Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.

Abstract. In the 80’s D. Eisenbud and J. Harris considered the following problem: “What are the limits of Weierstrass points in families of curves degenerating to stable curves? ” But for the case of stable curves of compact type, treated by them, this problem remained wide open since then. In the present article, we propose a concrete approach to this problem, and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position. Contents

Abstract. Each finite dimensional irreducible rational representation V of the symplectic group Sp 2g(Q) determines a generically defined local system V over the moduli space Mg of genus g smooth projective curves. We study H 2 (Mg; V) and the mixed Hodge structure on it. Specifically, we prove that if g ≥ 6, then the natural map IH 2 ( ˜ Mg; V) → H 2 (Mg; V) is an isomorphism where ˜ Mg is the Satake compactification of Mg. Using the work of Saito we conclude that the mixed Hodge structure on H 2 (Mg; V) is pure of weight 2 + r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H 2 (Mg; V) for 3 ≤ g < 6. Results of this article can be applied in the study of relations in the Torelli group Tg. The moduli space Mg of smooth projective curves of genus g is a quasiprojective variety over C. Its points correspond to isomorphism classes of smooth projective complex curves of genus g. It has only finite quotient

noncommutative geometry I: Riemann surfaces

Moduli of complex curves and noncommutative geometry: Riemann surfaces and dimension groups

noncommutative geometry I: classification of

algebras and torsion points of elliptic curves with complex multiplication

algebras and torsion points of elliptic curves with complex multiplication

conjecture on the torsion points of elliptic curves with the complex multiplication