In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « - Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. We will discuss localization of its eigenfunctions as well as the behavior of the local statistics of its eigenvalues. Around large energies, we will see that the eigenfunctions are localized and follow a universal shape given by the exponential of a Brownian motion plus a drift, a behavior already observed by Rifkind and Virag in tridiagonal matrix models. Based on joint works with Cyril Labbé.

We forecast the realized covariance matrix of asset returns in the U.S. equity market by exploiting the predictive information of graphs in volatility and correlation. Specifically, we augment the Heterogeneous Autoregressive (HAR) model via neighborhood aggregation on these graphs. Our proposed method allows for the modeling of interdependence in volatility (also known as spillover effect) and correlation, while maintaining parsimony and interpretability. We explore various graph construction methods, including sector membership and graphical LASSO (for modeling volatility), and line graph (for modeling correlation). The results generally suggest that the augmented model incorporating graph information yields both statistically and economically significant improvements for out-of-sample performance over the traditional models. Such improvements remain significant over horizons up to one month ahead, but decay in time. The robustness tests demonstrate that the forecast improvements are obtained consistently over the different out-of-sample sub-periods, and are insensitive to measurement errors of volatilities.

In this talk, we consider differential equations perturbed by multiplicative fractional Brownian noise. Depending on the value of the Hurst parameter $H$, the resulting equation is pathwise viewed as an ordinary ($H>1$), Young  ($H \in (1/2, 1)$) or rough  ($H \in (1/3, 1/2)$) differential equation. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness  for equations with irregular drifts: strong existence and path-by-path uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case.

In the rough regime $H\in(1/3,1/2)$ we assume positive but arbitrarily small drift regularity for strong 
well-posedness, while for distributional drift we obtain weak existence. 

This is a joint work with Máté Gerencsér.

A finite W-algebra is a gadget associated to each nilpotent orbit in a complex semisimple Lie algebra g. There is a functor from W-modules to a full subcategory of g-modules, known as Skryabin’s equivalence, and every primitive ideals of the enveloping algebra U(g) as the annihilator of a module obtained in this way. This gives a convenient way of organising together primitive ideals in terms of nilpotent orbits, and this approach has led to a resurgence of interest in some hard open problems which lay dormant for some 20 years. The primitive ideals of U(g) which come from one-dimensional representations of W-algebras are especially nice, and we shall call them Losev—Premet ideals. The goal of this talk is to explain my recent work which seeks to: (1) describe the structure of the space of the dimensional representations of a finite W-algebra and (2) classify the Losev—Premet ideals.