ECM Seminar: Tomorrow’s Mathematicians Today 2023 Winners

Part of the IMA Early Career Mathematicians Seminar Series.

Maria Esteban Casadevall (Heriot-Watt) won the IMA Prize for the best presentation and Veronica Bitonti (University College London) won the IMA Popular Vote Prize.

Maria’s talk: Geometric methods for community detection: Discrete notions of curvature in graphs and clustering in the Hyperbolic Space

In recent years, a wide number of geometrical tools have been developed to exploit the intrinsic properties of complex networks. In this talk we will explore two of these notions and understand its application to community detection problems.

The first part of the talk will focus on the Olliver Ricci curvature, a discrete notion of edge curvature based on the Wasserstein distance between probability distributions around neighbouring nodes. We will begin with a definition and description of such notion, together with a justification of its connection to the Ricci curvature in manifolds. A discrete version of the Ricci Flow can then be defined iteratively to flatten the curvature of the graph, bringing important applications to community detection.

The second part of the talk will explore the Hyperbolic space, which has lately been receiving increasing attention due to its success in representing learning for hierarchical data. We will begin with a definition and description of such spaces, together with a justification of why its geometric properties are particularly fit for hierarchical data. We will then discuss how the K-Means Clustering Algorithm can be defined in the Hyperbolic Space and we will see an implementation of such to a real-world network.

Veronica’s talk: The quest for structure: Continued Fractions in Enumerative Combinatorics

Dealing with patterns and how they can be formed is one of the main objects of studies in enumerative combinatorics, and an essential tool to understand such properties is through the majestic method, used already by Euler in 1746, to represent ordinary generating functions as a continued fraction. In particular, in 1980, Flajolet analysed in great detail the connection between Stieltjes-type and Jacobi-type continued fractions and generating functions for Dyck and Motzkin Paths with height-dependent weights. Further developments in this direction has expanded such knowledge to Thron-type continued fractions (T-fractions) which can be interpreted in terms of Schröder Paths. Recently such ideas were extended by Pétréolle, Shu, Sokal in 2018 to what they call “branched continued fractions”. In this presentation, the aim is to introduce the main concepts of the subject by looking at some classes of labelled increasing trees.