Automorphic forms, fake monster algebras and hyperbolic reflection groups

We construct 2 families of automorphic forms related to twisted fake monster algebras and calculate their Fourier expansions. This gives a new proof of their denominator identities and shows that they define automorphic forms of singular weight. We also obtain new infinite product identities which are the denominator identities of generalized Kac-Moody superalgebras. Finally we describe the reflection groups of the root lattices of these algebras.Comment: latex2e, 23 pages, 4 figure

Vertex Algebras, Lie Algebras and Superstrings

Certain vertex algebras and Lie algebras arising in superstring theory are investigated. We show that the Fock space of a compactified Neveu-Schwarz superstring, i.e. a Neveu-Schwarz superstring moving on a torus, carries the structure of a vertex superalgebra with a Neveu-Schwarz element. This implies that the physical states of such a string form a Lie algebra. The same is true for the GSO-projected states. The structure of these Lie algebras is investigated in detail. In particular there is a natural invariant form on them. In case that the torus has Lorentzian signature the quotient of these Lie algebras by the kernel of this form is a generalized Kac-Moody algebra. The roots can be easily described. If the dimension of space-time is smaller than or equal to 10 we can even determine their multiplicities

Harmonic theta series and the kodaira dimension of a6

We construct a basis of the space S14(Sp12(ℤ)) of Siegel cusp forms of degree 6 and weight 14 consisting of harmonic theta series. One of these functions has vanishing order 2 at the boundary which implies that the Kodaira dimension of A6 is nonnegative

The invariants of the Weil representation of SL2(Z)\mathrm{SL}_2(\mathbb{Z})

The transformation behaviour of the vector valued theta function of a positive definite even lattice under the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ is described by the Weil representation. This representation plays an important role in the theory of automorphic forms. We show that its invariants are induced from $5$ fundamental invariants

Endometrial cancer - reduce to the minimum. A new paradigm for adjuvant treatments?

<p>Abstract</p> <p>Background</p> <p>Up to now, the role of adjuvant radiation therapy and the extent of lymph node dissection for early stage endometrial cancer are controversial. In order to clarify the current position of the given adjuvant treatment options, a systematic review was performed.</p> <p>Materials and methods</p> <p>Both, Pubmed and ISI Web of Knowledge database were searched using the following keywords and MESH headings: "Endometrial cancer", "Endometrial Neoplasms", "Endometrial Neoplasms/radiotherapy", "External beam radiation therapy", "Brachytherapy" and adequate combinations.</p> <p>Conclusion</p> <p>Recent data from randomized trials indicate that external beam radiation therapy - particularly in combination with extended lymph node dissection - or radical lymph node dissection increases toxicity without any improvement of overall survival rates. Thus, reduced surgical aggressiveness and limitation of radiotherapy to vaginal-vault-brachytherapy only is sufficient for most cases of early stage endometrial cancer.</p

Reflective modular varieties and their cusps

We classify reflective automorphic products of singular weight under certain regularity assumptions. Using obstruction theory we show that there are exactly 11 such functions. They are naturally related to certain conjugacy classes in Conway's group $\text{Co}_0$. The corresponding modular varieties have a very rich geometry. We establish a bijection between their $1$-dimensional type-$0$ cusps and the root systems in Schellekens' list. We also describe a $1$-dimensional cusp along which the restriction of the automorphic product is given by the eta product of the corresponding class in $\text{Co}_0$. Finally we apply our results to give a complex-geometric proof of Schellekens' list.Comment: 71 page