Curriculum Vitae (including a description of my research).
I am currently a research fellow in the Department of Mathematics of the University of Bristol. The year before last I taught the 3rd year logic class here.
My email address is Andrew.Brooke-Taylor(at)bristol.ac.uk.
I obtained my doctorate at the Kurt Gödel Research Center for Mathematical logic at the Universität Wien (University of Vienna), working with Professor Sy Friedman.
My main area of interest is in set theory, specifically, large cardinal axioms and forcing, and this forms the focus of most of my research. I am also very interested in connections between set theory and other fields of mathematics, particular algebraic topology, which I spent a significant amount of time working on as a graduate student at MIT.
My work focuses mostly on forcing various combinatorial principles reminiscent of Gödel's constructible universe L to hold while preserving large cardinals, thus obtaining L-like outer models containing these large cardinals. In particular, I have worked on forcing morasses to exist at every regular cardinal, and was able to modify the standard forcing used for this in such a way that all large cardinals of a variety of kinds are preserved. I also came up with an iterated forcing for making the universe have a definable well-order while still enjoying the generalised continuum hypothesis. With this forcing one can preserve a proper class of measurable, Woodin, n-superstrong, n-huge, and other similar cardinals.
I have also been working with Joan Bagaria and Carles Casacuberta of the University of Barcelona on applications of large cardinal axioms from set theory to category theory and algebraic topology. Specifically, a large cardinal axiom known as Vopenka's Principle is known to be intimately tied to structural results for important kinds of categories, and we are investigating these connections.
Another line of research for me has been my investigation with Sheila Miller of the algebraic structures that arise from rank-to-rank embeddings, the existence of which is one of the strongest large cardinal axioms that we have (that is, that is thought to be consistent, or at the very least, that isn't currently known to be inconsistent). Despite the incredible strength of such assumptions, they are in some cases the only known way to prove certain very finitary, algebraic results.
In a completely different vein, I have also been doing work with Damiano Testa in finite model theory. Specifically, we have been considering a simplicial complex analogue of the (countably infinite) random graph, and generalisations of this. Amongst other things, this gives rise to a new 0-1 law for finite simplicial complexes, different from the known one presented by Blass & Harary.
My doctoral thesis (PDF format; also available in PS format). For a bit of light relief, I included a couple of figures (PS format).
Large cardinals and definable well-orders on the universe, Journal of Symbolic Logic 74, no. 2 (June 2009) pp 641-654. (via Project Euclid, or for those without a subscription)
(With Sy-David Friedman) Large cardinals and gap-1 morasses, Annals of Pure and Applied Logic 159, no. 1-2 (2009), pp 71-99. (via ScienceDirect, or for those without a subscription)
I gave a talk (pdf slides) at the Logic Colloquium 2006 about forcing morasses to exist. There, I gave particular attention to the interesting case of trying to preserve 1-extendible cardinals (and eventually succeeding!).
More recently, I gave a talk (pdf slides) at the Logic Colloquium 2007 about the definable well-order part of my thesis.
I was an invited speaker at the 16th Set Theory and it Neighbours meeting in London in December 2007, but didn't use computer slides for that one.