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.

Placing M-theory on a non-compact Calabi-Yau threefold allows us to construct low energy field theories in 5d with minimal supersymmetry, in a limit in which gravity is decoupled.  We venture into this topic by introducing all the building blocks we hope to capture in a 5d SCFT. Next, from the geometric perspective we realise the 5d gauge theory data from the objects within the Calabi-Yau geometry, given by curves, divisors, rulings, and singularities. After seeing how the geometry captures all the possible field theory data, we illustrate how to build some simple 5d SCFTs by placing M-theory on toric Calabi-Yau threefolds.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.