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Way back at the end of last year I put out a call to mathematicians I know: hop on Skype and chat to me for a while about the work you’re doing at the moment. The first person to answer was David Roberts, a pure mathematician from Adelaide.
We had a fascinating talk about one thread of David’s current work, which involves all sorts of objects I know no more about than their names. I had intended to release this as a podcast, but the quality of my recording was very poor and it turns out I’m terrible at audio editing, so instead here’s a transcription. Assume all mistakes are mine, not David’s.
If you’ve ever wanted to know what it’s like to work in the far reaches of really abstract maths, this is an excellent glimpse of it.
DR: I’m David Roberts, I’m a pure mathematician, currently between jobs. I work – as far as research goes – generally on geometry and category theory, and the interplay between those two. And also a little bit of logic stuff, which I thought I’d talk about.
The announcement of the Shaw Prize was posted on 23rd May, reading:
The Shaw Prize in Mathematical Sciences 2017 is awarded in equal shares to János Kollár and Claire Voisin for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing
problems that had appeared out of reach.
The prize is awarded annually to “individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence”.
The two joint winners this year, Kollár and Voisin, are both professors of algebraic geometry, at Princeton and Collège de France respectively, and have made major contributions to the effort to characterise rational varieties – solution sets of polynomials which differ from a projective space only by a low-dimensional subset.
Kollár’s work relates to the Minimal Model Program, which concerns moduli of higher-dimensional varieties – spaces whose points represent equivalence classes of varieties. These spaces, which Kollár has extensively worked on and developed the field dramatically, have applications in topology, combinatorics and physics. Voisin’s achievements have included solving the Kodaira problem (on complex projective manifolds), developing a technique for showing that a variety is not rational, and even finding a counterexample to an extension of the Hodge conjecture (one of the Clay prize problems), which rules out several approaches to the main conjecture.