My background is in algebraic number theory, although recently I have started to work in the area of formal proof verification. I believe that computers will be able to help humans with proofs within my lifetime, and I am actively trying to make this happen sooner by (a) helping to build a database of modern mathematical theorems and definitions and (b) trying to teach mathematics undergraduates how to use the software. See also Tom Hales' Formal Abstracts project.
If you are a mathematician (or a future mathematician) interested in starting to learn about how to prove theorems using computers, you could try the natural number game! Want some more advanced mathematics? Read about how Commelin, Massot and I told a computer what a perfectoid space was!
Some of my recent lectures have been videoed. For example:
* What computers can't do (lay intro to P v NP at Royal Institution).
* The Future of Mathematics (2019 talk at Microsoft Research).
* Will Computers Outsmart Mathematicians? (2021 talk at Gresham College).
My personal website (papers, teaching etc) is here.
Buzzard K, Verberkmoes A, 2018, Stably uniform affinoids are sheafy, Journal fur die Reine und Angewandte Mathematik, Vol:740, ISSN:0075-4102, Pages:25-39
Buzzard K, Ciere M, 2014, Playing simple loony dots and boxes endgames optimally, Integers: Electronic Journal of Combinatorial Number Theory, Vol:14, ISSN:1553-1732
Buzzard K, Gee T, 2014, The conjectural connections between automorphic representations and Galois representations, Automorphic Forms and Galois Representations, Vol 1, Vol:414, ISSN:0076-0552, Pages:135-187
Buzzard K, Diamond F, Jarvis F, 2010, ON SERRE'S CONJECTURE FOR MOD l GALOIS REPRESENTATIONS OVER TOTALLY REAL FIELDS, Duke Mathematical Journal, Vol:155, ISSN:0012-7094, Pages:105-161
Buzzard KM, 2007, Eigenvarieties, L-Functions and Galois Representations, Editor(s): Burns, Buzzard, Nekovar, Cambridge University Press, Pages:59-120, ISBN:9780521694155
Buzzard K, 2011, Computing weight one modular forms over $\C$ and $\Fpbar$, Computations with Modular Forms 2011