Multivariate point processes are used to model event-type data in a wide range of domains. One interesting application is to model the emission of electric impulses of biological neurons. In this context, the point process model needs to capture the time-dependencies and interactions between neurons, which can be of two kinds: exciting or inhibiting. Estimating these interactions, and in particular the functional connectivity of the neurons are problems that have gained a lot of attention recently. The general nonlinear Hawkes process is a powerful model for events occurring at multiple locations in interaction. Although there is an extensive literature on the analysis of the linear model, the probabilistic and statistical properties of the nonlinear model are still mainly unknown. In this paper, we consider nonlinear Hawkes models and, in a Bayesian nonparametric inference framework, derive concentration rates for the posterior distribution.  We also infer the graph of interactions between the dimensions of the process and prove that the posterior distribution is consistent on the graph adjacency matrix.

In this talk I will present an efficient algorithm to produce a provably dense sample of a smooth compact algebraic variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample.

An algebra $R$ is said to satisfy the Dixmier-Moeglin equivalence if a prime ideal $P$ of $R$ is primitive if and only if it is rational, if and only if it is locally closed, and a commonly studied problem in non-commutative algebra is to classify rings satisfying this equivalence, e.g. $U(\mathfrak g)$ for a finite dimensional Lie algebra $\mathfrak g$. We explore methods of generalising this to a $p$-adic setting, where we need to weaken the statement. Specifically, if $\hat R$ is the $p$-adic completion of a $\mathbb Q_p$-algebra $R$, rather than approaching the Dixmier-Moeglin equivalence for $\hat R$ directly, we instead compare the classes of primitive, rational and locally closed prime ideals of $\hat R$ within suitable "deformations". The case we focus on is where $R=U(L)$ for a $\mathbb Z_p$-Lie algebra $L$, and the deformations have the form $\hat U(p^n L)$, and we aim to prove a version of the equivalence in the instance where $L$ is nilpotent.

The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting.

We introduce the notion of δ-excursion, defined as a path which deviates by δ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into δ-excursions, which is useful for scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss and drawdown. As δ is decreased to zero, properties of this decomposition relate to the local time of the path. When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent δ-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursion properties match those observed in empirical data.

Joint work with Anna Ananova and Rama Cont: https://ssrn.com/abstract=3723980

Stochastic quantisation is, broadly speaking, the use of a stochastic differential equation to construct a given probability distribution. Usually this refers to Markovian Langevin evolution with given invariant measure. However we will show that it is possible to construct other kind of equations (elliptic stochastic partial differential equations) whose solutions have prescribed marginals. This connection was discovered in the '80 by Parisi and Sourlas in the context of dimensional reduction of statistical field theories in random external fields. This purely probabilistic results has a proof which depends on a supersymmetric formulation of the problem, i.e. a formulation involving a non-commutative random field defined on a non-commutative space. This talk is based on joint work with S. Albeverio and F. C. de Vecchi.

We analyze the regularity of solutions and discrete-time approximations of extended mean-field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics with coefficients involving mean-field interactions both on the  state and actions, and where objectives are optimized over
open-loop strategies.

We show for a large class of extended MFC problems that the unique optimal open-loop control is 1/2-Hölder continuous in time. Based on the regularity of the solution, we prove that the value functions of such extended MFC problems can be approximated by those with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping up to an order 1/2 error, which is optimal in our setting. Further, we show that any epsilon-optimal control of these discrete-time problems
converge to the optimal control of the original problems.

To establish the time regularity of optimal controls and the convergence of time discretizations, we extend the canonical path regularity results to general coupled 
McKean-Vlasov forward-backward stochastic differential equations, which are of independent interest.

This is based on join work joint work with C. Reisinger and Y. Zhang.

Martin Gallauer (North): "Algebraic algebraic geometry"
If a space is described by algebraic equations, its algebraic invariants are endowed with additional structure. I will illustrate this with some simple examples, and speculate on the meaning of the title of my talk.

Zhaohe Dai (South): "Two-dimensional material bubbles"
Two-dimensional (2D) materials are a relatively new class of thin sheets consisting of a single layer of covalently bonded atoms and have shown a host of unique electronic properties. In 2D material electronic devices, however, bubbles often form spontaneously due to the trapping of air or ambient contaminants (such as water molecules and hydrocarbons) at sheet-substrate interfaces. Though they have been considered to be a nuisance, I will discuss that bubbles can be used to characterize 2D materials' bending rigidity after the pressure inside being well controlled. I will then focus on bubbles of relatively large deformations so that the elastic tension could drive the radial slippage of the sheet on its substrate. Finally, I will discuss that the consideration of such slippage is vital to characterize the sheet's stretching stiffness and gives new opportunities to understand the adhesive and frictional interactions between the sheet and various substrates that it contacts.