Current Research Activities
Upper bounds for Grothendieck constants, quantum correlation matrices and
completely correlation preserving functions
Recently, we have completed our own contribution to a thorough analysis of a most fascinating and beautiful open problem that combines both, pure mathematics and applied mathematics (if such a separation is possible within mathematics at all). It has deep applications in the philosophy and foundations of quantum mechanics, quantum information and even large data analysis. The results will be reflected in the forthcoming volume 2349 of the Lecture Notes in Mathematics series that is about to be published around August 2024 (cf. list of publications).
Our approach is built on a strong inclusion and interplay of linear algebra, matrix analysis, real and complex analysis, some facts from harmonic analysis, theory and applications of special functions (in particular Hermite polynomials and Gaussian hypergeometric functions), integer programming with Boolean-valued functions, convex optimisation including semidefinite programming, probability theory, stochastic analysis, and even statistics including dependence modelling with copulas and computational complexity theory (implying a curse of dimensionality; also due to high-dimensional integration over finite-dimensional spheres). We have developed some new facts which hopefully support our goal to shed light on the rich structure of the convex set of all correlation matrices, including B. S. Tsirel'sons quantum correlation matrices. To this end, we have introduced certain non-linear mappings between them, which map each correlation matrix of any size to a correlation matrix of the same
size (entrywise, by means of the Hadamard product); the so called "completely correlation preserving functions (CCP)". A famous and important example of such a CCP function is J. L. Krivine's $\frac{2}{\pi}\arcsin : [-1,1]
\longrightarrow [-1,1]$.
That nice problem is the determination of the exact value of the real and complex Grothendieck constant in Grothendieck's famous inequality.
Here is a beautiful and crystal clear talk of Professor Gadadhar Misra from the Indian Institute of Science (IISc), presented at the International Centre for Theoretical Sciences (ICTS) of the Tata Institute of Fundamental Research in Bengaluru, India in 2019. In particular, he sheds further light on the significance of Grothendieck's inequality, visualises its recent applications in computer science and complexity theory (maximum cut of graphs*) and presents Krivine's original proof step by step: The Grothendieck inequality.
It seems that we only see a tip of an iceberg here and that even an approximation of the real and complex Grothendieck constant already requires a persistent banging of our head against very massively constructed entrance doors...
At least also our approach which is - strongly - built on the possibly unavoidable inversion of suitable Taylor series enables a rather quick reproduction of Krivine's constant as an upper bound of the real Grothendieck constant as well as Haagerup's constant as an upper bound of the complex Grothendieck constant. Ideally, given our generalisation, it should reveal a possible path how to obtain a smaller value of these upper bounds. To this end, we follow a few parts of a highly fascinating and further groundbreaking, yet technically quite demanding paper. Nevertheless, our approach via CCP functions analysis also allows a shortening and a simplification of parts of the sophisticated proof.
In passing, we will provide a further proof of Grothendieck's inequality (for both fields).
* Michel Goemans and David Williamson receive 2022 Steele Prize for Seminal Contribution to Research
Primary MSC
- 15A45 - Miscellaneous inequalities involving matrices
- 15A60 - Applications of functional analysis to matrix theory
- 15A63 - Quadratic and bilinear forms, inner products
- 15B48 - Positive matrices and their generalizations; cones of matrices
- 30B10 - Power series (including lacunary series) in one complex variable
- 30B40 - Analytic continuation of functions of one complex variable
- 33C20 - Generalized hypergeometric series, pFq
- 33C45 - Orthogonal polynomials and functions of hypergeometric type
- 33C60 - Hypergeometric integrals and functions defined by them
- 33E20 - Functions defined by series and integrals
- 44A05 - General integral transforms
- 46N10 - Applications of functional analysis in optimization and programming
- 46N50 - Applications of functional analysis in quantum physics
- 47D07 - Markov semigroups of linear operators and applications to diffusion processes
- 52A20 - Convex sets in n dimensions (including convex hypersurfaces)
- 52A40 - Inequalities and extremum problems (convex geometry)
- 60A10 - Probabilistic measure theory
- 60E05 - General theory of probability distributions
- 60E10 - Transforms of probability distributions
- 60G07 - General theory of stochastic processes
- 60H05 - Stochastic integrals
- 60H30 - Applications of stochastic analysis (to PDE, etc.)
- 60G44 - Martingales with continuous parameter
- 60G46 - Martingales and classical analysis
- 60G48 - Generalizations of martingales
- 60G51 - Processes with independent increments; Lévy processes
- 60G57 - Random measures
- 62E10 - Characterization and structure theory of statistical distributions
- 62E15 - Exact distribution theory in statistics
- 62H05 - Characterization and structure theory (multivariate analysis)
- 62H10 - Multivariate distributions of statistics
- 62H99 - Multivariate analysis
- 81P05 - General and philosophical topics in quantum theory
- 81P15 - Quantum measurement theory
- 81P40 - Quantum coherence, entanglement, quantum correlations
- 81P45 - Quantum information, communication, networks
- 90C09 - Boolean programming
- 90C20 - Quadratic programming
- 90C22 - Semidefinite programming
- 90C25 - Convex programming
- 90C27 - Combinatorial optimization
- 94C10 - Switching theory, application of Boolean algebra; Boolean functions
Secondary MSC
- 05D40 - Probabilistic methods in combinatorics
- 42A61 - Probabilistic methods in Fourier analysis
- 46B07 - Local theory of Banach spaces
- 46B28 - Spaces of operators; tensor products; approximation properties
- 46G10 - Vector-valued measures and integration
- 46G12 - Measures and integration on abstract linear spaces
- 46L07 - Operator spaces and completely bounded maps
- 47L20 - Operator ideals
- 52A07 - Convex sets in topological vector spaces (convex geometry)
- 60C05 - Combinatorial probability
- 60E07 - Infinitely divisible distributions; stable distributions
- 60H07 - Stochastic calculus of variations and the Malliavin calculus
- 60J65 - Brownian motion
- 62A01 - Foundations and philosophical topics in statistics
- 68Q25 - Analysis of algorithms and problem complexity
- 81P68 - Quantum computation
- 90C35 - Programming involving graphs or networks
- 91G20 - Derivative securities
- 91G30 - Interest rates (stochastic models)
- 91G40 - Credit risk
Keywords Primary: Analysis of Boolean functions; Bell’s inequality; Bell’s theorem; CHSH inequality; elliptope; foundations of quantum mechanics; Gaussian measure; Grothendieck’s inequality; Grothendieck constant; Hadamard matrix; quantum information; measure theory; MAX-CUT; positive semidefinite; Schoenberg; SDP; semimartingales; special functions
Keywords Secondary: Geometry of Banach spaces; Malliavin calculus; mathematical finance; mathematical foundations of machine learning; operator algebras; operator ideals; operator spaces; POVMs; statistics of machine learning; stochastic volatility models; tensor norms; vector measures
Membership in Research Networks
Most Recent Research Related Activities
Online Databases for Mathematical Research
Further Interests
- Differential Geometry
- Number Theory
- Cryptology (i.e., Cryptography and Cryptoanalysis)
[Many thanks to Peter Wagner for pointing this highly fascinating subject out to me!]
Current Version: February 23, 2024