Mathematical Colloquium

In 1902 Burnside asked whether any finitely generated torsion group is necessarily finite. By now there is a long line of negative answers, albeit not necessarily accessible. I will explain the basics of small cancellation theory and our approach to the Burnside problem.
(j/w A. Atkarskaya and E. Rips)

Accurate and timely inversion of tsunami source parameters from observational data is essential for developing effective early warning systems, where decisions must be made under uncertainty. The detection of tsunami sources from observational data, such as buoy and hydrophone measurements, poses significant challenges in Bayesian inverse modeling. The forward problem involves solving complex hyperbolic systems, often modeled using the shallow water equations.
I will discuss the discretization of the forward model using an ADER-DG method, which is capable of handling wetting and drying processes. Additionally, I will outline how to efficiently perform the inversion using a fully parallelized multi-level Markov Chain Monte Carlo (MLMCMC) method. A key challenge lies in balancing computational feasibility with accuracy, particularly in high-dimensional parameter spaces. To address this, I will demonstrate the effect of incorporating a Gaussian Process surrogate model as a computationally inexpensive coarse-level approximation.

In enumerative geometry, we fix geometric objects and conditions and count how many objects satisfy the conditions. For example, there are 2 plane conics passing through 4 points and tangent to a given line. Tropical geometry can be viewed as a degenerate version of algebraic geometry and has proved to be a successful tool for enumerative problems. We review tropical curve counting problems. In particular, we show how tropical methods can be applied for quadratically enriched counts, which can be viewed as generalizations that allow results over any ground field.

Over the past decade or so, tools from algebraic topology have been shown to be very useful for the analysis and characterization of networks, in particular for exploring the relation of structure to function. I will describe some of these tools and illustrate their utility in neuroscience, primarily in the framework of a collaboration with the Blue Brain Project.