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The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a whole. The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly. *

In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper, and in particular include the Mathieu moonshine observed by Eguchi-Ooguri-Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.

Numerical Ranges II is a sequel to Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras written by the same authors and published in this series in 1971. The present volume reflects the progress made in the subject, expanding and discussing topics under the general headings of spatial and algebra numerical ranges and further ranges.

Abstract. We study a self-dual N = 1 super vertex operator algebra and prove that the full symmetry group is Conway's largest sporadic simple group. We verify a uniqueness result which is analogous to that conjectured to characterize the Moonshine vertex operator algebra. The action of the automorphism group is sufficiently transparent that one can derive explicit expressions for all the McKay-Thompson series. A corollary of the construction is that the perfect double cover of the Conway group may be characterized as a pointstabilizer in a spin module for the Spin group associated to a 24 dimensional Euclidean space. IntroductionThe preeminent example of the structure of vertex operator algebra (VOA) is the Moonshine VOA denoted V ♮ which was first constructed in [FLM88] and whose full automorphism group is the Monster sporadic group.Following [Höh96] we say that a VOA is nice when it is C 2 -cofinite and satisfies a certain natural grading condition (see §2.1), and we make a similar definition for super vertex operator algebras (SVOAs). We say that a VOA is rational when all of its modules are completely reducible (see §2.2). Then conjecturally V ♮ may be characterized among nice rational VOAs by the following properties:• self-dual • rank 24• no small elements where a self-dual VOA is one that has no non-trivial irreducible modules other than itself, and we write "no small elements" to mean no non-trivial vectors in the degree one subspace, since in a nice VOA this is the L(0)-homogeneous subspace with smallest degree that can be trivial. In this note we study what may be viewed as a super analogue of V ♮ . More specifically, we study an object A f ♮ characterized among nice rational N = 1 SVOAs by the following properties.• self-dual • rank 12• no small elements An N = 1 SVOA is an SVOA which admits a representation of the Neveu-Schwarz superalgebra, and now "no small elements" means no non-trivial vectors with degree 1/2. (We define an SVOA to be rational when its even sub-VOA is rational, and a self-dual SVOA is an SVOA with no irreducible modules other than itself.) The earliest evidence in the mathematical literature that there might be an object such as A f ♮ was given in [FLM85] where it was suggested to study a Z/2-L is a self-dual SVOA, and one finds that the graded character ofand is a Hauptmodul for a certain genus zero subgroup of the Modular groupΓ = PSL(2, Z). One can check that, but for the constant term, these coefficients exhibit moonshine phenomena for Co 1 . For example, we have that 276 is the dimension of an irreducible module for Co 1 , and 2048 = 1+276+1771 is a possible decomposition of the degree 3/2 subspace into irreducibles for Co 1 . The only problem being that there is no irreducible representation of Co 1 of dimension 8, and the space corresponding to the constant term in the character of V f L would have to be a sum of trivial modules were it a Co 1 -module at all. As observed in; that is, a space with the correct character for Co 1 . We note here that the existence of V f ♮ has...

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant, and used this expression -the first Rademacher sum -to verify its modular invariance. By generalizing Rademacher's approach we construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for groups commensurable with the modular group. Our methods provide explicit expressions for the Fourier expansions of the Rademacher sums we construct at arbitrary cusps, and illuminate various aspects of the structure of the spaces of automorphic integrals, including the actions of Hecke operators. We give a moduli interpretation for a class of groups commensurable with the modular group which includes all those that are associated to the Monster via monstrous moonshine.We show that within this class the monstrous groups can be characterized just in terms of the behavior of their Rademacher sums. In particular, the genus zero property of monstrous moonshine is encoded naturally in the properties of Rademacher sums.Just as the elliptic modular invariant gives the graded dimension of the moonshine module, the exponential generating function of the Rademacher sums associated to the modular group furnishes the bi-graded dimension of the Verma module for the Monster Lie algebra. This result generalizes naturally to all the groups of monstrous moonshine, and recovers a certain family of monstrous Lie algebras recently introduced by Carnahan.Our constructions suggest conjectures relating monstrous moonshine to a distinguished family of chiral three-dimensional (3D) quantum gravities, and relating monstrous Lie algebras and their Verma modules to the second quantization of this family of chiral 3D quantum gravities.

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