Conference Service Mandl

abstract: Analytic number theory and Diophantine approximation are important and closely connected areas of number theory. The two topics have roots that are as old as mathematics, the first one being motivated initially by the study of prime numbers, while the second concerns the nature of numbers. In the last ten years there have been several striking developments in both fields (for example, Green and Tao's proof of arbitrarily long arithmetic progressions of primes and the work of Rivoal and others on zeta values at odd integers). This summer school will be an occasion for graduate students and young researchers to learn some of the latest developments in these rapidly developing fields and eventually contribute to them.

abstract: Lie algebras and their representations form an extremely rich and important field of mathematics. Geometric representation theory is a relatively new field which has attracted much attention. The general idea is to use geometric methods to construct classically algebraic objects, such as representations of Lie groups and Lie algebras. Geometric techniques have proven to be particularly well suited to establishing positivity and integrality results, as these are often easy consequences of the geometric nature of the objects involved. One is also often able to use the representation theory of objects such as (affine) Kac-Moody algebras to organize and better understand the homology of various interesting spaces appearing in the constructions, like Hilbert schemes, flag varieties, Steinberg varieties, affine Grassmannians, and quiver varieties. In addition to its obvious connections to representation theory, geometric representation theory has been found to be intimately related to combinatorics (crystals, quivers), cluster algebras, mathematical physics, and many other subjects.