ar X iv : 0 80 7 . 35 76 v 1 [ m at h . A G ] 2 3 Ju l 2 00 8 LINEAR RELATIONS BETWEEN POLYNOMIAL ORBITS

ar X iv : 0 80 7 . 00 58 v 2 [ m at h . D G ] 1 6 Ju l 2 00 8 EQUIVARIANT DIFFERENTIAL CHARACTERS AND SYMPLECTIC REDUCTION

We describe equivariant differential characters (classifying equi-variant circle bundles with connections), their prequantization, and reduction.

ar X iv : m at h - ph / 0 60 80 01 v 1 3 1 Ju l 2 00 6 1 Modular Invariants and Fischer - Griess Monster 1

We show interesting relations between extremal partition functions of a family of conformal field theories and dimensions of the irreducible representations of the Fischer-Griess Monster sporadic group. We argue that these relations can be interpreted as an extension of Monster moonshine.

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 : 0 80 7 . 23 71 v 1 [ m at h . A C ] 1 5 Ju l 2 00 8 The type of the base ring associated to a transversal

In this paper we determine the type, a − invariant, Hilbert f unction and Hilbert series of the base ring of some transversal polymatroid such that its associated cone has dimension n and n + 1 facets.

ar X iv : m at h / 06 02 50 8 v 2 [ m at h . G R ] 1 9 N ov 2 00 7 PRESENTATIONS OF FINITE SIMPLE GROUPS : A QUANTITATIVE APPROACH

There is a constant C 0 such that all nonabelian finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2 G 2 (3 2e+1), have presentations with at most C 0 generators and relations and total length at most C 0 (log n + log q). As a corollary, we deduce a conjecture of Holt: there is a constant C such that dim H 2 (G, M) ≤ C dim M for every finite simple group G, e...