ar X iv : 1 50 2 . 06 47 2 v 1 [ m at h . R A ] 8 N ov 2 01 4 Gröbner - Shirshov bases and PBW theorems ∗

ar X iv : m at h / 04 05 17 6 v 3 [ m at h . R T ] 1 7 N ov 2 00 4 QUANTIZED SYMPLECTIC OSCILLATOR ALGEBRAS OF RANK ONE

A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with Uq(sl2). We study its representation theory, and in particular, its category O.

ar X iv : h ep - p h / 06 06 24 7 v 1 2 2 Ju n 20 06 S - bases as a tool to solve reduction problems for Feynman integrals

We suggest a mathematical definition of the notion of master integrals and present a brief review of algorithmic methods to solve reduction problems for Feynman integrals based on integration by parts relations. In particular, we discuss a recently suggested reduction algorithm which uses Gröbner bases. New results obtained with its help for a family of three-loop Feynman integrals are outlined.

ar X iv : m at h / 04 11 35 1 v 2 [ m at h . A T ] 1 7 N ov 2 00 4 POINCARÉ SUBMERSIONS

We prove two kinds of fibering theorems for maps X → P , where X and P are Poincaré spaces. The special case of P = S yields a Poincaré duality analogue of the fibering theorem of Browder and Levine.

ar X iv : h ep - e x / 02 11 06 9 v 1 2 8 N ov 2 00 2 Jet Production at CDF Mario Mart́ınez

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...