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WS 2024/25

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27.11.2024 um 17:15 Uhr in Raum 69/117

Dr. Aida Maraj (MPI-CBG Dresden)

Understanding Multivariate Gaussian Models via Toric Geometry

Algebraic geometry has recently provided a new approach to advancing problems in multivariate Gaussian models. This is achieved by identifying Gaussian distributions with symmetric matrices and analyzing the polynomials that vanish on these matrices, known as ideals. The talk will focus on Brownian motion tree (BMT) models, a type of Gaussian model used in phylogenetics. BMT models have a hidden toric geometry, which we use to provide formulas on the maximum likelihood degree and its dual. Finally, the need to classify statistical models with toric geometry motivates us to introduce the symmetry Lie group of an ideal. This group can detect when an ideal is toric under a linear transformation. No prior knowledge of toric ideals or BMT models is required.

04.12.2024 um 17:15 Uhr in Raum 69/117

Prof. Dr. Alexander Drewitz (Universtität Köln)

A Journey Through Percolation - From Independence to Long-Range Correlations

Percolation models have been playing a fundamental role in statistical physics for several decades by now. They had initially been investigated as a model for the gelation of polymers during the 1940s by chemistry Nobel laureate Flory and Stockmayer. From a mathematical point of view, the birth of percolation theory was the introduction of Bernoulli percolation by Broadbent and Hammersley in 1957, motivated by research on gas masks for coal miners. One of the key features of this model is the inherent stochastic independence which simplifies its investigation, and which has lead to deep mathematical results. During recent years, there has been a growing interest in investigating percolation models with long-range correlations, aiming to capture a more realistic and complex scenario.  We will survey parts of the development of percolation theory, and then discuss some recent progress for the Gaussian free field with a particular focus on the understanding of the critical parameters in the associated percolation models.