| EPSRC Reference: |
EP/N022653/1 |
| Title: |
Workshop: Hilbert's Sixth Problem |
| Principal Investigator: |
Gorban, Professor A |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
Mathematics |
| Organisation: |
University of Leicester |
| Scheme: |
Standard Research |
| Starts: |
01 May 2016 |
Ends: |
31 July 2016 |
Value (£): |
16,410
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| EPSRC Research Topic Classifications: |
| Algebra & Geometry |
Logic & Combinatorics |
| Mathematical Physics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Panel History: |
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Summary on Grant Application Form |
In the year 1900 Hilbert presented his problems to the International Congress of Mathematicians (he presented 10 problems at the talk; the full list of 23 problems was published later). The list of Hilbert's 23 problems was very influential for 20th century mathematics. The sixth problem concerns the axiomatization of those parts of physics which are ready for a rigorous mathematical approach.
The original formulation (in English translation) was:
"6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics." This is definitely "a programmatic call" for the axiomatization of existent physical theories.
In a further explanation Hilbert proposed two specific problems: (i) axiomatic treatment of probability with limit theorems for the foundation of statistical physics, and (ii) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua":
The Sixth Problem has inspired several waves of research. Its mathematical content changes in time what is very natural for a programmatic call. In the 1930s, the axiomatic foundation of probability seemed to be finalized by Kolmogorov on the basis of measure theory. Nevertheless, in the 1960s Kolmogorov and Solomonoff stimulated new interest in the foundation of probability (Algorithmic Probability).
Hilbert, Chapman and Enskog created asymptotic expansions for the hydrodynamic limit of the Boltzmann equations. The higher terms of the Chapman-Enskog expansion are singular and truncation of this expansion does not have rigorous sense (Bobylev). Golse, Bardos, Levermore and Saint-Raymond proved rigorously the Euler limit of the Boltzmann equation in the scaling limit of very smooth flows, but recently Slemrod used the exact results of Karlin and Gorban and proposed a new, Korteweg asymptotic of the Boltzmann equation.
It seems that Hilbert presumed the kinetic level of description (the "Boltzmann level") as a compulsory intermediate step between the atomistic view and continuum mechanics. Nevertheless, this intermediate description may be omitted. Now, L. Saint-Raymond with co-authors is developing a new approach to the problem "from the atomistic view to the laws of motion of continua" without the intermediate kinetic equations.
Quantum mechanics was invented after the Hilbert problems were stated. The first attempt at formalization of quantum mechanics was performed by von Neumann (who was Hilbert's assistant). Subsequently, the axiomatic approach to the quantum world has been developed by many researchers and there are many versions of its axiomatization. Ideas and methods of quantum computing and quantum cryptography have transformed research in the foundation of quantum mechanics into an applied discipline with a potential for engineering applications. Many new mathematical structures and methods have been invented.
Work on Hilbert's Sixth Problem involves many areas of mathematics: mathematical logic, algebra, functional analysis, differential equations, geometry, probability theory, theory of algorithms, and many others. It remains one of the most seminal areas of interdisciplinary dialog in mathematics and mathematical physics.
The proposed workshop aims to gather top experts in Hilbert's Sixth Problem, to review the current achievements in the solutions of this problem, and to formulate the main mathematical challenges and problems which have arisen from 115 years of work. This renewed programmatic call should be disseminated and explained together with modern achievements to interdisciplinary research community, to a new generation of mathematicians, and to the public so that they can better appreciate the contribution of mathematics to science and technology.
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.le.ac.uk |