Talk (Work in Progress): Frank Oertel
Completely correlation preserving mappings: another way to tackle the upper bound of the real and complex Grothendieck constant?
In 1953 the famous mathematician A. Grothendieck shaped a non-trivial result that he called The Fundamental Theorem on the Metric Theory of Tensor Products. This result is known today as Grothendieck's inequality, respectively Grothendieck's Theorem.
Grothendieck formulated this deep statement in the rather abstract language of tensor norms on tensor products of Banach spaces. To this end he described how to generate new tensor norms from known ones and unfolded a powerful duality theory between tensor norms. Only in 1968, thanks to J. Lindenstrauss and A. Pelczynski Grothendieck's inequality was decoded and equivalently rewritten – in matrix form - leading to its strong impact on the geometry of Banach spaces after 1968.
Since the appearance of Grothendieck's paper in 1953 there exists the (still) - open - problem to determine the smallest possible constant - called the Grothendieck constant – which can be used in Grothendieck's inequality. The Grothendieck constant depends on the choice of the real or complex field as the underlying field of scalars.
In addition to its reformulation in functional analysis Grothendieck's inequality admits further equivalent representations - each one of them reflecting deep and surprising links to different scientific branches, such as semidefinite programming in convex optimisation, NP-hard combinatorial optimisation, maximum cuts in graphs, the P = NP problem, the Ising model for spin glasses, communication complexity, private data analysis, geomathematics and - due to the pioneering work of B. S. Tsirelson - foundations and philosophy of quantum mechanics.
Based on block matrix analysis and techniques from Gaussian multivariate analysis we will present a further equivalent representation of Grothendieck's inequality, where we have to allocate the set of all correlation matrices (the so called elliptope) to a (strict) subset of all of its extreme points, given by the subset of all correlation matrices of rank 1. Both sets are linked in expectation by Grothendieck's identity (in the real case) and Haagerup's identity (in the complex case).
We will indicate that actually both equalities share a common, somewhat hidden underlying structure which again shows how Grothendieck's inequality is linked with some significant representation theorems, coined by I. J. Schoenberg.
By introducing a certain class of non-linear mappings which map correlation matrices of any size into correlation matrices of the same size, we immediately reobtain Krivine's famous upper bound of the real Grothendieck constant as a particular case of our approach, so that such correlation preservers might become a further useful tool to tackle the hard Grothendieck constant problem.
However, in both cases, the real and the complex one, we are lead to power series involving multi-dimensional Gaussian integrals (similarly appearing in quantum field theory) which seemingly cannot be calculated explicitly due to a required integration of non-smooth functions of modulus 1, so that suitable numerical approximation techniques very likely have to be applied.
First we present the main ideas in the real case and conclude our talk with a short glimpse at the complex case.
Despite the subject's intrinsic technical nature, we will present our thoughts in a rather mitigated way, addressed to a mixed audience and non-experts in functional analysis.