The Ax-Lindemann(-Weierstrass) theorem is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. For algebraic torus over C this is the analogue of the classical Lindemann-Weierstrass theorem about transcendental numbers to the functional case. This theorem is a key step to prove the André-Oort/Manin-Mumford conjecture by the method of Pila-Zannier. In this talk I will briefly introduce the history of the theorem, explain how to view it as a bi-algebraicity statement and (if time permits) discuss its relationship with the André-Oort/Manin-Mumford conjecture (the latter one known as Raynaud’s Theoerem).

We construct affine Kac-Moody symmetric spaces. These spaces are associated to affine Kac-Moody algebras in a similar way as finite dimensional Riemannian symmetric spaces are associated to finite dimensional simple Lie algebras. Affine Kac-Moody symmetric spaces are constructed as tame Fréchet manifolds and equipped with a weak Lorentzian metric. We describe their classification and explain their geometry.

(joint work with M. Christandl and B. Sahinoglu, http://arxiv.org/abs/1210.0463 [5])

The von Neumann entropy is an extension of the classical Shannon entropy to quantum theory. It plays a fundamental role in quantum statistical mechanics and quantum information theory. Mathematically, given a quantum state described by a positive-semidefinite « density matrix », the Neumann entropy equals the Shannon entropy of the eigenvalues.

In this talk I will describe an approach to studying eigenvalues and entropies of quantum states that is based on the representation theory of the symmetric group. Its irreducible representations are labeled by Young diagrams, which can be understood as discretizations, or « quantizations », of the spectra of quantum states. In this spirit, I will show that the existence of a quantum state of three particles with given eigenvalues for its reduced density matrices is determined by the asymptotic behavior of a representation-theoretic quantity: the recoupling coefficient, which measures the overlap between two « incompatible » decompositions of a triple tensor product of irreducible representations. As an application, the strong subadditivity of the von Neumann entropy can be deduced solely from symmetry properties of this coefficient. If time permits, I will also discuss the connection of our work to Wigner’s observation that the semiclassical behavior of the 6j-symbols for SU(2) — basic to the quantum theory of angular momentum — is governed by the existence of Euclidean tetrahedra, and more generally to Horn’s problem.

I will survey the theory of free entropy introduced by Voiculescu, with applications to von Neumann algebra theory, and I will outline a new approach which solves some open questions pertaining to this quantity.