The classical moment problem for the half-line and the real line were formulated and solved by Stieltjes (1894) and Hamburger (1920) respectively, and the theory has been treated in influential monographs by Shohat-Tamarkin (1943) and Akhiezer (1965). The refined theory of the indeterminate moment problem, i.e., the case where two and hence infinitely many different measures have the same moments, is a delicate blend of complex function theory and spectral theory. It has gained renewed interest with the study of orthogonal polynomials associated with q-basic hypergeometric functions important for the representation theory of quantum groups, because many of these new orthogonal polynomials come from indeterminate moment problems. Also several remarkable formulas of Ramanujan can be understood in the light of indeterminate moment problems. It is only during the last two decades that one has been able to make rather complete calculations of all the relevant ”characters” appearing in a concrete indeterminate moment problem.The log-normal distribution is an example of an indeterminate measure. We will give a review of the theory together with new results about the relation between the growth of P and summability properties of the sequence (Pn(z)). The order of the function P is called the order of the moment problem.The multidimensional moment problem will also be discussed.

To a probability distribution \mu with moments of any order (1) s_n = \int x^n d?(x), n = 0,1,..., we consider the orthonormal polynomials P_n, i.e., \int P_n(x)P_m(x) d\mu(x) = \delta_{nm}. They satisfy P_2(z) := \sum_n |P_n(z)|^2<\infty for all complex z precisely in the indeterminate case, where there are different probability measures with the same moments (1). This leads to a study of entire functions like K(z,w) = \sum_n P_n(z)P_n(w), z,w \in C and L(z) = \sum_n z_n/\sqrt{s_{2n}}. During the last 20 years there has been a general study of these entire function as well as many concrete examples, often related to q-series. We will give a review of some of these results together with new results about the relation between the growth of P and summability properties of P_n. The order of the function P is called the order of the moment problem. It is shown that under suitable conditions on the recurrence coefficients in the three term recurrence relation zPn(z)=bnPn+1(z) +anPn(z)+bn?1Pn?1(z), the order of the moment problem is equal to the exponent of convergence of the sequence (bn). Similar results are obtained for logarithmic order and for more general types of slow growth. The new results are based on joint work with Ryszard Szwarc, Wroclaw.

A. Duran and the speaker have studied some non-linear transformations from Hausdorff moment sequences (an) to Stieltjes moment sequences (sn), namely sn= (a0a1···an)?1, sn= (a0+a1+···+an)?1. Several examples will be given leading to surprising new moment sequences. The “sum” transformation has a fixed point (mn) defined by the recursive equation (m0+m1+···+mn) mn= 1, n \geq 0. The representing measure \omega of (mn) has an increasing and convex density on ]0,1[ and is best characterized via its Bernstein transform f. It turns out that f has a meromorphic extension to the whole complex plane satisfying the functional equation \psi(f(z+ 1)) = f(z), and that f can be completely described in terms of iterations of the rational function \psi(z) = z?1/z. The function f is also characterized by a theorem analogous to the Bohr-Mollerup theorem for the Gamma function.

Let F(X,Y) \in \mathbb{Z}(X,Y) be a form of degree r >=3, irreducible over irrationals and having at most s+1 non-zero coefficients. Let h be a non-zero integer. Siegel proposed that the number N_F(h) of integer solutions to the Thue inequality |F(X,Y)|<=h may be bounded only in terms of s and h. In this talk, we present some contributions in this direction.

We shall indicate a solution to a problem posed by Euclid: Prove or disprove the parallel postulate assuming only the first 4 axioms of Euclidean geometry. The solution leads to a new geometry called hyperbolic geometry. We shall only assume some familiarity with calculus.

Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely gem -complexity and regular genus. In this talk, we prove that, for any closed connected PL 4-manifold M, its gem-complexity k(M) and its regular genus G(M) satisfy:k(M)?3\chi(M) + 10m?6 and G(M)?2\chi(M) + 5m ? 4,wherer k(\pi_1(M)) =m.These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of PL 4-manifolds and a large class of PL4-manifolds satisfy these bounds. We also provide upper bounds for regular genus and gem-complexity of manifold bundles over circle. The bound for regular genus is attained by an orientable (resp., non-orientable) manifold bundle over circle when the manifold is S^n, RP^3, S^2×S^1 or g(S1×S1)×S1 for n,g \geq 1. In particular, we prove that there exists an orientable (S2×S1)-bundle over S^1 with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product RP3×S1 among closed connected prime orientable 4-manifolds.

The order bound is a general method to obtain a lower bound for the minimum distance of an evaluation code. It is a very good bound in case the code is defined using Goppa's construction of codes from curves. In my talk I will outline the main ideas behind the order bound and make them more explicit in the case of one-point algebraic-geometry codes

Among the mathematical development inspired by Arnold's conjecture, about the lower bound of fixed points of a symplectomorphism, Conley-Zehnder's work of 1983 is notable. Their idea gave rise to many further developments like the Floer homology. The proposed commentary will lead to the idea of the proof by Conley-Zehnder.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

Let f be a smooth function on R. The divided difference matrices whose (i,j) entries are [f(\lambda_i)-f(\lambda_j)]/[\lambda_i-\lambda_j], \lambda_1,...,\lambda_n \in R are called Loewner matrices. In a seminal paper published in 1934 Loewner used properties of these matrices to characterise operator monotone functions. In the same paper he established connections between this matrix problem, complex analytic functions, and harmonic analysis. These elegant connections sent Loewner matrices into the background. Some recent work has brought them back into focus. In particular, characterisation of operator convex functions in terms of Loewner matrices has been obtained. In this talk we describe some of this work.