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 Chernov Prize
10.01.2008
The Foundation for the Advancement of Research in Finsler Geometry
announces a special prize
for solving the following mathematical problem:
In a linear four-dimensional Finsler space with a Chernoff metric function, it is necessary to construct transformations that could have a physical interpretation of transitions from one timelike worldline in the form of a straight line to another similar one.
The metric of Chernoff space in an isotropic basis has the form of a symmetric polynomial in four variables of the third degree:
S3=x1x2x3+x1x2x4+x1x3x4+x2x3x4.
In a basis similar to the orthonormal one, obtained by the following linear transformation of the isotropic basis:
x1=ct+x+y+z,
x2=ct+x-y-z,
x3=ct-x+y-z,
x4=ct-x-y+z
The Chernoff space metric takes the form:
S3=4ct(c2t2-x2-y2-z2)+8xyz.
To be awarded the prize, the solution must be presented on the forum pages:
http://www.scientific.ru/dforum/altern
and found satisfactory by the jury, represented by V.M. Chernov.
The prize amount is 25,000 (twenty-five thousand) rubles.
The competition period is until December 31, 2008.
If desired, the author will be given the opportunity to publish the solution in the journal "Hypercomplex Numbers in Geometry and Physics."
 | Finsler Prize
10.01.2008
Non-Commercial Foundation for Finsler Geometry Research
and Moscow Bauman State Technical University
establish the Prize for the solution of the following problem:
“Construct a unified geometrical theory
of the gravitational and electromagnetic fields
on the base of
the 4-dimensional Finsler space
with the Berwald-Moor metric,
or prove the impossibility of such theory”.
The goal of the Constitutors of the Prize is to stimulate the research exploiting the hyper complex algebras as the universal codes of Geometry and Physics. First of all, the Constitutors are interested in the research connected with the poly-numbers – the commutative associative algebras – as the natural generalizations of real and complex numbers preserving their main arithmetic properties. Recently, it turned out that the poly-numbers are closely connected with various Finsler geometries [5]. These geometries are the generalizations of Riemannian geometries that are the base of General Relativity Theory and other modern geometric field theories. One of the poly-numbers’ classes leads to the Finsler geometry with Berwald-Moor metric (ds4 =
dξ1dξ2dξ3dξ4). Its basic invariants have the powers higher than two, and this makes the difference with the Riemannian and other common geometries. The change of the quadratic metric to the higher order one implies the qualitatively new geometrical ideas, and the Constitutors express the belief that this opens the unexpected and fruitful perspectives in fundamental Physics.
Although Finsler geometry did not demonstrate its advantages over Riemannian constructions in Physics for almost a hundred years of its existence, it is this geometry that promises a lot now. This is due to the generalization of one of the main objects in many geometries – the scalar product which becomes not the bilinear symmetrical form but the poly-linear form [4]. The natural consequence of such generalization is the need to pass from the common two-index Finsler metric tensor depending on point and direction to its poly-index generalization depending only on point.
It is known that the attempts to unify several fundamental interactions on the base of geometry with the Riemann metric (Einstein, Weyl [1]) failed, while the more successful ones (Kaluza-Klein, superstrings, branes etc.) use more than four dimensions. If the problem of the unification of the gravitation and electromagnetism appears to be solvable in four dimensions on the base of a Finsler metric, this fact will undoubtedly stimulate both the Finsler geometry research in Physics and the theory of hypercomplex numbers in Mathematics.
According to the competition conditions, the fundamental metric should be that of Berwald-Moor. This metrics provides the limit transitions not only to the Galilean space but also to the Minkowsky space, thus, making probable the corresponding unified field theory contain both classical and relativistic Physics. Besides, the geometry with this metric can be classified as poly-metric one, and the role of poly-metric geometries among all Finsler geometries was underlined by P.K.Rashevski [3]. This role is partly revealed in the case of complex plane that is an example of a bimetric space.
Competition conditions
1. It is principal that the competitive paper provides the following:
- the unification of the above mentioned fundamental interactions must be purely geometrical (that is must go along with Einstein, Weyl, Kaluza theories) and ensue from the same principles;
- the basic space-time geometry arises from the change of Minkowski metric to the Finsler type Berwald-Moor metric which has the form of (ds4 =
dξ1dξ2dξ3dξ4) in a special basis;
- the space-time must be 4-dimensional;
- the author who proves the impossibility of such theory can also apply for the Prize.
| The third students competition
30.04.2006
In 2006 the third All-Russia competition of student’s abstracts on a theme
“Hypercomplex numbers and their relation with the geometry of linear Finsler Spaces” appears.
The purpose and conditions of competition:
| Second Student Competition
15.06.2005 The results of the Second All-Russian Student Essay Competition on the topic
"Hypercomplex Numbers and Their Connection with the Geometry of Linear Finsler Spaces" have been announced.
1st place - V. V. Mnogoletniy, Faculty of Management and Management of Moscow Institute of Physics and Technology
2nd place - I. V. Kutuzov, Ulyanovsk State University
3rd place - V. A. Ivanov, Faculty of
Mathematics and Informatics, Krasnoyarsk State University (first year)
4th place - V. V. Serkin, Institute of Mathematics and Economics, Irkutsk State University.
All laureates received the right to visit Moscow in July 2005 to attend the international PIRT Conference (July 4-7, 2005) and meet leading experts in Finsler geometry. Travel and three-day accommodations are covered.
In addition, the top three winners received prizes. The first place winner received an annual stipend of 5,000 rubles per month, the second place winner received 3,000 rubles per month, and the third place winner received 2,000 rubles per month.
| The second student competition
25.10.2004 In 2005 the second All-Russia competition of student’s abstracts on a theme
“Hypercomplex numbers and their relation with the geometry of linear Finsler Spaces”
appears.
The purpose and conditions of competition:
| Results of the Second Competition (2003–2004)
25.10.2004
Dear colleagues!
In summing up the results of the Second Competition on Hypercomplex Numbers and Related Spaces, despite the reduced number of participants, I would like to note the increased professionalism of the submitted papers. Unfortunately, due to the organizers' fault, the competition conditions did not sufficiently emphasize that the competition was intended to consider papers exclusively dealing with spaces whose metric functions cannot be reduced to ordinary quadratic relations. As a result, for purely formal reasons, we had to disqualify a number of very interesting and insightful papers, which undoubtedly represent a significant contribution to the development of Riemannian geometry. Of the six papers accepted for the competition, the jury considered the following to be the best and deserving of the announced prize:
1. G. S. Asanov, "Finsleroid Space Equipped with an Angle."
2. V. V. Kassandrov, "Algebrodynamics: Pre-Light, Caustic Particles, and Time Flow."
3. G. I. Garas'ko, "Generalized Analytic Functions of a Polynumerical Variable."
4. S. V. Lebedev, "Surface of Simultaneity in H3."
The prize fund was distributed among the winners in full. No distinction was made between the entries based on the prize placement, although the prize amounts varied somewhat.
To allow website visitors to express their opinions on the jury's decision, all competition entries are posted on a separate page.
| Terms of the Second Competition for the Best Research Paper on the Topic "Associative-Commutative Hypercomplex Numbers and Their Applications"
25.10.2004
Summing up the results of the past competition, we are pleased to note that the proposed topic found an interested audience. I would like to express my sincere gratitude to everyone who participated in the competition or showed interest in its content. Although the number of participants was relatively small, it became clear that, on the one hand, the problem is relevant and requires further development, and on the other hand, it requires greater specificity.
The works submitted to the competition touched on a wide variety of aspects of hypercomplex number theory. It should be noted, however, that none of them solved the main goal: to clearly and convincingly demonstrate the right of associative-commutative hypercomplex numbers to occupy a significant place in mathematics on par with real and complex numbers. In this regard, the competition committee decided not to award the first prize in 2002 and to add the remaining undistributed portion of the prize fund to the fund allocated for the next competition on a similar topic.
Second prize was awarded to research that, in the opinion of the competition committee members, best met the competition conditions announced a year ago. However, this does not mean that the papers that were not awarded did not contain interesting ideas, and the authors are welcome to implement them in the Second Competition.
The start of the Second Research Competition on Associative-Commutative Hypercomplex Numbers and Their Applications is hereby announced, as well as the establishment of a prize fund of 500,000 (five hundred thousand) rubles.
Competition Terms:
| Results of the First Competition (2002–2003)
13.10.2004
First prize was not awarded.
Second prize was awarded to the following papers:
1. S. V. Lebedev, "Some Properties of Associative-Commutative Hypercomplex Numbers."
2. A. I. Aleksandrovich and S. V. Skubilin-Burtsev, "Special Algebraic Structures and Spatial Problems of Mathematical Physics."
3. G. I. Garas'ko, "Three-Numbers Whose Norm Cube Is a Non-Degenerate Three-Form."
The remainder of the prize fund, $5,400, will be used to increase the competition fund, which will be announced in 2003.
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