L4

1pm Jonathan Tam: Markov decision processes with observation costs

We present a framework for a controlled Markov chain where the state of the chain is only given at chosen observation times and of a cost. Optimal strategies therefore involve the choice of observation times as well as the subsequent control values. We show that the corresponding value function satisfies a dynamic programming principle, which leads to a system of quasi-variational inequalities (QVIs). Next, we give an extension where the model parameters are not known a priori but are inferred from the costly observations by Bayesian updates. We then prove a comparison principle for a larger class of QVIs, which implies uniqueness of solutions to our proposed problem. We utilise penalty methods to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate our framework.

Preprint at https://arxiv.org/abs/2201.07908

1.45pm Remy Messadene: signature asymptotics, empirical processes, and optimal transport

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning, deep learning and more recently in kernel methods. In this work, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path.

2.30-3.00 Tea & coffee in the mezzananie

3-4pm Julien Berestycki: Extremal point process of the branching Brownian motion

Let X be a smooth projective curve over a finite field equipped with an action of a finite group G. I’ll first briefly introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result will be a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic if the projection X → X/G is at most weakly ramified. This generalises a theorem of Chinburg for the tamely ramified case. I’ll end with some speculations in the arbitrarily wildly ramified case. This is joint work with Helena Fischbacher-Weitz and with Adriano Marmora.

Lahars and huaicos are potent natural hazards that threaten lives and livelihoods. They comprise debris-laden fluid that flows rapidly down slopes, bulking up considerably as they progress. Owing to their rapid onset and the significant threat that they pose to communities and infrastructures, it is important to be able to predict their motion in order to assess quantitatively some of the impacts that they may cause. In this seminar I will present mathematical models of these flows and apply them to various natural settings, drawing on examples from Peru and the Philippines.  Along the way I will show some informative, idealised solutions, the susceptibility of these flows to roll wave instabilities, ways to prevent ill-posedness and how to include measured topography in the computation.

MHD equilibrium is an important topic for fusion (and other MHD applications). A tokamak, in principle, is a toroidally symmetric fusion device and so MHD equilibrium can be reduced to solving the time independent MHD equations in axisymmetry. This produces the Grad-Shafranov equation (a two dimensional, nonlinear PDE) which has been solved using various techniques in the fusion community including finite difference, finite elements and spectral methods. A similar PDE exists if there is a plasma column with helical symmetry. Non-axisymmetric plasmas do occur in tokamaks as a result of instabilities and applied fields. However, if there is no symmetry angle there is no PDE to be solved. The current workhorse for finding non-axisymmetric equilibria uses energy minimization to find the equilibrium. New approaches to this problem that can use state of the art techniques are desirable. The speaker has formulated a coupled set of PDEs for the non-axisymmetric MHD equilibrium problem assuming that flux surfaces are nested (i.e. there are no magnetic islands) and has written this in weak form to use finite element method to solve the equations. The questions are around whether there is an optimal way to try to formulate the problem for FEM and to couple the equations, what sort of elements to use, if other solution techniques would be better suited and so on.

Moving from persistent homology in one parameter to multiparameter persistence comes at a significant increase in complexity. In particular, the notion of a barcode does not generalize straightforwardly. However, in this talk, I will show how it is possible to assign a unique barcode to a multiparameter persistence module if one is willing to take Z-linear combinations of intervals. The theoretical discussion will be complemented by numerical experiments. This is joint work with Steffen Oppermann and Steve Oudot.

The AdS/CFT correspondence provides a rich setup to study the properties of gauge theories and the dual theories of gravity, in particular their thermodynamic properties. On RxS3, the maximally supersymmetric Yang-Mills theory with gauge group U(N) exhibits a phase transition that resembles the confinement-deconfinement transition of QCD. For infinite N, this transition is characterized by Hagedorn behavior. We show how the corresponding Hagedorn temperature can be calculated at any value of the ’t Hooft coupling via integrability. For large but finite N, we show how the Hagedorn behavior is replaced by Lee-Yang behavior.

This will be a zoom seminar with communal viewing in L4

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. If time allows I will also discuss some applications of this result. (Joint with Chieu-Minh Tran and Ruixiang Zhang)