Remarks on Schur's Conjecture

Remarks on a conjecture of Fontaine and Mazur

Remarks on Frankl's conjecture

First a few reformulations of Frankl’s conjecture are given, in terms of reduced families or matrices, or analogously in terms of lattices. These lead naturally to a stronger conjecture with a neat formulation which might be easier to attack than Frankl’s. To this end we prove an inequality which might help in proving the stronger conjecture.

Further Remarks on a Problem of Moser and a Conjecture of Mawhin

Multiplicity results for both periodic and anti-periodic forced oscillations for the pendulum equation are given on the background of a detailed exposition of the results related to the so-called problem of Moser and to the Mawhin conjecture.

Alperin’s Conjecture for Algebraic Groups

We prove analogues for reductive algebraic groups of some results for finite groups due to Knörr and Robinson from ‘Some remarks on a conjecture of Alperin’, J. London Math. Soc (2) 39 (1989), 48–60, which play a central rôle in their reformulation of Alperin’s conjecture for finite groups.

Proof of the Kurlberg-rudnick Rate Conjecture 0.4 Geometric Approach 0.5 Side Remarks

In this paper we give a proof of the Hecke quantum unique ergodicity rate conjecture for the Berry-Hannay model. A model of quantum mechanics on the two dimensional torus. This conjecture was stated in Z. Rudnick’s lectures at MSRI, Berkeley 1999 and ECM, Barcelona 2000.