ar X iv : m at h / 04 09 06 8 v 3 [ m at h . G N ] 1 7 M ay 2 00 7 THE COMBINATORICS OF τ - COVERS

ar X iv : m at h / 04 09 06 8 v 2 [ m at h . G N ] 5 J ul 2 00 5 THE COMBINATORICS OF τ - COVERS

We solve four out of the six open problems concerning critical cardi-nalities of topological diagonalization properties involving τ-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based o...

ar X iv : m at h / 06 05 35 5 v 1 [ m at h . G R ] 1 3 M ay 2 00 6 Axes in Outer Space

ar X iv : h ep - p h / 06 05 13 6 v 1 1 1 M ay 2 00 6 Lepton Flavor Violating τ and μ decays

ar X iv : h ep - t h / 06 05 17 1 v 1 1 7 M ay 2 00 6 M - Theory of Matrix Models

Small M-theories unify various models of a given family in the same way as the M-theory unifies a variety of superstring models. We consider this idea in application to the family of eigenvalue matrix models: their M-theory unifies various branches of Hermitean matrix model (including Dijkgraaf-Vafa partition functions) with Kontsevich τ-function. Moreover, the corresponding duality relations l...

ar X iv : m at h / 06 07 71 6 v 1 [ m at h . D G ] 2 7 Ju l 2 00 6 THE SPINORIAL τ - INVARIANT AND 0 - DIMENSIONAL SURGERY

Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ + 1 (g) be the first non-negative eigenvalue of the Dirac operator on (M, g, χ). We set τ (M, χ) := sup inf λ + 1 (g) where the infimum runs over all metrics g of volume 1 in a conformal class [g 0 ] on M and where the supremum runs over all conformal classes [g 0 ] on M. Let (M # , χ #) be obtained from (M, χ)...