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We are pleased to announce the launch of the London Heilbronn Colloquia, a new lecture series sponsored by the Heilbronn Institute for Mathematical Research. The talks will be held at HIMR’s partner institutions in London, and given by distinguished mathematicians and theoretical physicists at the forefront of current research. They are aimed at a general mathematical audience, and each will be followed by a reception offering more opportunity for informal interaction with the speaker.
The first lecture will be given by Chandrashekhar Khare (UCLA) at King’s College London on 26 May 2023, followed by talks by Tadashi Tokieda (Stanford) at UCL on 1 June 2023, and Mihalis Dafermos (Cambridge, Princeton) at Imperial on 16 June 2023. You can find more information about the speakers and talks on the official event website.
Details of the first colloquium are below:-
Speaker: Chandrashekhar Khare, FRS (UCLA)
Time: 26 May 2023, 3pm
Location: King’s Building K6.29 (Anatomy Lecture Theatre)
Title: The Shimura-Taniyama-Weil conjecture and beyond
Abstract from the speaker: 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!