L6

RFRS groups were introduced by Ian Agol in connection with virtual fibering of 3-manifolds. Notably, the class of RFRS groups contains all compact special groups, which are groups with particularly nice cocompact actions on cube complexes. In this talk, I will give an introduction to ℓ²-Betti numbers from an algebraic perspective and discuss what group theoretic properties we can conclude from the (non)vanishing of the ℓ²-Betti numbers of a RFRS group.

We will try to solve the isomorphism problem amongst Coxeter groups by looking at finite quotients.  Some success is found in the classes of affine and right-angled Coxeter groups.  Based on joint work with Samuel Corson, Philip Moeller, and Olga Varghese.

This talk will introduce Khovanov and Knot Floer Homology as tools for studying knots. I will then cover some applications to problems in knot theory including distinguishing embedded surfaces and how they can be used in the context of ribbon concordances. No prior knowledge of either will be necessary and lots of pictures are included.

In 2006, Jan Dymara introduced the concept of weighted \(\ell^2\) Betti numbers as a method of computing regular \(\ell^2\) Betti numbers of buildings. This notion of dimension is measured by using Hecke algebras associated to the relevant Coxeter groups. I will briefly introduce buildings and then give a comparison between the regular \(\ell^2\) Betti numbers and the weighted ones.

It is a standard result that mapping class groups of high genus do not surject the integers. This is easily shown by computing the abelianization of the mapping class group using a presentation. Once we pass to finite index subgroups, this becomes a conjecture of Ivanov. More generally, we can ask which groups admit epimorphisms from finite index subgroups of the mapping class group. In this talk, I will present a geometric approach to this question, using harmonic maps, and explain some recent results.

We study networks of firms in which inputs for production are not easily substitutable, as in several real-world supply chains. Building on Robert May's original argument for large ecosystems, we argue that such networks generically become dysfunctional when their size increases, when the heterogeneity between firms becomes too strong, or when substitutability of their production inputs is reduced. At marginal stability and for large heterogeneities, crises can be triggered by small idiosyncratic shocks, which lead to “avalanches” of defaults. This scenario would naturally explain the well-known “small shocks, large business cycles” puzzle, as anticipated long ago by Bak, Chen, Scheinkman, and Woodford. However, an out-of-equilibrium version of the model suggests that other scenarios are possible, in particular that of `turbulent economies’.

The ground state of N noninteracting Fermions in a rotating harmonic trap enjoys a one-to-one mapping to the complex Ginibre ensemble. This setup is equivalent to electrons in a magnetic field described by Landau levels. The mean, variance and higher order cumulants of the number of particles in a circular domain can be computed exactly for finite N and in three different large-N limits. In the bulk and at the edge of the spectrum the result is universal for a large class of rotationally invariant potentials. In the bulk the variance and entanglement entropy are proportional and satisfy an area law. The same universality can be proven for the quaternionic Ginibre ensemble and its corresponding generalisation. For the real Ginibre ensemble we determine the large-N limit at the origin and conjecture its universality in the bulk and at the edge.

The scattering matrix in quantum mechanics must be unitary to ensure the conservation of the number of particles, hence their 
eigenvalues are unimodular.  In systems with fully developed Quantum Chaos  the statistics of those unimodular 
eigenvalues  is well described by  the Poisson kernel.
However, in real experiments  the associated scattering matrix is sub-unitary due to intrinsic losses,  and
 the moduli of S-matrix eigenvalues become non-trivial,  yet the corresponding theory is not well-developed in general.  
 I will present some results for the mean density of those moduli in the framework of random matrix models for the case of broken time-reversal invariance,
and discuss a way to get a generalization of the Poisson kernel to systems with uniform losses.

Let X and H be large, and consider n ranging from 1 to X. For an arithmetic function f(n), what is the best mean square approximation of f(n) by a restricted divisor sum (a function of the sort sum_{d|n, d < H} a_d)? I hope to explain how for a wide variety of arithmetic functions, when X grows and H grows like a power of X, a solution of this problem is connected to the evaluation of random matrix integrals. The problem is connected to some combinatorial formula for computing high moments of traces of random unitary matrices and I hope to discuss this also.