Multiparameter persistent homology is a generalization of persistent homology that allows for more than a single filtration function. Such constructions arise naturally when considering data with outliers or variations in density, time-varying data, or functional data. Even though its algebraic roots are substantially more complicated, several new invariants have been proposed recently. In this talk, I will go over such invariants, as well as their stability, vectorizations and implementations in statistical machine learning.

In the 1970s, Serre conjectured that any continuous, irreducible and odd mod p representation of the absolute Galois group G_Q is modular. Serre furthermore conjectured that there should be an explicit minimal weight "k" such that the Galois representation is modular of this weight, and that this weight only depends on the restriction of the Galois representation to the inertial subgroup I_p. This is often called the weight part of Serre's conjecture. Both the weight part, and the modularity part, of the Serre's conjecture are nowadays known to be true. In this talk, I want to explain how to rephrase the conjecture in representation theoretic terms (for k >= 2), so that the weight k is replaced by a certain (mod p) irreducible representation of GL_2(F_p), and how upon rephrasing the conjecture one can realize it as a statement about local-global compatibility with the mod p local Langlands correspondence.