I will present joint work with Alessandro Fazzari in which we prove precise conditional estimates for the third (non-absolute) moment of the logarithm of the Riemann zeta function, beyond the Selberg central limit theorem, both for the real and imaginary part. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which captures the interaction of a prime power with Montgomery's pair correlation function. This conjecture can be proved on a certain subrange unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.

In the first half of the talk, I will briefly survey the theory of matroids with coefficients, which was introduced by Andreas Dress and Walter Wenzel in the 1980s and refined by the speaker and Nathan Bowler in 2016. This theory provides a unification of vector subspaces, matroids, valuated matroids, and oriented matroids. Then, in the second half, I will outline an intriguing connection between Lorentzian polynomials, as defined by Petter Brändén and June Huh, and matroids with coefficients.  The second part of the talk represents joint work with June Huh, Mario Kummer, and Oliver Lorscheid.

We present an exponential-integrator-based multi-index Monte Carlo (MIMC) method for the weak approximation of mild solutions to semilinear stochastic partial differential equations (SPDEs). Theoretical results on multi-index coupled solutions of the SPDE are provided, demonstrating their stability and the satisfaction of multiplicative error estimates. Leveraging this theory, we develop a tractable MIMC algorithm. Numerical experiments illustrate that MIMC outperforms alternative approaches, such as multilevel Monte Carlo, particularly in low-regularity settings.