The « very good » property for algebraic stacks was introduced by Beilinson and Drinfeld in their paper « The Quantization of Hitchin’s Integrable System and Hecke Eigensheaves ». They proved that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is « very good » as long as X has genus g > 1. We will introduce the « very good » property in the context of a group action on an algebraic variety, and prove it for a moduli space of parabolic bundles on P1 arising from quiver representations. As a special case, we will consider the « very good » property for the diagonal action of the group PGL(n) on a product of partial flag varieties and its relationship with the space of solutions to the additive Deligne-Simpson problem.

The principal approximation methods used to compute the inspiral of compact binary systems are the post-Newtonian (pN) expansion, in which an orbital angular velocity MΩ serves as the expansion parameter; and the self-force or extreme-mass-ratio-inspiral approach, in which the small parameter is the mass ratio m/M of the binary’s two components. We work in an overlapping regime where both approximations are valid and find numerical values of pN coefficients at orders beyond the reach of current analytical work. In this talk we present a novel analytic extraction of high-order pN parameters that govern quasi-circular binary systems using ultra-high accuracy numerical computations.

The problem reads: Is there a general principle(s) that underlies the processes of human learning the following?

1. Native languages.

2. Mathematics.

3. Bipedal locomotion.

We shall bring (speculative) evidence for the existence of such principle(s) and indicate a possible direction of (quasi)mathematical modeling of (conjectural) universal learning.

Starting from the known unfaithful spinorial representations of the compact subalgebra KE(10) of the hyperbolic Kac Moody algebra E(10), new fermionic `higher spin’ representations are constructed in a second quantized framework.

Alternative Splicing is an important source of variety in the process of protein expression. It is deregulated in human diseases including cancer. The precise mechanism through which a splicing variant is selected for expression is not known at present. Here, we demonstrate that alternative splicing involves the RNA interference protein machinery, previously discovered as important for gene regulation at a distinct step, i. e. at the translational level.

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.