**Appendix 3**

**POSSIBLE TRENDS IN THE COMING DECADES**

**By Michael Gromov**

Here are a few brief remarks on possible trends in mathematics for the coming
decades.

1. Classical mathematics is a quest for structural harmony. It began with
the realization by ancient Greek geometers that our 3-dimensional continuum
possessed a remarkable (rotational and translational) symmetry (groups O(3) and
(R3) which permeates the essential properties of the physical world. (We stay
intellectually blind to this symmetry no matter how often we encounter and use
it in everyday life while generating or experiencing mechanical motion, e.g.
walking. This is partly due to non-commutability of O(3) which is hard to
grasp.) Then, deeper (non-commutative) symmetries were discovered: Lorentz and
Poincare in relativity, gauge groups for elementary particles, Galois symmetry
in the algebraic geometry and number theory, etc. And similar mathematics
appears once again on a less fundamental level, e.g., in crystals and
quasicrystals, in selfsimilarity for fractals, dynamical systems and statistical
mechanics, in monodromies for differential equations, etc.

The search for symmetries and regularities in the structure of the world will
stay at the core of the pure mathematics (and physics). Occasionally (and often
unexpectedly) some symmetric patterns discovered by mathematicians will have
practical as well as theoretical applications. We saw this happening many times
in the past; for example, integral geometry lays at the base of the x-ray
tomography (CAT scan), the arithmetic over prime numbers leads to generation of
perfect codes and infinite dimensional representations of groups suggest a
design of large economically efficient networks of a high connectivity.

2. As the body of mathematics grew, it became itself subject to a logical
and mathematical analysis. This has led to the creation of mathematical logic
and then of the theoretical computer science. The latter is now coming of age.
It absorbs ideas from the classical mathematics and benefits from the
technological progress in the computer hardware which leads to a practical
implementation of theoretically devised algorithms. (Fast Fourier transform and
fast multipole algorithm are striking examples of the impact of pure mathematics
on numerical methods used every day by engineers.) And the logical
computational ideas interact with other fields, such as the quantum computer
project, DNA-based molecular design, pattern formation in biology, the dynamics
of the brain, etc. One expects that in several decades computer science will
develop ideas on even deeper mathematical levels which will be followed by
radical progress in the industrial application of computers, e.g., a (long
overdue) breakthrough in artificial intelligence and robotics.

3. There is a wide class of problems, typically coming from experimental
science (biology, chemistry, geophysics, medical science, etc.) where one has to
deal with huge amounts of loosely structured data. Traditional mathematics,
probability theory, and mathematical statistics, work pretty well when the
structure in question is essentially absent. (Paradoxically, the lack of
structural organization and of correlations on the local level lead to high
degree of overall symmetry. Thus the Gauss law emerges in the sums of random
variables.) But often we have to encounter structured data where classical
probability does not apply. For example, mineralogical formations or
microscopic images of living tissues harbor (unknown) correlations which have to
be taken into account. (What we ordinarily "see" is not the "true image" but
the result of the scattering of some wave: light, x-ray, ultrasound, seismic
wave, etc.) More theoretical examples appear in percolation theory, in
selfavoiding random walk (modelling long molecular chains in solvents), etc.
Such problems, stretching between clean symmetry and pure chaos, await the
emergence of a new brand of mathematics. To make progress one needs radical
theoretical ideas, as well as new ways of doing mathematics with computers and
closer collaboration with scientists in order to match mathematical theories
with available experimental data. (The wavelet analysis of signals and images,
context dependent inverse scattering techniques, geometric scale analysis, and
x-ray diffraction analysis of large molecules in crystallized form indicate
certain possibilities.)

Both the theoretical and industrial impacts of this development will be
enormous. For example, an efficient inverse scattering algorithm would
revolutionize medical diagnostics, making ultrasonic devices at least as
efficient as current x-ray analysis.

4. As the power of computers approaches the theoretical limit and as we turn
to more realistic (and thus more complicated) problems, we face the "curse of
dimension" which stands in the way of successful implementations of numerics in
science and engineering. Here one needs a much higher level of mathematical
sophistication in computer architecture as well as in computer programming,
along with the ideas indicated above in 2. and 3. Successes here may provide
theoretical means for performing computations with high power growing arrays of
data.

5. We must do a better job of educating and communicating ideas. The
volume, depth, and structural complexity of the present body of mathematics make
it imperative to find new approaches for communicating mathematical discoveries
from one domain to another and drastically improving the accessibility of
mathematical ideas to non-mathematicians. As matters stand now, we,
mathematicians, often have little idea of what is going on in science and
engineering, while experimental scientists and engineers are in many cases
unaware of opportunities offered by progress in pure mathematics. This
dangerous imbalance must be restored by bringing more science to the education
of mathematicians and by exposing future scientists and engineers to core
mathematics. This will require new curricula and a great effort on the part of
mathematicians to bring fundamental mathematical techniques and ideas
(especially those developed in the last decades) to a broader audience. We
shall need for this the creation of a new breed of mathematical professionals
able to mediate between pure mathematics and applied science. The
cross-fertilization of ideas is crucial for the health of the science and
mathematics.

6. We must strengthen financing of mathematical research. As we use more
computer power and tighten collaboration with science and industry, we need more
resources to support the dynamic state of mathematics. Even so, we shall need
significantly less than other branches of science, so that the ratio of
profit/investment remains highest for mathematics, especially if we make a
significant effort to popularize and apply our ideas. So it is important for us
to make society well aware of the full potential of mathematical research and of
the crucial role of mathematics in near and long-term industrial
development.