March 1 - August 31, 2007
Gil Kalai (The Hebrew University of Jerusalem)
Polytopes have intrigued mathematicians since ancient times. The ancient Egyptians knew quite a bit of the geometry of polytopes, and the pyramids are, of course, a special type of polytopes. The ancient Greeks discovered the five platonic solids. The five platonic solids: note that the Icosahedron is dual to the Dodecahedron, the Cube is dual to the Octahedron and the Tetrahdron is self-dual.
Euler, who can be regarded as the father of modern graph theory, proved a remarkable formula that explains the relationship between combinatorics and polytopes. Euler's formula asserts that: for every polytope in space with V vertices, E edges and F faces: V - E + F = 2
For example, for the cube, V = 8, E = 12, and F = 6 and indeed 8 - 12 + 16 = 2.
Euler's formula is one of the most important formulae is mathematics and can be regarded as a starting point for topology.
Polytopes in dimensions higher than three have been studied since the 19th century. The first rigorous proof of an extension of Euler's formula for higher dimension was obtained by Poincaré. Poincaré used tools from algabraic topology, a new subject of study that he himself developed. It turns out that Euler's formula is closely related to topology, an important part of geometry.
The research group will explore the following topics: the important and mysterious notion of "duality" between polytopes; the notion of "valuations" of convex sets; random polytopes and complexes; the relationships between combinatorics and topology; the "rigidity" of graphs; and metric aspects of polytopes.
[RG #107] Charity and Piety in the Middle East in Late Antiquity and the Middle Ages: Continuity and Transformation
September 1, 2006 - February 28, 2007
Miriam Frenkel (Ben-Zvi Institute)
Yaacov Lev (Bar-Ilan University)
Charity practices create and give shape to individual social institutions. They may have a crucial impact upon rulers' policies and public image, and affect patterns of social solidarity, stratification and social control. They are capable of impinging upon the social position of individuals, the place ascribed to family, religious institutions and civil society, as well as influencing economic and daily life and certain aspects of the life cycle.
At the discursive level charity may both reflect and shape worldviews and concepts. It is a field in which social values and norms are competing and being tested. This discourse is conveyed in theological, liturgical, literary and documentary texts which may express the image of the ideal society, the ways in which societies treat the "other", and how they interpret such basic aspects of life as wealth, poverty, work, destiny, individuality etc.
We will ask the basic questions that might assist us in analyzing charity from various perspectives: What were the motivations for giving charity? Who were the recipients of charity? Who were the agents of charity distribution? What was the place of charity in society, its relation to religious institutions, gender, family structures, etc.? These questions have been presented in the past but only sporadically, and they were never applied to a number of interrelated cultures over a vast span of time. In dealing with these questions we will attempt to bridge over eras and cultures that are normally perceived as distinct and separate.