Mathematics Subject Classification

Abstract: Let A be the generator of an analytic semigroup Ton L2(Ω), where Ω is a homogeneous space with doubling property. We prove maximal Lp-Lp a—priori estimates for the solution of the parabolic evolution equation u'(t)=Au(t)+f(t), u(0)=0 provided Tmay be represented by a heat—kernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification:35K22, 58D25, 47D06

Abstract: Denote S to be the class of functions which are analytic, normalised and univalent in the open unit disc D = {z : |z| < 1}. The important subclasses of S are the class of starlike and convex functions, which we denote by S and C. This paper focuses on attaining sharp upper bounds for the functional |a2a4−a3| for functions f(z) = z + ∑∞ n=2anz n belonging to S and C. Mathematics Subject Classification: Primary 30C45

Abstract: We give an unconditional proof of the Andre-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang’s conjecture for torsion points in powers of the multiplicative group. The second includes the Manin-Mumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink’s relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of Andre-Oort–Manin-Mumford type by Pila. 2010 Mathematics Subject Classification: 11G18, 03C64

Abstract: Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories. 1991 Mathematics Subject Classification: 32G34, 35Q15, 35Q53, 20F55, 53B50 WDVV equations of associativity is the problem of finding of a quasihomogeneous, up to at most quadratic polynomial, function F (t) of the variables t = (t, . . . , t) and of a constant nondegenerate symmetric matrix (