Fri, 21 Nov 2014

14:30 - 15:45
L2

The History of Mathematics in 300 Stamps

Robin Wilson
(Open University)
Abstract

The entire history of mathematics in one hour, as illustrated by around 300 postage stamps featuring mathematics and mathematicians from across the world.

From Euclid to Euler, from Pythagoras to Poincaré, and from Fibonacci to the Fields Medals, all are featured in attractive, charming and sometimes bizarre stamps. No knowledge of mathematics or philately required.

Mon, 10 Nov 2014

12:00 - 13:00
L5

Lessons from crossing symmetry at large N

Tomasz Lukowski
(Oxford)
Abstract
In this talk I will discuss how to construct all solutions consistent with crossing symmetry in the limit of large central charge $c ~ N^2$, starting from the four-point correlator of the stress tensor multiplet in N=4 SYM. Unitarity forces the introduction of a scale $\Delta_{gap}$ and these solutions organize as a double expansion in 1/c and $1/\Delta_{gap}$. These solutions are valid to leading order in 1/c and to all orders in $1/\Delta_{gap}$ and reproduce, in particular, instanton corrections previously found. Comparison with such instanton computations allows to fix $\Delta_{gap}$. Using this gap scale one can explain the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge. Furthermore, I will present connections between such upper bounds and positivity constraints arising from causality in flat space and I will discuss how certain relations derived from causality constraints for scattering in AdS follow from crossing symmetry.
 
Mon, 10 Nov 2014
14:15
L5

Tropical moment maps for toric log symplectic manifolds

Marco Gualtieri
(Toronto)
Abstract

I will describe a generalization of toric symplectic geometry to a new class of Poisson manifolds which are
symplectic away from a collection of hypersurfaces forming a normal crossing configuration.  Using a "tropical
moment map",  I will describe the classification of such manifolds in terms of decorated log affine polytopes,
in analogy with the Delzant classification of toric symplectic manifolds. 

Mon, 17 Nov 2014
14:15
L5

The Horn inequalities and tropical analysis

Andras Szenes
(Geneva)
Abstract

 I will report on recent work on a tropical/symplectic approach to the Horn inequalities. These describe the possible spectra of Hermitian matrices which may be obtained as the sum of two Hermitian matrices with fixed spectra. This is joint work with Anton Alekseev and Maria Podkopaeva.

Fri, 21 Nov 2014
16:30
L2

The Mathematics of Non-Locality and Contextuality

Samson Abramsky
(Dept of Computer Science - University of Oxford)
Abstract

Quantum Mechanics presents a radically different perspective on physical reality compared with the world of classical physics. In particular, results such as the Bell and Kochen-Specker theorems highlight the essentially non-local and contextual nature of quantum mechanics. The rapidly developing field of quantum information seeks to exploit these non-classical features of quantum physics to transcend classical bounds on information processing tasks.

In this talk, we shall explore the rich mathematical structures underlying these results. The study of non-locality and contextuality can be expressed in a unified and generalised form in the language of sheaves or bundles, in terms of obstructions to global sections. These obstructions can, in many cases, be witnessed by cohomology invariants. There are also strong connections with logic. For example, Bell inequalities, one of the major tools of quantum information and foundations, arise systematically from logical consistency conditions.

These general mathematical characterisations of non-locality and contextuality also allow precise connections to be made with a number of seemingly unrelated topics, in classical computation, logic, and natural language semantics. By varying the semiring in which distributions are valued, the same structures and results can be recognised in databases and constraint satisfaction as in probability models arising from quantum mechanics. A rich field of contextual semantics, applicable to many of the situations where the pervasive phenomenon of contextuality arises, promises to emerge.

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