this also gives new proofs of some well known and new results in the field in low dimensions the acc conjecture holds for a lc thresholds of surfaces the acc conjecture holds for lc thresholds of 3 folds termination of 3 fold log flips holds for effective pairs and the existence of 4 fold pl flips holds

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group

twenty three constructions for the leech lattice in the previous chapter we classified the points at maximal distance from the leech lattice the deep holes in that lattice and showed that there are 23 classes of such holes the classes being in one to one correspondence with the 23 niemeier lattices in 24 dimensions we now present 23 constructions for the leech lattice one for each class of hole or niemeier lattice two of these are the usual constructions of the leech lattice from the golay codes over f 2 and f 3