Given a random elliptic or hyperelliptic curve of genus g over Q, how many rational points do we expect the curve to have? Equivalently, how often do we expect a random polynomial of degree n to take a square value over the rational numbers?  In this talk, we give an overview of recent conjectures and theorems giving some answers and partial answers to this question.

The talk will consist of two loosely connected parts: set-theoretic and computer science. We give an overview (no technical details) of our results on Galois-Tukey connections as a general framework for problem reduction. Boolean structure of absolutely divergent series gives rise to several Boolean-like asymptotic structures. Second part deals with applications of many valued logic to preference modeling, querying top-k answers and learning each individual user preferences from behaviour data (especially we mention lack of real world benchmarks).
 

In the first half of this talk we will review several notions of coarse or weak
Ricci Curvature on metric measure spaces which include the work of Yann
Ollivier. The discussion of the notion of coarse Ricci curvature will serve as
motivation for developing a method to estimate the Ricci curvature of a an
embedded submaifold of Euclidean space from a point cloud which has applications
to the Manifold Learning Problem. Our method is based on combining the notion of
``Carre du Champ" introduced by Bakry-Emery with a result of Belkin and Niyogi
which shows that it is possible to recover the rough laplacian of embedded
submanifolds of the Euclidean space from point clouds. This is joint work with
Micah Warren.

Over the past two decades, Algebraic Multigrid (AMG) has become a prominent tool for the rapid solution of unstructured-grid problems. For nearly 20 years after its inception, AMG was largely ignored, due to the overhead and setup costs necessary to create the components of the algorithm. So long as geometrically structured grids were the norm, this condition persisted. In recent years, however, as unstructured grids became more common and as the problem sizes grew to demand large-scale massively parallel computers, AMG has emerged as a widely used methodology. In this talk we describe the basic components and theory of AMG. Here, the focus is on the underlying philosophy of the method. We then concentrate on the more recent advances in AMG, and in particular on the development of the Bootstrap AMG framework.  

In 1956, the Dutch graphic artist M.C. Escher made an unusual
lithograph with the title `Print Gallery'. It shows a young man
viewing a print in an exhibition gallery. Amongst the buildings
depicted on the print, he sees paradoxically the very same gallery
that he is standing in. A lot is known about the way in which
Escher made his lithograph. It is not nearly as well known that it
contains a hidden `Droste effect', or infinite repetition; but
this is brought to light by a mathematical analysis of the studies
used by Escher. On the basis of this discovery, a team of
mathematicians at Leiden produced a series of hallucinating
computer animations. These show, among others, what happens
inside the mysterious spot in the middle of the lithograph that
Escher left blank.