Week 22.05.2023 – 28.05.2023

Ulf Lindsröm, a Leverhulme visiting professor at Imperial College will discuss sigma models, which are maps from a domain to a target space T. The geometry of the target space is determined by the dimension of the domain and symmetries of the model. When it has isometries that can be gauged, the quotient space, i.e., the space of orbits under the isometries, supports a new sigma model. The target space geometry of the new model is the quotient of the T by the isometry group.

This is first described for a bosonic sigma model and it is pointed out that we need to understand supersymmetric sigma models, their isometries and gauging as well as the quotient in order to apply the scheme to models with extended supersymmetry. We then look at these issues. The final goal is to construct new hyperkähler geometries from hyperkähler geometries with isometries, so making sure that the quotient construction preserves the symmetries etc.

Please visit https://lonti.weebly.com/spring-2023-series.html for more information.

If E/Q is an elliptic curve, and d is a squarefree integer, then the 2-torsion modules of E and its quadratic twist E_d are isomorphic. In particular their 2-Selmer groups can be made to lie in the same space. Poonen-Rains provide a heuristic model for the behaviour of these 2-Selmer groups individually, as E varies, but how independent are they? We'll present results in this direction.

The Shimura-Taniyama-Weil modularity conjecture asserts that all elliptic curves over Q arise as images of quotients of the Poincare upper half plane by congruence subgroups of the modular group SL2(Z). Wiles proved Fermat's Last Theorem by establishing the modularity of semistable elliptic curves over Q. Subsequent work of Breuil-Conrad-Diamond-Taylor established the modularity of elliptic curves over Q in full generality. My work with J-P. Wintenberger gave a proof of the generalized Shimura-Taniyama-Weil conjecture which asserts that all "odd, rank 2 motives over Q" are modular. This is a corollary of our proof of Serre's modularity conjecture.


Very little is known when one looks at the same question over finite extensions of Q. I will talk about the recent beautiful work of Ana Caraiani and James Newton which proves modularity of all elliptic curves over Q(i). An input into their proof is a result, proved in joint work with Patrick Allen and Jack Thorne, that proves the analog of Serre's conjecture for mod 3 representations that arise from elliptic curves over Q(i).

My talk will give a general introduction to this circle of ideas centred around the modularity conjecture for motives and Galois representations over number fields. We know only fragments of what is conjectured, but what little we know is already quite remarkable!