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Let \(p\) be a real polynomial in $n$ variables of even degree \(2d\). A fundamental computational task, with applications in optimization and real algebraic geometry, is to decide whether \(p\) can be written as a sum of squares of polynomials. That is, whether there exist polynomials \(q_1, \ldots, q_m\) such that \(p = q_1^2 + q_2^2 + \cdots + q_m^2\). In this talk, I will discuss the computational complexity of this question. (based on joint work with Nikolas Gärtner and Victor Magron)

Suppose we are given some data, and we hypothesize a structural causal model to describe them: how can we narrow the set of causal graphs compatible with our observations? The theory of identifiability aims to answer this question. We show that, in the case of additive noise models, the score function of the data contains all the information about the causal graph. However, this requires strong and, crucially, hard-to-verify modeling assumptions, like additivity of the noise. When direct experiments to infer causality are not feasible, this raises the question: how can we move past these restrictions? Borrowing ideas from independent component analysis, we show how multiple environments (read: non i.i.d. data) can overcome these limitations: for structural causal models with arbitrary causal mechanisms, data from only three environments uniquely identify the causal graph from the Jacobian of the score function. Thus, non-i.i.d.-ness turns from a curse into a blessing for causal discovery.

In this talk, we investigate mathematically how capillary-driven viscous thin fluid films evolve on microscopic length scales, in which case thermal noise due to fluctuations of the fluid particles comes into play. The underlying stochastic partial differential equation (SPDE) is a stochastic thin-film equation, a fourth-order degenerate-paraolic PDE driven by nonlinear gradient noise. This equation was first suggested in the physical literature approximately 20 years ago and existence of solutions for nonlinear noise was only established very recently. The key observation is that the Stratonovich formulation of the equation is the physically correct mathematical formulation, leading to a suitable balance of fluctuations and dissipation of the underlying physics and the correct balance in the energy-entropy dissipation relations. Specifically we are able to establish existence of nonnegative martingale solutions for nonlinear mobilities and we further prove existence of measure-valued solutions for initial values with non-full support. The latter forms a first step towards proving finite speed of propagation and for investigating contact-line dynamics on microscopic length scales. This talk is based on joint works with Konstantinos Dareiotis (Leeds), Benjamin Gess (TU Berlin and Max Planck Institute MiS, Leipzig), Günther Grün (Erlangen), and Max Sauerbrey (formerly TU Delft, now Max Planck Institute MiS, Leipzig).

We analyze the qualitative behavior of stochastic partial differential equations (SPDEs) with a particular focus on bifurcations. To this aim we investigate a change of sign in the finite-time Lyapunov exponents (FTLEs) of the SPDE in a small noise regime and close to a phase transition. Under suitable assumptions, the FTLEs are positive and thus indicate a change of stability. These results are applied to the stochastic Allen-Cahn and Burgers equation with non-Markovian noise and to singular SPDEs. Moreover, we also discuss properties of FTLEs, in particular large deviations type results.