Our project will focus on the study of the patterns of transmission to, and appropriation by, medieval Jewish cultures of Greek-Arabic thought, with special emphasis on a comparison with the parallel processes in the Muslim-Arabic and Christian-Latin cultures. The group will study different aspects of the absorption of originally Greek knowledge (mainly but not only scientific and philosophical ideas) within the different medieval Jewish cultures in the Mediterranean between the 8th and the 15th centuries, and examine the role played by Jews in knowledge transfer from Europe to the Ottoman Empire in the 16th century. These processes are worthy of study, not only in and of themselves, but also as a reexamination, comparatively speaking, of the varying accounts offered for the Muslim-Arabic and Christian-Latin cases, based on the role of institutions of learning. The absence of similar institutions in Jewish cultures affords the possibility of "controlling" the thesis that what allowed Western Europe to lead the way from medieval science to the scientific revolution was the institutionalization of learning within that society.

 

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.