Geometry, Number Theory and Logic

July 6th 2016, at the Mathematical Institute: An afternoon of talks by Frances Kirwan, James Maynard and Angus MacIntyre on the occasion of the retirements of Nigel Hitchin, Roger Heath-Brown and Boris Zilber. 

Conference poster:  HHZ.pdf

Titles and abstracts

2.30 pm Frances Kirwan: Hyperkähler quotients and hyperkähler implosion

Abstract: One of the many areas of geometry profoundly influenced by the work of Nigel Hitchin is that of hyperkähler geometry: geometry based on the quaternions. In particular, in a joint paper with three physicists, he introduced the hyperkähler quotient construction which allows us to construct new hyperkähler spaces from suitable group actions on hyperkähler manifolds. This construction is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Hyperkähler implosion is in turn an analogue of symplectic implosion (introduced in a 2002 paper of Guillemin, Jeffrey and Sjamaar) which is related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry.

3.30 pm James Maynard: Primes with missing digits

Abstract: A major challenge in analytic number theory is showing the existence of primes in various `thin' sets of integers (sets which contain at most X^{1-c} elements less than X). Typically we only succeed when such sets are `linear' or `multiplicative' in some way (the sparsest known sets where we can get an asymptotic estimate for the number of primes being due to Heath-Brown and Heath-Brown respectively).
We show that there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a thin set of numbers, but the set has less additive or multiplicative structure than previous examples. The proof relies on decorrelating 'digit conditions' which say when the Fourier transform of numbers with restricted digits is large, from 'Diophantine conditions' which say when the Fourier transform of the primes is large.

4.45 pm Angus MacIntyre: The Zilber exponential.

Abstract: Zilber, in a paper from 2005, used very sophisticated model theory to define an exponential function on the complex field C. The exponential satisfies Schanuel's Conjecture, and has a very natural Nullstellensatz for systems of exponential polynomials. Zilber has conjectured that this exponential field structure on C, defined without any analysis or topology, is isomorphic to the classical complex exponential. I will report on various results inspired by the conjecture, in particular some linking Schanuel's Conjecture to classical conjectures on common zeros of complex exponential functions. The conjecture remains open, and is deeply connected to the problem of whether the real field has a first-order definition in terms of the complex exponential function.

5.30pm Reception hosted by OUP