Hilbert Problems

for the Geosciences in the 21st Century

M. Ghil

Dept. of Atmospheric Sciences & IGPP, UCLA

http://www.atmos.ucla.edu/tcd/

Here are some interesting problems in the Geosciences for the 21st century:

  1. What is the coarse-grained structure of low-frequency atmospheric variability, and what is the connection between its episodic and oscillatory description?
  2. What can we predict beyond one week, for how long, and by what methods?
  3. What are the respective roles of intrinsic ocean variability, coupled ocean-atmosphere modes, and atmospheric forcing in seasonal-to-interannual variability? What are the implications for climate prediction?
  4. How does the oceans’ thermohaline circulation change on interdecadal and longer time scales, and what is the role of the atmosphere and of sea ice in such changes?
  5. What is the role of chemical cycles and biological changes in affecting climate on slow time scales, and how are these affected in turn by climate variations?
  6. What can we learn about these problems from the atmospheres and oceans (if any) of other planets and their satellites?
  7. Given the answer to the above questions, what is the role of humans in modifying climate, and can we achieve enlightened climate control of our planet?

 

Mathematical Problems

Lecture delivered before the

2nd International Congress of Mathematicians at Paris in 1900

by Prof. David Hilbert

 

Introduction. Philosophy of problems, relationship between mathematics and science, role of proofs, axioms and formalism.

"Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading [geoscientific] spirits of coming generations will strive? What new methods and new facts in the wide and rich field of [geoscientific] thought will the new centuries disclose?"

 

Problem 1. Cantor's problem of the cardinal number of the continuum. (The

continuum hypothesis.)

K. Gödel. The consistency of the axiom of choice and of the generalized continuum hypothesis. Princeton Univ. Press, Princeton, 1940.

Problem 2. The compatibility of the arithmetical axioms.

Problem 3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes.

V. G. Boltianskii. Hilbert's Third Problem Winston, Halsted Press, Washington, New York, 1978; C. H. Sah. Hilbert's Third Problem: Scissors Congruence. Pitman, London 1979.

Problem 4. Problem of the straight line as the shortest distance between two points. (Alternative geometries.)

Problem 5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. (Are continuous groups automatically differential groups?)

Montgomery and Zippin. Topological Transformation Groups. Wiley, New York, 1955; Kaplansky. Lie Algebras and Locally Compact Groups. Chicago Univ. Press, Chicago, 1971.

Problem 6. Mathematical treatment of the axioms of physics.

L. Corry: "Hilbert and the axiomatization of physics (1894—1905)" Arch. History Exact Sciences, 51 (1997).

Problem 7. Irrationality and transcendence of certain numbers.

N.I.Feldman. Hilbert's Seventh Problem (in Russian), Moscow State

Univ, 1982, 312pp. MR 85b:11001.

Problem 8. Problems of prime numbers. (The distribution of primes and the Riemann hypothesis.)

Problem 9. Proof of the most general law of reciprocity in any number field.

Problem 10. Determination of the solvability of a diophantine equation.

S. Chowla. The Riemann Hypothesis and Hilbert's Tenth Problem.

Gordon & Breach, New York, 1965; Yu. V. Matiyasevich. Hilbert's Tenth Problem. MIT Press, Cambridge, Mass.,1993, available on the web.

Problem 11. Quadratic forms with any algebraic numerical coefficients.

Problem 12. Extension of Kronecker's theorem on abelian fields to any algebraic realm of rationality.

R.-P. Holzapfel. The Ball and Some Hilbert Problems. Springer-Verlag, New York, 1995.

Problem 13. Impossibility of the solution of the general equation of the 7-th degree by means of functions of only two arguments. (Generalizes the impossibility of solving 5-th degree equations by radicals.)

Problem 14. Proof of the finiteness of certain complete systems of functions.

M. Nagata. Lectures on the 14th Problem of Hilbert. Tata Institute of Fundamental Research, Bombay, 1965.

Problem 15. Rigorous foundation of Schubert's enumerative calculus.

Problem 16. Problem of the topology of algebraic curves and surfaces.

Yu. Ilyashenko & S. Yakovenko, eds. Concerning the Hilbert 16th Problem. American Mathematical Society, Providence, R.I., 1995; B.L.J. Braaksma, G.K. Immink, & M. van der Put, eds. The Stokes Phenomenon and Hilbert's 16th Problem. World Scientific, London, 1996.

Problem 17. Expression of definite forms by squares.

Problem 18. Building up of space from congruent polyhedra. (n-dimensional crystallography groups, fundamental domains, sphere packing problem.)

Comments on the theory of analytic functions.

Problem 19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?

Problem 20. The general problem of boundary values. (Variational problems.)

Problem 21. Proof of the existence of linear differential equations having a prescribed monodromic group.

Problem 22. Uniformization of analytic relations by means of automorphic functions.

Problem 23. Further development of the methods of the calculus of variations.

Final comments.

"The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!"

Thanks to David E. Joyce, Clark Univ., Worcester, Mass.; Fabio D’Andrea, LMD/ENS, Paris; & K. Ide, UCLA.

The Pace of Progress in Climate Dynamics

1. 1970s - Aerosols up (some)

- Temperatures down (some)

- Future: down a lot?

2. 1980s - Greenhouse gases (GHGs) up (a lot)

- Temperatures up (a little)

- Future: up a lot?

3. System - complicated - many feedbacks,

both positive and negative

- nonlinear - small pushes, big effects?

  1. The answer for (D T)/(D CO2) has not changed much in 30 years or so, and it has gone the wrong way:

    2. (1.5—4.5 K)/(2*CO2) then to [(-? K)—5 K]/(2*CO2) now!

  2. The question does deserve a good answer, but it’s slow in coming.

http://www.atmos.ucla.edu/tcd/


"Waves vs. Particles" in Atmospheric Low-Frequency Variability

  • Are the regimes but slow phases of the oscillations?

    • Kimoto & Ghil

    (JAS, 1993 a, b)

     

  • Are the oscillations but instabilities of particular equilibria?

    • Legras & Ghil

    (JAS, 1985)
  • How about both: "chaotic itinerancy" (Itoh & Kimoto, JAS, 1999)
  • How about neither? Null hypotheses:

    • a) It’s all due to interference of linear waves, e.g.,

    neutrally stable Rossby waves;

    Lindzen et al.

    (JAS, 1982)

    b) It’s all due to red noise – Hasselmann (Tellus, 1976), Mitchell (Quatern. Res., 1976), Penland & co. (Magorian, Sardeshmukh, 1990’s).

     

    Concluding Remarks

    1. There are some good questions left in the Geosciences for the 21st century: not too easy, so as to be stimulating, and not too hard, so as to be attractive.
    2. We’ll have better tools: computers, satellites, instruments, lab devices, etc.
    3. We need a few more good, young people to make the best use of these tools and to sharpen them further.
    4. To get this help, we need to make the Geosciences intellectually exciting, as well as relevant to society.
    5. The degree of historical maturity of a broad area of the sciences, like the Geosciences, is highly correlated with its mathematical sophistication.
    6. This sophistication can be measured by
    1. the predictive power of the area’s theories;
    2. its use of the most advanced mathematical tools;
    3. its creating new & even more powerful tools.