Week 05.03.2023 – 11.03.2023

Speaker: Alex Best, 14:00-14:20

Title: p-adic integration and points on curves

Abstract: The problem of algorithmically determining the set of rational and integral points on curves has seen impressive progress in the last 20 years, especially in the area surrounding Chabauty's method.
I'll give an overview of recent work involving $p$-adic integration applied to such problems and their strengths and limitations.

Speaker: Sebastian Monnet, 14:30-14:50

Title: S4-quartics with prescribed norms.

Abstract: Let $K$ be a number field with $\mathbb{Q}$-basis $\{e_1, ..., e_n\}$, and let $\alpha$ be a rational number. It is natural to ask whether the "norm equation" $N_{K/Q}(x_1e_1 + ... + x_ne_n) = \alpha$ has rational solutions. Since the answer depends only on $K$, we may ask how often this norm equation has rational points as we vary $K$. The case of abelian number fields was solved by Frei-Loughran-Newton, and in this talk we present one of the simplest non-abelian cases: $S_4$-quartics.

The fuzzy approach to clustering arises to cope with situations where objects have not a clear assignment. Unlike the hard/standard approach where each object can only belong to exactly one cluster, in a fuzzy setting, the assignment is soft\DSEMIC that is, each object is assigned to all clusters with certain membership degrees varying in the unit interval. The best known fuzzy clustering algorithm is the fuzzy k-means (FkM), or fuzzy c-means. It is a generalization of the classical k-means method. Starting from the FkM algorithm, and in more than 40 years, several variants have been proposed. The peculiarity of such different proposals depends on the type of data to deal with, and on the cluster shape. The aim is to show fuzzy clustering alternatives to manage different kinds of data, ranging from numerical, categorical or mixed data to more complex data structures, such as interval-valued, fuzzy-valued or functional data, together with some robust methods. Furthermore, the case of two-mode clustering is illustrated in a fuzzy setting.