Arnold's seminar
September 12th, 2000

V.I.Arnold
Plans of the seminar for autumn semester, 2000

Among other things, we are planning to discuss the following works
and topics.

1. Ya.Dymarski. Membranes, ellipsoids and symmetric operators with 
multiple eigenvalues.

2. P.Biran. 
   Symplectic packing in dim 4. Geom. Funct. Anal. 1997:7, 420--437.
   Constructing new ample divisors out of flows. 1997 (?)

3. R.Low. Casuality in terms of linking in light ray space.
E.g. in: Classic and Quantum Gravity 7 (1990), p.177,
11 (1994), p. 453.
These papers and pther e-prints related to this topic are available from 
Robert Low's home-page: http://www.mis.coventry.ac.uk/~mtx014/.

4. M.Haiman. Papers where the number (n+1)^{n-1} appears
in different siuations, E.g. J.Alg.Geom 3 (1994), 17--76,
J.Comb.Th. 82 (1998), 74--111.
These papers and other e-prints related to this topic are available 
from Mark Haiman's home-page: http://math.ucsd.edu/~mhaiman/.

5. Old works of Swedish mathematicians and astronomers on continued 
fractions. Do they contain a proof of Kuzmin's probability theorem, 
for example?
A.Wiman. Ueber eine Wahrscheinliche Aufgabe bei Kettenbruchentwicklungen.
Abh. Foehr (?) Stockholm 57 (1900), S. 589--846.
H.Gylden. Fractions continues. CR Acad Sci Paris 197 (1888), 1584--1587.

6. S.Newcomb. Note on the frequency of digits.
Amer Math. Monthly 4 (1881), 39--40.
[Note. There's an error here, because AMM begins in 1894.
Can anybody find the correct reference?]
 
7. Recent developments related to Lagrange's theorem about
continued fractions. For example, Tsuchiashi--Korkina's theorem.

8. Statistics of numbers that appear in the expansions into continued
fractions. For example, 
   take a random number (p,q) \in Z^2 with p^2+q^2<=R^2, 
   write p/q = a_1+1/(a_2+...),
   for a fixed n compute the probability of a_i=n as i -> \infty
   take the limit as R -> \infty
Conjecture. The limit is the same Gaussian number p_n which gives
the corresponding probability for almost all irrational numbers.

9. V.Chernov. New results on Vassiliev invariants.

10. Complexity according to J.Milnor (ask A.V.Chernavsky).

11. Theses of Napolitano, Ricardo Uribe, Mauricio Garay.