Meet the new PhD students in (Applied) CATS

Deep, but not difficult: Neurovarieties of polynomial residual networks - Isabel Dahlgren.

Problem: Find a toy model of a neural network that is analytically tractable, yet flexible enough to yield insights about neural networks used in practise. One solution attempt: replace all activations with polynomials. For networks with polynomial activation functions, understanding neural networks comes down to algebraic geometry: questions about expressivity and training bias translate naturally into questions about fibers and singularities. I’ll tell you more about this correspondence and exemplify it with some results on residual networks.

Kunz's Theorem and Deformation of F-injectivity - Andrea Minna

In this seminar I will introduce myself and talk about my Master’s thesis. In commutative ring theory, one of the strongest properties of a ring is regularity. A fundamental characterization of regularity in prime characteristic was given by Ernst Kunz in 1969. His theorem led to the development of a vibrant area of research: F-singularities. I will briefly describe Kunz’s characterization and introduce the problem of deformation of F-injectivity, showing some known cases.  

From local to global in directed random graphs - Léon Burkhardt

Random graphs are a fundamental tool for studying complex interaction networks arising in biology, communication systems, or social science. Empirical biological networks, such as gene regulatory networks, exhibit distinctive structural features, including power-law degree distributions and the over-representation of specific local motifs, most notably feed-forward loops.
Motivated by these observations, we investigate how local attachment rules influence the global topology of directed random graphs. We discuss a family of spatial preferential attachment models based on "the age-dependent random connection model" introduced by Gracar, Grauer, Lüchtrath, and Mörters in their 2019 paper. We show how suitable parameter choices allow them to reproduce key statistical and topological properties observed in empirical networks.