ar X iv : m at h / 03 07 25 1 v 1 [ m at h . D G ] 1 7 Ju l 2 00 3 PERTURBATIONS OF DIRAC OPERATORS

ar X iv : m at h / 03 09 40 8 v 2 [ m at h . D G ] 1 2 Ju l 2 00 4 EINSTEIN METRICS ON SPHERES

ar X iv : 0 70 7 . 33 03 v 1 [ m at h . O A ] 2 3 Ju l 2 00 7 OPERATOR - VALUED FRAMES

ar X iv : m at h / 02 07 25 7 v 1 [ m at h . A G ] 2 7 Ju l 2 00 2 RATIONAL CURVES ON HYPERSURFACES OF LOW DEGREE , II

This is a continuation of [7] in which we proved irreducibility of spaces of rational curves on a general hypersurface X d ⊂ P n of degree d < n+1 2. In this paper, we prove that if d 2 + d + 2 ≤ n and if d ≥ 3, then the spaces of rational curves are themselves rationally connected.

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)...

ar X iv : h ep - l at / 0 00 30 13 v 1 1 7 M ar 2 00 0 BUTP / 2000 - 6 The construction of generalized Dirac operators on the lattice

We discuss the steps to construct Dirac operators which have arbitrary fermion offsets, gauge paths, a general structure in Dirac space and satisfy the basic symmetries (gauge symmetry, hermiticity condition , charge conjugation, hypercubic rotations and reflections) on the lattice. We give an extensive set of examples and offer help to add further structures.