Week 14.01.2024 – 20.01.2024

The convergence of stochastic integrals driven by a sequence of Wiener processes $W_n\to W$ (with convergence in $C_t$) is crucial in the analysis of SPDEs. In this talk I shall focus on convergence of stochastic integrals of the form $\int_0^T V_n\, {\rm d} W_n \to \int_0^T V\,{\rm d} W$. Standard methods do not directly apply when $V_n$ converges to $V$ only weakly in the temporal variable. I shall discuss several convergence results that address the need to take limits of stochastic integrals when strong temporal convergence is absent. The key ingredient is an additional condition in the form of a uniform $L^1$ time translation estimate that is often available in SPDE settings but in itself insufficient to imply strong temporal compactness. This discussion will be in the context of applications to semilinear stochastic transport equations and stochastic conservation laws.

I will study the property of Chaos with Magnetic Field in spin glasses. I will report the results for Spin Glasses on Random Regular Graphs and 4D lattices. I will compare the simulation with analytic predictions obtained generating random trees according to the Replica Symmetry Breaking theory. I will show that using the Overalp Probability function P(q) as input one can quantitatively predict the degree of decorrelation as the field increases. One can also compute the finite volume effects in the magnetization and the susceptibility as a function of the field.

Many important examples of Banach algebras are also dual Banach spaces, and over the last thirty years a theory of so-called dual Banach algebras has emerged. An interesting and important way to study a dual Banach algebra is by studying (or even classifying) its weak*-closed (left/right/two-sided) ideals. It also turns out that weak*-closed ideals have connections to other topics as well, such as asymptotic properties of group representations.

In this talk we shall give an introduction to this topic, with our main examples being the measure algebra M(G) of a locally compact group G, and algebras of convolution operators on l^p spaces. We shall describe classifications of the weak*-closed (left/right/two-sided) ideals in M(G) for certain classes of groups G. Finally, we shall outline some new results about weak*-simplicity of M(G) and algebras of convolution operators.

We propose a new method for changepoint estimation in partially-observed, high-dimensional time series that undergo a simultaneous change in mean in a sparse subset of coordinates. Our first methodological contribution is to introduce a 'MissCUSUM' transformation (a generalisation of the popular Cumulative Sum statistics), that captures the interaction between the signal strength and the level of missingness in each coordinate. In order to borrow strength across the coordinates, we propose to project these MissCUSUM statistics along a direction found as the solution to a penalised optimisation problem tailored to the specific sparsity structure. The changepoint can then be estimated as the location of the peak of the absolute value of the projected univariate series. In a model that allows different missingness probabilities in different component series, we identify that the key interaction between the missingness and the signal is a weighted sum of squares of the signal change in each coordinate, with weights given by the observation probabilities. More specifically, we prove that the angle between the estimated and oracle projection directions, as well as the changepoint location error, are controlled with high probability by the sum of two terms, both involving this weighted sum of squares, and representing the error incurred due to noise and the error due to missingness respectively. A lower bound confirms that our changepoint estimator, which we call 'MissInspect', is optimal up to a logarithmic factor. The striking effectiveness of the MissInspect methodology is further demonstrated both on simulated data, and on an oceanographic data set covering the Neogene period.

After a PhD in cosmology, Vincent co-founded B12 Consulting with two other physicists. This company specializes in developing custom AI solutions, tailored for various business and organizational needs. The team now comprises over 40 individuals, primarily from science backgrounds including mathematics, physics, and engineering, with half holding PhDs. This diverse expertise fuels innovative approaches and solutions in their projects. In this presentation, Vincent will discuss his personal journey to becoming an entrepreneur, focusing on the challenges of bridging academic research with the world of business.

https://www.b12-consulting.com/en/team/vincent-boucher/