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\title{Polymathematics : \\ is mathematics a single science or a set of arts?}
\author{V.I.~Arnold\\
	{\small Steklov Mathematics Institute, Moscow} \\ 
	{\small CEREMADE, Universit\'e Paris-Dauphine}
	\thanks{Partially supported by the Russian Basic Research Foundation,
		project 99-01-01109, and by the Institut Universitaire de France.}
	\thanks{The author thanks I.~Mazella and J.-O.~Moussafir for the typing help.}}
\date{}

\begin{document}

\maketitle

\hspace{1.6in}
\parbox[b]{3in}{\it
	The shaft becomes too deep \ldots
	and if new veins would not be discovered,
	places for geometry in the Academy
	will become what already are
	the Arabic chairs at the Universities.

	Lagrange to D'Alembert, 1781}

\vspace{40pt}

All mathematics is divided into three parts: cryptography (paid for by CIA,
KGB and the like), hydrodynamics (supported by manufacturers of atomic
submarines) and celestial mechanics (financed by military and by other
institutions dealing with missiles, such as NASA.).

\medskip
Cryptography has generated number theory, algebraic geometry over finite
fields, algebra \footnote{The creator of modern algebra, Vi\`ete, was the
cryptographer of King Henry~I\/V of France.}, combinatorics and computers.

Hydrodynamics procreated complex analysis, partial derivative
equations, Lie groups and algebra theory, cohomology theory and 
scientific computing. 

Celestial mechanics is the origin of dynamical systems, linear algebra, 
topology, variational calculus and symplectic geometry.

\medskip
The existence of mysterious relations between all these different domains
is the most striking and delightful feature of mathematics 
(having no rational explanation).

The experience of the past centuries shows that the development of
mathematics was due not to technical progress (consuming most of the efforts
of mathematicians at any given moment), but rather to discoveries of
unexpected interrelations between different domains (which were made
possible by these efforts).

The precious technical reports of the present state of different
mathematical domains reminds us of the trench warfare.
The description of the front line with its bends and everyday moves 
is of course highly important for the battle participants. 

But the pernicious character of the separating 
tendencies (to which the growing specialisation of mathematicians and the
fragmentation of mathematics into small domains leads) becomes 
evident when one tries to understand the development of mathematics in the 
past with all its meanders.

\medskip
Sylvester (1876) already described as an astonishing intellectual
phenomenon, the fact that {\it general statements are simpler than their
particular cases}. The antibourbakist conclusion that he drew from this
observation is even more striking. According to Sylvester, {\it a
mathematical idea should not be petrified in a formalised axiomatic setting,
but should be considered instead as flowing as a river}. One should always
be ready to change the axioms, preserving the informal idea.

\medskip
Consider for instance the idea of a number. It is impossible to discover
quaternions trying to generalise real, rational, complex or algebraic number fields.

The possibility of such {\it informal generalisation} of all mathematics, for
which we have no ready axioms, seems to me the most appealing dream.

The informal complexification, quaternionization, symplectization,
contactization etc., described below, are acting not on such small things, 
as points, functions, varieties, categories or functors, but on the whole of mathematics.

\medskip
I have successfully used these ideas many times as a method to guess
new results. I hope therefore that in the future this method of the multiplication of
mathematics will be as standard, as is now the transition from finite-dimensional
linear algebra to the theory of integral equations and to functional analysis.

\medskip
Perhaps, the simplest example of this multiplication paradigm is provided by the
Killing-Coxeter theory of the reflection groups. Linear algebra is essentially
the theory of the special roots systems $A_k$. The basic facts of linear algebra
(like the eigenvalues and the Jordan blocks theory) can be reformulated in terms
of the roots, making the statements meaningful for other systems of roots. These
new statements miraculously occur to be correct (while suitably modified).
The theories of the roots systems $B_k$, $C_k$, $D_k$ (corresponding to the
Euclidean and symplectic spaces geometry) are from this point of view {\it rather
the sisters than the daughters} of the usual vector space geometry (even though they are, 
of course, the geometries of the usual vector spaces endowed with additional structures).

\medskip
By the way, many of the results of these theories are in fact also true not
only for the exceptional crystallographic groups ($E_6$, $E_7$, $E_8$, $F_4$, $G_2$) but
also for the non-crystallographic Euclidean reflection groups ($I_2(p)$, $H_3$, $H_4$). 
For example, starting from the ideas of O.~Scherbak (1985), A.B.~Givental (1988)
discovered in the theory of Lagrange and Legendre projections in symplectic
and contact geometry a geometrical problem, whose solutions are in a natural 
one-to-one correspondence with the Euclidean reflection groups
(and not just with the crystallographic ones as in my preceding theory).
This is the problem of classification of the simple projections (having no continuous
invariants) of the (virtually singular) Lagrange and Legendre subvarieties.

\medskip
The main applications of the mathematics multiplication idea are not 
to be found in finite dimensional algebra but rather in infinite-dimensional 
calculus, where the Killing classification of the simple Lie algebras is 
replaced by the Cartan classification of the simple Lie pseudo-groups.

For example the symplectization idea suggests that all differential
geometry and topology notions and results should have symplectic 
versions -- the symplectic geometry and the symplectic topology.

In some cases these versions are quite obvious. The ordinary vector fields
correspond to the Hamiltonian ones, their Poisson brackets algebra to the 
Poisson brackets of the Hamilton functions, the strings should move from 
the configuration space to the phase space and so on. In other cases this 
generalisation is less evident. For instance, the submanifolds of the ordinary
geometry correspond to the Lagrangian submanifolds of symplectic manifolds
(A.~Weinstein's Principle).

\medskip
Trying to symplectize the Euler-Poincar\'e theorem on the sum of the
indices of the singular points of vector fields, I was lead in 1965 to the 
conclusion that the symplectization should be an astonishing extension of 
the Morse theory. I formulated ``Arnold's conjectures'' on the fixed points of
symplectomorphisms, on the intersections of the Lagrangian submanifolds
and on the linkings of the Legendrian ones.

The simplest of these conjectures states that {\it the number of fixed points of
an area preserving diffeomorphism of a two-torus to itself, ``preserving the mass centre'',
has at least $4$ fixed points (taking the multiplicities into account)}.

In the more general case of a compact symplectic manifold, the 
diffeomorphism should be the time one map of a flow defined by time-dependent 
Hamilton function, and the number of fixed points is minorated
by the minimal number of the critical points of a function on the manifold
(the fixed and critical points being counted either both with their multiplicities, 
or in both cases geometrically).

These conjectures, generalising the ``last theorem'' of Poincar\'e on the
circular annulus mapping, have been later studied in a series of brilliant 
works of many authors (Ya.~Eliashberg, P.~Rabinowitz, C.~Conley and E.~Zehnder, 
M.~Chaperon, J.-C.~Sikorav, F.~Laudenbach, Yu.~Chekanov, A.~Floer, H.~Hofer, 
C.~Viterbo, A.~Weinstein, D.~Salomon, A.~Givental, M.~Gromov, and others). 
{\it Quantum and Floer cohomologies} and Gromov's {\it pseudoholomorphic curves} theories are 
well-known byproducts of this development.

The {\it Lagrange intersection theory} is a far-reaching extension of the Morse theory,
replacing the functions by such genuinely multivalued function as $\sqrt x$.
Another extension replaces the functions by the closed $1$-forms and 
Morse theory by Novikov's complex. The corresponding version of the Lagrange
submanifolds intersection theory is due to J.-C.~Sikorav.

\medskip
I have heard that my initial conjectures, which triggered all these theories, are now
proved (by Fukaya, Ono, Salomon, Ruan and others). Unfortunately I was unable to
understand the technical details of these proofs. Kontsevich was unable to
explain these details to me, while all the proofs are based on his theory of stable
mappings of curves.

\medskip
As far as I understand, all these proofs minorate the number of fixed
points by the sums of the Betti numbers while I conjectured that it is 
minorated by the Morse number (or by the minimal number of geometrically 
different critical points if the fixed points are counted geometrically).

In my dreams of the 60s the symplectomorphisms fixed points number
minoration was followed by the similar study of the symplectic correspondences, 
which are not graphs of symplectomorphisms.

My idea of symplectic and contact topology was from the very beginning
different from that of Gromov and Eliashberg,
who were the first to explore the new domains discovered by the 
symplectization and contactization methods.

In their opinion, the symplectic (or contact) topology objects should have the
``symplectic (or contact) homeomorphism invariance'' property : they should persist under the
$C^0$-small symplectic (or contact) diffeomorphisms.

To me the word  {\it ``topology'' with any adjective } is the study of
the {\it discrete} invariants of the continuous objects of the corresponding branch of
geometry, be they homeomorphism invariant or not.

\medskip
Thus, I include into {\it projective topology} the {\it M{\"o}bius theorem
on the three inflection points of a projective line deformations} (claiming that a
noncontractible circle, embedded generically into the real projective
plane, has at least three inflection points, the dual curve having at least three cusps).

A recent conjecture in projective topology (due to F.~Aicardi and D.~Panov)
extends the M\"obius theorem to the generic surfaces in $\rp^3$, smoothly close to a
plane. The conjecture claims, that {\it the parabolic line of such a surface is
tangent to the asymptotic direction  of the surface at least at $6$ points
(generating $6$ swallowtails of the dual surface) and that there are at least $4$ parabolic
lines if there are only $6$ tangency points}.

\medskip
The recent spectacular achievements of {\it contact topology} include the
creation by Yu.~Chekanov (and also by Ya.~Eliashberg and H.~Hofer) of the {\it contact
homology} theory. One may hope that the further development of these ideas would
provide the proof of the conjectures on the necessity of four cusps of any wave
front eversion in the plane.

This conjecture extends to the genuine multivalued function (of $\sqrt x$
type) the {\it Sturm theorem on periodic functions}, which itself extends the Morse
theory of critical points to higher derivatives. This theorem claims that {\it the
number of zeros of a function on a circle is minorated by the number of zeros of its first
non vanishing Fourier harmonic} (providing a confirmation of the general
topological economy principle of algebraic objects). 

The relation of all this domain to cyclic homology and characteristic classes, 
discovered by M.~Kazarian, provides some hope that Morse theory can be extended 
to the higher derivatives (or to the more general case of nonholonomic constraints).

This extension should contain the (genuine) multivalued functions Morse
theory -- the theory of Lagrange intersections and Legendre linking (describing the
causality in the relativistic physics, according to R.~Low and R.~Penrose) as the first
derivative case.

Symplectic and contact topologies, created as the symplectization and the
contactization of differential topology, are today well settled domains of
mathematics.

\medskip
Another attempt to multiply mathematical results is the {\it complexification 
and quaternionization dream}. The first spectacular success of this idea was the
proof the Gudkov's conjecture in real algebraic geometry.

One of the most fundamental problems of mathematics is {\it the problem of the
topological structures of the real curves \footnote{The Russian way to formulate
problems is to mention the first nontrivial case (in a way that no one
would be able to simplify it). The French way is to formulate it in the most general form
making impossible any further generalisation.} defined by algebraic equations of degree
$n$}.

This problem has been solved for $n = 2$ by the Ancients  (ellipses, hyperbolas,
parabolas\dots). The cases $n = 3$ or $4$ were settled by Descartes and Newton. 
Hilbert has included the case $n = 6$ in his 16th problem. For $n = 8$, the answer 
is still unknown. 

These {\it real problems} are too difficult for the modern algebraic geometers
and for present day computers. 

\medskip
According to Harnack, the number of connected components of a
curve of degree $n$ in the real projective plane does not exceed $g+1$, where 
$g = (n-1) (n-2)/2$ is the genus of the curve. Curves with $g+1$ components do
exist, they were called $M$-curves ($M$ for ``maximal'') by I.G.~Petrovsky.

Hilbert announced that he had proved that only two arrangements of the
$11$ ovals of an $M$-curve of degree $6$ were possible : only one oval contains other ovals
inside its disc, the number of the interior ovals being either $1$ or $9$.

\medskip
A Nizhni Novgorod mathematician, D.~Gudkov, submitted in 1970 his Thesis,
proving that Hilbert was wrong. There exists one (and only one) more possible
arrangement, for which the number of the interior ovals is $5$. I.G.~Petrovsky asked me
to check this paper, contradicting both the Hilbert's statement and the preceding
paper by Gudkov, in which he was proving the (wrong) Hilbert's claim.

\medskip
Mistakes are an important and instructive part of mathematics, perhaps as
important a part as the proofs. {\it Proofs are to mathematics what
spelling (or even calligraphy) is to poetry}. Mathematical works do
consist of proofs, just as poems do consist of characters.

Leibniz started his calculus studies from  the formula $d(uv) = dudv$. 
Cauchy, the $\epsilon-\delta$ inventor, proved in his calculus course the 
continuity of the limit of the (nonuniformly) convergent series of continued
functions. Lagrange's mistake in the linear ODE's theory hampered the
development of linear algebra and of Jordan's form theory. The story of the
Poincar{\'e} conjecture started from his confusion of homotopy with homology.

His ``New methods of celestial mechanics'' are the by-product of the attempts to
prove a wrong statement (the by-product -- the creation of the Dynamical Systems
theory -- being by far more important than this wrong statement ``solving'' a prize problem).
The Burnside celebrated theorem on the groups of order $p^a q^b$ was first
formulated by him wrongly. Leray told me, that his works on the hyperbolic PDE's were
motivated by the remarkable paper of Petrovsky, who wrongly used the triviality of the
cotangent bundle of every sphere (the paper was later modernised by
Atiyah, Bott and G\aa rding).  Kolmogorov's initial definition of the entropy of a
dynamical system was wrong, as well as the Pontriagin's and Rokhlin's calculations of
homotopy groups of spheres. I would be able to provide dozens of more recent examples of
mistakes in celebrated papers if I did not fear for my life. I shall only mention
a wrong symplectic reduction theorem in the first edition of the ``Mathematical
Methods of Classical Mechanics'' (appendix 5,B).

\medskip
The mistake of Hilbert, discovered by Gudkov, has lead to the foundation of the
modern real algebraic geometry. Trying to understand Gudkov's very complicated
paper, I have observed that in all his examples of $M$-curves of
degree $2k$, bounding orientable surfaces $B$, the following {\it ``Gudkov congruence''}
for the Euler characteristics holded
$$ \chi (B) = k^2 {\rm mod} \ 8 . $$
Knowing the importance of the congruences  mod $8$ in the topology of
$4$-manifolds, I have decided to replace the real surface $B$ by its 
complexification of real dimension four.

The problem arose : {\it how to complexify a manifold with boundary ? }
This is a standard difficulty : the complexification is an informal operation for
which there are no axioms, we should try to guess.

\medskip
After several attempts I came to the following conclusion. Consider the real
manifold with boundary, defined by the inequality $f(x) \geq 0$, where $f$
is a real function. To complexify it, replace the inequality by the equality 
$f(x) = y^2$. We guess, that {\it the complexification of the real surface 
$B$ with boundary $f = 0$, is the two-fold covering  of the complement in 
$\cp^2$ of the Riemannian surface $f = 0$ (which complexifies the boundary curve), 
ramified along the Riemannian surface.}

Applying the $4$-dimensional topology result to this $4$-manifold, I was able to
prove the Gudkov congruence mod $4$ (Rokhlin later extended it to the
mod $8$ case).

\medskip
>From this moment on, the real algebraic geometry entered the modern mathematics. 
In a series of brilliant works by Kharlamov, Nikulin, Viro, Shustin, Polotovski,
Khovanski, Orevkov and others, a substantial progress was achieved. The
interrelations of this domain to the ``{\it geometry of the formulae}''
(that is to the toric varieties and Newton polyhedra theory), to the 
``fewnomials'' of Kushnirenko-Sevastianov-Khovanski and even to mathematical 
logic have been discovered.

\medskip
Applying these ideas, I once suggested to I.M.~Gelfand that they might
simplify the complicated formulae of representation theory and of the hypergeometric
functions theory : one should express their coefficients in terms of the convex
polyhedra geometry. Gelfand (with his collaborators) immediately used this
suggestion, finding new brilliant applications. He has always stressed (quite
correctly) that {\it mathematicians never appreciate new ideas}, only
the last step to the summit of the mountain counts in this mountain-climbing
\footnote{M.M.~Postnikov formulated an even more radical statement : 
{\it the science never accepts new ideas, it fights against them.} 
Most scientists at any given moment are working on horse-shoes and naturally 
do react negatively against the limousines ideas. See the curious attempt
of S.-T.~Yau to fight against Givental's theory in the present volume.}

\medskip
I shall now list some complexifications and quaternionizations of different
mathematical objects. The {\it trinities} (real version, complex version,
quaternionic versions) are listed below as the lines of a big informal
commutative diagram, whose verticals are (mostly nontrivial) operations, 
transforming one trinity into another. Some of these operations are described 
in the comments below.


\newlength{\arraylinespace}
\setlength{\arraylinespace}{6pt}
\newlength{\arraynegspace}
\setlength{\arraynegspace}{-2pt}

$$
\begin{array}{lccc}
(1) & 
\reel & \complexe & \quaternion \\[\arraylinespace]
(2) & 
\begin{array}{c} \text{Morse theory} \\[\arraynegspace] \text{anses attachment} \end{array} &
\begin{array}{c} \text{Picard-Lefschetz theory} \\[\arraynegspace] \text{Dehn twist} \end{array} &
\text{?} \\[\arraylinespace]
(3) &
\pi_0(\reel \setminus 0) = \entier_2 & 
\pi_1(\complexe \setminus 0) = \entier & 
\pi_3(\quaternion \setminus 0) = \entier \ \text{?} \\[\arraylinespace]
(4) &
\rp^n & \cp^n & \hp^n \\[\arraylinespace]
(5) &
\rp^{1} = S^1 & \cp^1 = S^2 & \hp^1 = S^4 \\[\arraylinespace]
(6) &
\rp^1/\aut \reel = S^1 & \cp^2/\aut \complexe = S^4 & \frac{\hp^4/\aut \quaternion}{\conj} = S^{13} \\[\arraylinespace]
(7) &
\text{Quadratic forms} & \text{Hermitian forms} & \text{Hyperhermitian forms} \\[\arraylinespace]
(8) &  
\begin{array}{c} \text{Von Neuman -- Wigner} \\[\arraynegspace] \text{eigenvalues repulsion} \end{array} &
\begin{array}{c} \text{Quantum Hall effect} \\[\arraynegspace] \text{and Berry phase} \end{array} &
\text{?}\\[\arraylinespace]
(9) &
\begin{array}{c} \text{M{\"o}bius $S^0$ bundle}  \\[-2pt] S^1 \rightarrow S^1 \end{array} &
\begin{array}{c} \text{Hopf $S^1$ bundle} \\[-2pt] S^3 \rightarrow S^2 \end{array} &
\begin{array}{c} \text{Hopf $S^3$ bundle} \\[-2pt] S^7 \rightarrow S^4 \end{array}\\[\arraylinespace]
(10) &
\begin{array}{c} \text{Monodromy of} \\[\arraynegspace] \text{a covering} \end{array} &
\begin{array}{c} \text{Curvature of} \\[\arraynegspace] \text{a connection} \end{array} &
\begin{array}{c} \text{Hypercurvature of} \\[\arraynegspace] \text{a hyperconnection?} \end{array} \\[\arraylinespace]
(11) &
w & c & p \\[\arraylinespace]
(12) &
O,\ SO & U,\ SU & Sp,\ \text{?} \\[\arraylinespace]
(13) &
\text{Tetrahedron} & \text{Octahedron} & \text{Icosahedron} \\[\arraylinespace]
(14) &
(4,4,6) & (6,8,12) & (12,20,30) \\[\arraylinespace]
(15) &
x^2 + y^3 + z^4 & x^2 + y^3 + y z^3 & x^2 + y^3 + z^5 \\[\arraylinespace]
(16) &
x^3 + y^3 + z^3 & x^2 + y^4 + z^4 & x^2 + y^3 + z^6 \\[\arraylinespace]
(17) &
(\pi/3, \pi/3, \pi/3) & (\pi/2, \pi/4, \pi/4) & (\pi/2, \pi/3, \pi/6) \\[\arraylinespace]
(18) &
A_3 \ \includegraphics{a3.eps} &
B_3 \ \includegraphics{b3.eps} & 
H_3 \ \includegraphics{h3.eps} \\[\arraylinespace]
(19) &
2(1+3+3+5)=24 & 2(1+5+7+11)=48 & 2(1+11+19+29)=120 \\[\arraylinespace]
(20) &
(2,4,4,6) & (2,6,8,12) & (2,12,20,30) \\[\arraylinespace]
(21) &
D_4 \ \raisebox{-8pt} {\includegraphics{d4.eps}} & 
F_4 \ \includegraphics{f4.eps} & 
H_4 \ \includegraphics{h4.eps} \\[\arraylinespace]
(22) & 
E_6 \ \raisebox{-10pt}{\includegraphics{e6.eps}} & 
E_7 \ \raisebox{-10pt}{\includegraphics{e7.eps}} & 
E_8 \ \raisebox{-10pt}{\includegraphics{e8.eps}} \\[\arraylinespace]
(23) &
\complexe[t] & \complexe[t, t^{-1}] & \complexe[t, t^{-1}, (1-t)^{-1}] \\[\arraylinespace]
(24) &
\text{Numbers} & \text{Trigonometric numbers} & \text{Elliptic numbers} \\[\arraylinespace]
(25) &
H & K & Ell
\end{array}
$$

Few comments to this self-explanatory table follow.

\medskip
(2) $ \rightarrow$ (3) : {\it Morse theory} describes the modification of
the real level hypersurface of a smooth function, due to the jump of the noncritical
value from one of the two connected components of the noncritical values set to
the other. {\it Picard-Lefschetz theory} describes the Dehn twist of the 
complex level hypersurface of an holomorphic function, due to the motion of the
noncritical value around the critical one.

\medskip
(5) $\rightarrow$ (6) : I have used the diffeomorphism $\cp^2 / {\rm Conj} \approx S^4$ 
in my 1971 paper on real algebraic geometry as a well known fact. But
Rokhlin told me that the proof of this fact, known to Pontriagin in the 30s, had never
been published. I therefore asked experts (including Kuiper, who was visiting Moscow 
-- which I was unable to leave -- for the IMU buisness) whether
they were aware of this fact, and later Kuiper and Massey published their proofs.

My original (1971) proof, based on the {\it theory of the hyperbolic
PDE's}, provides also the {\it real and the quaternionic versions of the Pontriagin's
theorem} : the proofs of the three versions are identical.

\medskip
(6) $ \rightarrow $ (7) : Hermitian (hyperhermitian) quadratic forms in complex
(quaternionic) vector spaces are {\it real quadratic forms}, invariant under the
action of the multiplication of the vectors by complex numbers
(quaternions) of norm $1$. This definition is missing in the algebra textbooks.

The proof of (6) is based on the {\it hyperbolicity of the cone of the
degenerated real quadratic forms}. The cones of the degenerated hermitian
(hyperhermitian)  forms are also hyperbolic, providing such generalisations of (6), as
$$
S^{11}/U(2) = \cp^5/SU(2) = \hp^{2}/S^{1} = S^{7}, \quad S^{23}/Sp(2) = S^{13}.
$$

The three hyperbolic cones (corresponding to the real quadratic, Hermitian and
hyperhermitian forms) are in fact {\it universal} varieties, providing the
``Schur index'' of representation theory.

\medskip
(7) $\rightarrow $ (8) : The Wigner-von Newman theorem of the eigenvalues
repulsion is based on the fact that {\it the codimension of the variety of quadratic forms
with multiple eigenvalues (in the space of the quadratic forms in an Euclidean
space) is equal to $2$}. 

In my 1972 paper ``Modes and quasimodes'' I have studied the monodromy of the 
{\it eigenvectors fibration} over the complement of this
codimension two variety as well as the Hermitian case (where the
codimension of the variety of the forms having a multiple eigenvalue is equal to $3$).

As S.P.~Novikow later explained to me, the complexification of this
study implies the {\it topological theory of the (integer) quantum Hall effect.} I had also
missed in 1972 the {\it Berry phase theory, describing the natural adiabatic
connection of the eigenvectors fibration in the Hermitian case}.

The present situation of the hyperhermitian case is in a sense similar to
that of the Hermitian case in 1972. The mathematical theory is ready, 
but its physical applications are still to be named. These applications should be
interesting : the complexification of the monodromy being the connection notion, its 
quaternionization (which is also the complexification of the connection ) 
should provide a $4$-form, measuring the dependence of the complex connection 
from the complex structure.

\medskip
(8) $\rightarrow$ (9) : The restriction of the eigenvectors bundle to the {\it
link} $S^1$ of the variety of the quadratic forms having a multiple
eigenvalue (at its regular point) is the {\it M{\" o}bius bundle} (sending the boundary of
the M{\"o}bius band to its central circle). In the Hermitian (hyperhermitian) case the
transversal to the multiple eigenvalues variety forms is $\reel^3$ 
(respectively $\reel^5$), therefore the link in $S^2$ (respectively $S^4$). 
The eigenvectors bundle's restriction to this link is the {\it Hopf bundle}.

\medskip
(9) $ \rightarrow$ (10) : The ``hypercurvature'' should be a $4$-form,
providing the first Pontriagin class. Its geometrical description as of the 
{\it complexification of the curvature} might hopefully provide more 
topological information than its probable relation to the square of 
the curvature form and to the Chern-Simons theory.

The algebra of forms, generated by the Chern forms of the eigenvectors
bundles, has been studied recently by B.~Shapiro and his collaborators; 
they have also proved the degeneration of the spectral sequence, corresponding to the
stratification of the Hermitian forms space accordingly to the eigenvalues multiplicities.

The real and hyperhermitian cases seems to be unsettled in these theories.

\medskip
(1) $ \rightarrow $ (13) : It is difficult to believe that {\it the
octahedron is the complex version of the tetrahedron, and icosahedron -- of the octahedron}. I
first deduced it from the lecture of D.~Kazhdan at the 80th Gelfand's birthday
celebration at Rutgers (1993). The strange numerology below provides some
confirmation of the mysterious parallelisms of all mathematical
trinities, such as the parallelisms (1)$ \rightarrow $(13) or (1)$ \rightarrow $(22).

\medskip
(13)$ \rightarrow $ (14) : The numbers  of edges have the following property :
$$6 \ = \ 2.3 , \quad 12 \ = \ 3.4 , \quad 30 \ = \ 5.6 . $$
We recognise in the trinity $(2,3,5)$ the {\it codimensions of the
varieties of the multiple eigenvalues quadratic, Hermitian and hyperhermitian forms}. 
Of course, we have also
$$
2 = 1 + \dim \reel, \quad 3 = 1 + \dim \complexe, \quad 5 = 1+\dim \quaternion
$$
and the numbers
$$
3 = 2 + \dim \reel, \quad 4 = 2 + \dim \complexe, \quad 6 = 1+\dim \quaternion
$$
are {\it the codimensions of the varieties of the degenerated forms}.

\medskip
(13)$ \rightarrow $(15) : The rotation group $\Gamma \subset SO(3)$ of a regular
polyhedron (13) is covered twice by the ``binary group'' $\widetilde \Gamma
\subset {\rm Spin}(3) = SU(2)$, acting on $\complexe^2$. Its {\it orbit space} is the
surface of zeros of the polynomial (15).

\medskip
(15)$ \rightarrow $(16) : The simple singularities (15) are 
``{\it fenced}'' from the sea of the nonsimple ones by the ``{\it parabolic}'' 
(or ``affine $E_6, E_7, E^{''}_8$) singularities, of which the polynomials 
(16) are the simplest representatives.

\medskip
(16)$ \rightarrow $ (17) : Consider a {\it triangle which one can buy in a
stationery}, (17). The affine reflection group defined by the triangle acts
on the corresponding {\it elliptic curve}. The fencing singularity (16) can be obtained
from this action on the elliptic curve essentially by the same invariant theory
construction which builds the simple singularity (15) from the action of
the binary group of a regular polyhedron (13) on the rational curve $\cp^1$.

\medskip
(13)$ \rightarrow $ (18) : The symmetry groups (18) of regular polyhedra
(13) are generated by reflections.

\medskip
(18) $ \rightarrow $ (19) : The mirrors of a reflection group (18) of order
$|W|$ subdivide the $3$-space into $|W|$ simplicial cones, called the Weyl chambers.

The three boundary mirrors of a chamber subdivide the $3$-space into $8$ larger
pyramids -- the {\it Springer cones}. The summands in (19) represent the
numbers of the Weyl chambers in each Springer cone. Note the prime summands.

\medskip
(19) $ \rightarrow $ (20) $ \rightarrow $ (21) $ \rightarrow $ (13) : Adding
one to the numbers of line (19) one gets the {\it degrees} (20) {\it of the
invariants} of the reflection groups (21) and also the numbers of vertices, faces and
edges of the polyhedra (13).

\medskip
(22) $ \rightarrow $ (23) : The classical ``{\it folding}'', transforming
$E_8$ into $H_4$, transforms also $E_7$ into $F_4$ and $E_6$ into $D_4$
(generated by $e_1-e_2, e_2 - e_3, e_3- e_4$ and $2e_1$ instead of $e_1 + e_2$).

\medskip
(15)$ \rightarrow $ (22) : The monodromy groups of the simple singularities
(15) are the Euclidean reflection groups (22).

\medskip
(13)$ \rightarrow $(22) : The {\it J.~MacKay correspondence} describes the
Dynkin diagrams of the (extended) root systems (22) directly in terms of the
representation theory of the binary group of the corresponding regular polyhedron (13).

\medskip
(11) $ \rightarrow $ (23) : Gabrielov observed that in {\it the polylogarithmic
formulae for the characteristic classes} studied in his works with Gelfand,
Losik and Mc Pherson, the two poles case corresponded to the Chern class and the three
poles case to the Pontriagin class.

\medskip
(23)$ \rightarrow $ (24) : The Turaev-Frenkel theory of the {\it elliptic
numbers	relation to modular hypergeometric functions} is described in their paper in the
``Arnold-Gelfand Mathematical Seminars'', Birkh{\"a}user 1997.

\medskip
(23)$ \rightarrow $(25) : The trinity, consisting of {\it cohomology}, $K$-{\it
theory} and {\it elliptic theory}, has been suggested by A.B.~Givental.

\medskip
It is interesting that the complexification of a manifold is by no way unique,
depending on the structures we are interested in. Thus, the complexification
of $S^1 = \rp^1$ being $\cp^1 = S^2$ that of $S^1 = SO(2)$ is $SU(2) = S^3$. This is
nice, making it possible to complexify the M{\"o}bius bundle 
$$
S^0  \rightarrow (S^1 = SO(2)) \rightarrow (S^1 = \rp^1)
$$ 
to obtain the Hopf bundle 
$$
S^1 \rightarrow (S^3 = SU(2)) \rightarrow (S^2 = \cp^1).
$$
Trinity (6) suggests that the complexification of $S^1$ is sometimes also $S^4$.

\medskip
Trinity (3) suggests that {the complexification of $\entier_2$ is $\entier$}. This is
confirmed also by the following construction (originating from one of the attempts to {\it
complexify the braid groups} and {\it to quaternionize the permutation groups).}

\medskip
Consider the set $E$ of homotopy classes at mappings from a Lie group $G$
to itself preserving the identity. Define the (virtually noncommutative) ``addition''
in $E$ by $(f+g) (x) = f(x) g(x)$ and the ``multiplication'' by $(fg) (x) = f(g(x))$. The
resulting algebraic structure on $E$ seems to  have no name (the distributivity
holding only from one side), so I shall call this ``ring'' $E$ 
the {\it ellipse of} $G$.

The ellipse of $S^0$ is the ring $\entier_2$ of two elements, that of $S^1 =
SO(2)$ is the ring $\entier$. The ellipse of $S^3 = SU(2)$ is also $\entier$ (I would prefer to
get the Gauss numbers ring as the complexification of $\entier$).

\medskip
To complexify $\entier$ one can also consider it as the {\it died braid group}
$\ibr (2)$ where 
$$
\ibr (n) = \pi_1 (\complexe^n \setminus \complexe\text{-mirrors of } A_n),
$$ 
or as the {\it ordinary braid group} $\br(2)$, where 
$$
\br(n) = \pi_1(\text{Config}_n \complexe).
$$
The died braid group $\ibr(n)$ is the complexification of the symmetric group:
$$
\pi_0 (\reel^n \setminus \reel \text{-mirrors of } A_n) \approx S(n).
$$

\medskip
To {\it complexify the fundamental group} $\pi_1 (X)$ one is tempted
to write $\pi_1 (X) = \pi_0(\Omega X)$ and to complexify it to 
$\pi_1 (^{\complexe}\Omega \ ^{\complexe}X)$. 
The {\it complexification of the loop space} is perhaps related to the {\it
holomorphic} loops. Applying these ideas to $X = \complexe \setminus 0$, we should consider as
complex loops the polynomial mappings of $\complexe$ into $\complexe^2 \setminus 0$ (the
boundary conditions at infinity being provided by the highest degree terms).

The Zariski theorem reduces the calculation of the fundamental group of the
space of these pairs of polynomials to the calculation of the fundamental group of the
complement of a single rational curve in $\complexe^2$ (which is the image of a generic
polynomial mapping of $\complexe$ of a given degree). I guess that the final answer is
either $\entier$ or $\entier^2$, depending on whether the boundary conditions at infinity
correspond to the complexification of the died braid group $\entier \approx \ibr(2)$ 
or to the ordinary braid group $\br(2) \approx \entier$.

\medskip
The {\it quaternionic versions of the Coxeter and Shephard-Todd groups} are
still to be defined and classified. These versions might be Lie groups whose orbit
spaces are smooth, or groups generated by the ``quaternionic reflections''. One might
speculate that the existence of the sporadic simple groups is related to what remains
from the real and complex reflection groups in the quaternionic version of the reflection
groups trinity.

\medskip
The {\it complexification of the orientation} is of course a nowhere vanishing
holomorphic highest degree form. B.~Khesin and I.~Frenkel have suggested
that {\it the complexification of the cohomology theory} should be the Leray theory of
meromorphic differential forms. Using this theory, Khesin and Rosly were able to define the
{\it ``complex linking number''} of two complex curves in some complex $3$-manifolds 
(generalising the Gauss integral formula for the linking number of real
curves in the real $3$-space).

\medskip
In mathematics we always encounter mysterious analogies, and our trinities
represent only a small part of these miracles. I might mention, as an example, the
``strange duality'' of the Lobachevsky triangles, which I discovered in 1974 and
which is now explained by V.~Batyrev as the first manifestation of the general {\it mirror
symmetry} of physicists. As an example of a still puzzling mystery, I shall
mention the {\it R.~Faure's duality}, relating the particle moving with energy $E$
in the field with potential $U(z) = | dw / dz|^2$ with the particle moving with
energy $-1/E$ in the field with potential energy $V(w) = -|dz / dw|^2$, whatever be the
holomorphic function $w(z)$. The function $w = z^2$ provides the classical {\it
duality between the Hooke and gravitation or Coulomb forces}, but the above
duality is quite general and holds both in the classical and in the quantum version.

A commented list of several hundred of problems, originating from such
``experimental facts'', is prepared by my Moscow seminar participants and will
hopefully be published soon.

\medskip
The attempts to complexify and to quaternionize mathematical theories are making
clear the fundamental unity of all parts of mathematics.  The growing
specialisation and the bureaucratic subdivision of mathematics into small domains becomes an
obstacle to its development. The organisers of the International Congress of
Mathematicians in Berlin in 1998 have considered their bureaucratic
sections as scientifically independent unities. As a result, parallel talks, formally
belonging to different sections, were in fact devoted to the same subject, be it called
symplectic geometry, mathematical physics, differential topology, partial
differential equations, global analysis, quantum mathematics or infinite
dimensional Lie algebra theory.

Several invited speakers of the Berlin Congress told me that they
would rather have listened to each other's talk, but were unable to do it, speaking
simultaneously.

\medskip
This lack of understanding of the interrelations between different domains of
mathematics originates from the disastrous divorce of mathematics from
physics in the middle of the 20th century, and from the resulting degeometrisation of the
mathematical education.

Criticising my statement that mathematics is a part of physics, one of the
former best Bourbaki leaders \footnote{who declined the invitation to
participate in the present book, explaining that, according to his experience, 
all collective works are failures.} has written to me in June 1998 : 
``Mathematics is completly different from physics\dots Mathematicians should 
not write on such philosophical questions, since even the best of them 
can write pure nonsense''.

It is  interesting to compare this boomerang statements with the words of
Hilbert, who wrote in 1930 : ``Geometry is nothing more than a branch of
physics; the geometrical truths are not essentially different from the physical ones in any
aspect and are established in the same way'' (Naturerkennen und Logik,
Naturwissenschaften, 1930, 959-963).

\medskip
Hilbert tried to predict the future development of mathematics and to
influence it by his Problems. The development of mathematics in the 20th century has
followed a different path. The most important achievements -- the flourishing of homotopy
theory and of differential topology, the geometrisation of all branches of
mathematics, its fusion with the theoretical physics, the discovery of the
algorithmically undecidable problems and the appearance of computers -- all
this went in a different (if not opposite) direction.

\medskip
The influence of H.~Poincar{\'e} and of H.~Weyl on the science of the 20th
century was much deeper. To Poincar{\'e}, who created modern mathematics,
topology and dynamical systems theory, the future of mathematics lay in the
development of mathematical physics, oriented to the description of the relativistic and
quantum phenomena. Among other important things, Poincar{\'e} explained, that
{\it only non interesting problems might be formulated unambiguously and solved completely}.
According to Poincar{\'e}, one should rather try to understand, what may
be {\it changed} in the problem formulation. He had in mind first of all the
variation of the coefficients of the equations in bifurcation type problems and all kinds
of general position arguments -- the topics which are now called singularity theory,
global analysis and functional analysis. Interestingly enough, what is
now called {\it versal deformation theorem} had already been proved in his Thesis (for
the case of zero dimensional holomorphic complete intersections) as lemma 4, basis
for his bifurcation theory.

\medskip
The mathematics of the 20th century mostly followed the road shown by
Poincar{\'e} (the main difficulty being -- as A.~Weil once told me -- the fact that
too many good mathematicians have appeared, whereas  all valuable
mathematicians knew personnally each other at Poincar{\'e}'s time).

\medskip
According to Kolmogorov, Hilbert was seriously worried by what would happen to the
Mathematische Annalen cover in 500 years : he thought that the names of the former 
Editors would fill up all the space. 

Kolmogorov objected to Hilbert that our culture would probably not survive for such a long period : 
the united bureaucrats of all countries will soon be able to stop all kind of creativity, 
making further mathematical discoveries impossible, as are geographical discoveries today.

We can imagine that some of the most appealing domains of mathematics will be
transformed into wilderness preserves, where rich people will be able to
buy for an expensive price the pleasure to hunt one-two theorems, guided by the scientific
jaegermeisters.

\medskip
It is difficult to decide, which of these predictions is more likely to happen. 
It seems however rather clear that the centre of the humanity will soon
move from wealthy Europe and North America to hungry Asia, where our culture
has few chances to survive.

\medskip
I would like to hope that this prediction is as wrong as the others. Discussing
these perspectives, the optimist H.~Whitney insisted that America is still
producing excellent mathematicians in spite of the sad fact that its general
cultural and educational level is already almost as low as that of the future global attractor.

One may also hope that the coming nuclear civil wars and military
confrontations will lead to a better appreciation of science by society and to a paradoxical
flourishing of world mathematics (similar to the flourishing which occurred in Russia
after the awful Bolshevik revolution).

\vspace{.3in}
\noindent
\parbox[lt]{2.5in}
{CEREMADE, UMR CNRS ${\rm N}^o \ 7534$ \\
Universit\'e Paris-Dauphine \\
Place du Mar\'echal de Lattre de Tassigny \\
75775 PARIS CEDEX 16 (FRANCE)}

\vspace{.2in}

\noindent
\parbox[lt]{2.5in}
{Steklov Mathematical Institute \\
8, Gubkina Street \\ 
117966 MOSCOW,  GSP-1, (RUSSIA)}

\end{document}