We extend the popular Gibson and Schwartz (1990) and Schwartz and Smith (2000) two-factor models for the spot price of a commodity to include stochastic volatility and correlation. This generalization is based on the Wishart variance-covariance matrix process. For both of the extended models we present the joint characteristic functions of the two state variables. The original models are known to fit the term-structure of implied volatility in futures and options markets very well. However, the extended models are also able to match volatility smiles observed in these markets. Regarding the analysis of financial time series, the assumption of a constant correlation between the state variables is known to be too restrictive. Introducing time-varying correlation via the Wishart process allows us to study its empirical behaviour in commodity markets through the use of filtering techniques.

We consider a thin layer of an incompressible viscous Newtonian fluid coating the inner wall of a horizontal cylinder rotating with constant speed. Assuming the film height is small compared to the radius of the cylinder, we formally derive a closed equation for the height h(t, θ) > 0 of the liquid film by means of a lubrication approximation: ht + hθ + γ h3(hθθθ + hθ) θ = g h3 cos θ θ in (0, T ) × T This rimming flow equation is of fourth-order, degenerate-parabolic, and quasilinear. Compet- ing effects are observed between viscosity, the surface tension γ, and gravity g. For g = 0 and a fixed mass m, the two-dimensional manifold M(m) := m + a sin θ + b cos θ a2 + b2 < m2 is invariant. If 0 < g ∼ δ ≪ 1 is small, we show that solutions which are bounded away from zero converge exponentially fast to a δ-neighbourhood of M(m). Here, the existence of solutions on a large time scale t ∼ 1/δ2 can be shown. Moreover, such solutions evolve on two distinct time scales: On the fast time scale t they only rotate around the origin with the speed of the cylinder, while on the slow time scale τ = δ2t the dynamics are governed by an ODE in τ on M(m).

We study causality in systems that allow for feedback loops among the variables via models of cross-sectional data from a dynamical system. Specifically, we consider the set of distributions which appears as the steady-state distributions of a stochastic differential equation (SDE) where the drift matrix is parametrized by a directed graph. The nth-order cumulant of the steady state distribution satisfies the corresponding nth-order continuous Lyapunov equation. Under the assumption that the driving Lévy process of the SDE is not a Brownian motion (so the steady state distribution is non-Gaussian) and the coordinates are independent, we are able to prove generic identifiability for any connected graph from the second and third-order Lyapunov equations while allowing the cumulants of the driving process to be unknown diagonal. We propose a minimum distance estimator of the drift matrix, which we are able to prove is consistent and asymptotically normal by utilizing the identifiability result.

State-sum constructions have numerous applications in both mathematics and physics. In mathematics, they yield invariants for knots and manifolds and serve as a powerful organizing principle in representation theory. To illustrate this principle, we discuss equivariant Frobenius-Schur indicators. In the context of physics, we explain how state-sum models offer a conceptual framework for tensor network models, based on collaboration with Fuchs, Haegeman, Lootens, and Verstraete.

Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.

As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.

I will explain why noise-free recovery is dictated by the geometry of natural random sets: for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the random sets are 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 }.

I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).

The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem. _____________________________________________

Invited by Prof. Holger Rauhut.

This talk shall be concerned with the surface quasi-geostrophic equation driven by a generic additive noise process W. By means of convex integration techniques, we establish existence of weak solutions whenever the stochastic convolution z associated with W is well defined and fulfils certain regularity constraints. Quintessentially, we show that the so constructed solutions to the non-linear equation are controlled by z in a linear fashion. This allows us to deduce further properties of the so constructed solutions, without relying on structural probabilistic properties such as Gaussianity, Markovianity or a martingale property of the underlying noise W. This is joint work with Florian Bechtold (University of Bielefeld) and Jörn Wichmann (Monash University) (cf. [1]). This activity receives partial funding from the European Research Council (ERC) under the EU-HORIZON EUROPE ERC-2021-ADG research and innovation programme (project „Noise in Fluids“, grant agreement no. 101053472). [1] F. Bechtold, T. Lange, J. Wichmann, "On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise", Electronic Journal of Probability 29 (2024): 1-38

TBA

Most modern machine learning applications are based on overparameterized neural networks trained by variants of stochastic gradient descent. To explain the performance of these networks from a theoretical perspective (in particular the so-called "implicit bias"), it is necessary to understand the random dynamics of the optimization algorithms. Mathematically this amounts to the study of random dynamical systems with manifolds of equilibria. In this talk, I will give a brief introduction to machine learning theory and explain how the Lyapunov exponents of random matrix products can be used to characterize the set of possible limit points for stochastic gradient descent. The talk is based on joint work with Maxmilian Engel.

In the field of algebraic statistics, we view statistical models as part of an algebraic variety and use tools from algebra, geometry, and combinatorics to learn statistically relevant information about these models. In this talk, we discuss the algebraic interpretation of likelihood inference for discrete statistical models. We present recent work on the iterative proportional scaling (IPS) algorithm, which is used to compute the maximum likelihood estimate (MLE), and give algebraic conditions under which this algorithm outputs the exact MLE in one cycle. Next, we introduce quasi-independence models, which describe the joint distribution of two random variables where some combinations of their states cannot co-occur, but they are otherwise independent. We combinatorially classify the quasi-independence models whose MLEs are rational functions of the data. We show that each of these has a parametrization which satisfies the conditions that guarantee one-cycle convergence of the IPS algorithm.