Fragment of the letter from V.I.Arnold to V.A.Vassiliev
intended for the seminar
19.02.98
I do not know who is at Moscow at present, but
Sedykh has a list of my suggestions, to which I might add
some more, for instance:
my papers in FA 1997 and 1998 were not discussed at the seminar:
the first might be reported by Panov, the second by Sedykh (who
should compare it with the papers by Romero-Fuster et al).
Panov himself have nice results on surfaces of any genus.
There is an interesting paper by Biran on symplectic packings
on 4-manifolds (relations to continued fractions): for
instance, for any compact sympl.4-mnfld there exists an N such
that any larger # of equal balls of appropriate sise cover
it almost completly (for the disc it starts from 9 balls).
He is and you might ask for the file.
There is a preprint of Benjamin McCay "Special Lagrangian
Characteristic Classes" relating Borel-Fuks (his spelling !)
and Kazarian (UMN 95:4).
I think you got the paper by Napolitano on config.spaces.
It might be reported by Pushkar (it implies the 10 diameters result for
the 2-torus)>
The Paris Seminar has now its homepage related to the Moscow one at
and we are plunning to put there abstracts (up to now- just titles)
and files of preprints.
The Kontsevich talk (on his work with Barannikov) was EXTREMLY
interesting, providing an explanation of the whole Saito flat
coordinates and Dubrovin Frobenius mnflds story.They are writing
on this with Serezha and are and .
I have written a paper on the realization of g(x,y) as of the
curvature of z=f(x,y).The file shall be on the homepage soon.
(* it is already on http://www.botik.ru/~duzhin/arnold/ -- S.D. *)
And you have perhaps the paper by Ferrand on the arrangements
on circles in the plane (related to the J_- invariants): he define
some commutative semigroup of invariants , whose stabilized version is
the Grothendiek group Z+Z generated by ind and J_-, but the semigroup
itself (related to the Viro formula for J_- via resolution) being
not well understood- the subject deserves a discussion.
Of course, it is
even more important to report what Seminar people have just
discovered, conjectures and so on. I shall try to prepare the
usual list of problems (which I discussed at the Paris Seminar
yesterday).
By the way: please ask the Seminar (Anisov,Sedykh,Kazarian,...)
what is known on the boundary of the set of those curves in R^3
(or in RP^3) which have a convex projection from some point
(which depends on the curve).
... The most interesting thing, however, is to understand what are the
complexifications of the finite order invariants. One (perhaps
misleading, but interesting per se) way is to consider the space of loops
and to call the discriminant the set of loops having the points on the
complex hypersurface. Say, for a point in C we get the first order
invariants of a loop in the component to be a ind + b and what we
classify is essentially the braids with two strings space Z.
The braids are indexed by i=ind of the loop in the complement.
The invariants are functions on Z. The finite order
invariants are polynomials of corresponding degree in i.
This is good, of course, but in my philosophy the complex
hypersurface selfintersection strata should appear somewhere,
and it seems not to be the case in this loops approach.
I prefer rather to think of the complexified invariants as of
cohomology classes of dim=1, that is of real hypersurfaces of
the complement, their boundary lying on the hypersurface.
The jump is then rather the local intersection of two branches
of this hypersurfaces near the selfintersection stratum: a
codimension two surface (the boundary lying on the stratum).
It takes values on the two-tori linked with the real 2D
trace of the hypersurface on the real 4D transversal to the
selfintersection stratum.
Thus the complexified invariants ring should be rather
some kind of (co)homology graded ring, containing objects of
different dimensions! I guess it is related to the mixed
Hodge structure, but there are too much free choices in
the definitions, and we should perhaps have an open competition
to find the best choice. you might just put the mathematical part of
this message on the homepage to open this competition!
Best
V.Arnold