In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.

# Past Forthcoming Seminars

Equivariant cohomology is adapted from ordinary cohomology to better capture the action of a group on a topological space. In Floer theory, given an autonomous Hamiltonian, there is a natural action of the circle on 1-periodic flowlines given by time translation. Combining these two ideas leads to the definition of $S^1$-equivariant symplectic cohomology. In this talk, I will introduce these ideas and explain how they are related. I will not assume prior knowledge of Floer theory.

We discuss two Freidlin-Wentzell large deviation principles for McKean-Vlasov equations (MV-SDEs) in certain path space topologies. The equations have a drift of polynomial growth and an existence/uniqueness result is provided. We apply the Monte-Carlo methods for evaluating expectations of functionals of solutions to MV-SDE with drifts of super-linear growth. We assume that the MV-SDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the "complete measure change" approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the "decoupling" approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step.

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.

Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.

In laboratories around the world, scientists use magnetic stirrers to mix solutions and dissolve powders. It is well known that at high drive rates the stir bar jumps around erratically with poor mixing, leading to its nick-name 'flea'. Investigating this behaviour, we discovered a state in which the flea levitates stably above the base of the vessel, supported by magnetic repulsion between flea and drive magnet. The vertical motion is oscillatory and the angular motion a superposition of rotation and oscillation. By solving the coupled vertical and angular equations of motion, we characterised the flea’s behaviour in terms of two dimensionless quantities: (i) the normalized drive speed and (ii) the ratio of magnetic to viscous forces. However, Earnshaw’s theorem states that levitation via any arrangement of static magnets is only possible with additional stabilising forces. In our system, we find that these forces arise from the flea’s oscillations which pump fluid radially outwards, and are only present for a narrow range of Reynold's numbers. At slower, creeping flow speeds, only viscous forces are present, whereas at higher speeds, the flow reverses direction and the flea is no longer stable. We also use both the levitating and non-levitating states to measure rheological properties of the system.

We first consider multilevel Monte Carlo and stochastic collocation methods for determining statistical information about an output of interest that depends on the solution of a PDE with inputs that depend on random parameters. In our context, these methods connect a hierarchy of spatial grids to the amount of sampling done for a given grid, resulting in dramatic acceleration in the convergence of approximations. We then consider multifidelity methods for the same purpose which feature a variety of models that have different fidelities. For example, we could have coarser grid discretizations, reduced-order models, simplified physics, surrogates such as interpolants, and, in principle, even experimental data. No assumptions are made about the fidelity of the models relative to the “truth” model of interest so that unlike multilevel methods, there is no a priori model hierarchy available. However, our approach can still greatly accelerate the convergence of approximations.

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system. The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem. In this talk we present the rigorous justification of the quasineutral limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimensions 1, 2, and 3. Also, we discuss a recent result in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.

One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse PD_n subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This generalises work of Whyte and Mosher--Sageev--Whyte.