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.

Dynamical interface motions are important flow patterns and fundamental free boundary problems in fluid mechanics, and have attracted huge attention in the mathematical community. Such waves for purely inviscid fluids are subject to various instabilities such as Kelvin-Helmholtz and Rayleigh-Taylor instabilities unless other stabilizing effects such as surface tension, Taylor-sign conditions or dissipations are imposed. However, in the presence of magnetic fields, it has been known that tangential magnetic fields may have stabilizing effects for free surface waves such as plasma-vacuum or plasma-plasma interfaces (at least locally in time), yet whether transversal magnetic fields (which occurs often for interfacial waves for astrophysical plasmas) can stabilize typical free interfacial waves remain to be some open problems. In this talk, I will show the stabilizing effects of the transversal magnetic fields for some interfacial waves for both compressible and incompressible multi-dimensional magnetohydrodynamics (MHD).

First, I will present the local (in time) well-posedness in Sobolev space of multi- dimensional compressible MHD contact discontinuities, which are the most typical interfacial waves for astrophysical plasma and prototypical fundamental waves for systems of hyperbolic conservations. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which leads to a two-phase free boundary problem that may have nonlinear Rayleigh- Taylor instability and whose front symbols have no ellipticity. We overcome such difficulties by exploiting full the transversality of the magnetic fields and designing a nonlinear approximate problem, which yields the local well-posed without loss of derivatives and without any other conditions such as Rayleigh-Taylor sign conditions or surface tension. Second, I will discuss some results on the global well-posedness of free interface problems for the incompressible inviscid resistive MHD with transversal magnetic fields. Both plasma-vacuum and plasma-plasma interfaces are studied. The global in time well-posedness of both interface problems in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field near an equilibrium are established, which reveals the strong stabilizing effect of the transversal field as the global well- posedness of the free boundary incompressible Euler equations (without the irrotational assumptions) around an equilibrium is unknown. This talk is based on joint work with Professor Yanjin Wang. 

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.

 In this short course, we will discuss the Euler equations and applications in gas dynamics and geometry. First, the basic theory of Euler equations and mixed-type problems will be reviewed. Then we will present the results on the transonic flows past obstacles, transonic flows in the fluid dynamic formulation of isometric embeddings, and the transonic flows in nozzles. We will discuss global solutions and stability obtained through various techniques and approaches. The short course consists of three parts and is accessible to PhD students and young researchers.

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.

What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

A graph $X$ is defined inductively to be $(a_0, . . . , a_{n−1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius 1 around $v$ is an $(a_1, . . . , a_{n−1})$-regular graph. A family $F$ of graphs is said to be an expander family if there is a uniform lower bound on the Cheeger constant of all the graphs in $F$. 

After briefly (re)introducing Coxeter groups and their geometries, we will describe how they can be used to construct very regular polytopes, which in turn can yield highly regular graphs. We will then use the super-approximation machinery, whenever the Coxeter group is hyperbolic, to obtain the expansion of these families of graphs. As a result, we obtain interesting infinite families of highly regular expander graphs, some of which are related to the exceptional groups. 

The talk is based on work joint with Conder, Lubotzky, and Schillewaert.