Week 15.05.2023 – 21.05.2023

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.

The study of the classical theta operator was key to Edixhoven's proof of the weight part of Serre's modularity conjecture. Since then, a lot of work has been devoted to extending the construction of this operator to other Shimura varieties, with an eye towards generalisations of Serre's conjecture.
My goal is to give an overview of a family of generalised theta operators, on certain unitary Shimura varieties, that I constructed in my thesis and studied in subsequent work.

This is a meeting of the COW seminar:

http://cow.alggeo.xyz/nextmeeting.html

1:00pm Laura Schaposnik (University of Illinois Chicago/Oxford): A map of Higgs bundles, generalized hyperpolygons, and more

2:30pm Hal Schenck (Auburn/Oxford): Dynamical systems meet algebraic geometry: From Kuramoto oscillators to Segre varieties

4:00pm Liana Heuberger (Bath): Combinatorial Reid's recipe for consistent dimer models

Title: Geometric Serre weight conjectures, Hasse invariants and \Theta-operators

Abstract: Serre’s conjecture, now a theorem of Khare and Wintenberger, states that every odd, irreducible representation Gal(K/Q) --> GL(2,F_p) arises from a modular form. Furthermore it prescribes the minimal weight (at least 2) and level (prime to p) of such a form. I’ll recall this, along with a more geometric variant due to Edixhoven involving modular forms of weight one, and the relation with certain “weight-shifting” operators in characteristic p. Then I’ll discuss a generalization (joint with Sasaki) of the geometric variant and related weight-shifting phenomena in the context of Hilbert modular forms.