The
importance of analysis in mathematics cannot be overestimated. Many areas of
modern mathematics may be seen as parts of analysis while, at the same time,
many areas of mathematics heavily rely on analysis to built on and further
comment on their problems.
Indeed
many mathematicians, who consider themselves to be analysts, work on differential
equations and prove numerous and powerful results regarding their behavior
(enumerating these results, one could begin by mentioning the existence theorem
of solutions in the O.D.E's case, and various results concerning solutions of
boundary value problems in the P.D.E.'s case).
Such
was the success of analysis in geometry, that the term \global analysis" was
coined to describe the work of those mathematicians who use analytical techniques
to study geometrical and topological properties \in the large". The proof
of the Geometrization and Poincare conjectures is probably the most famous
result of using analytical tools (properties of the Ricci flow) to achieve a
geometrical result (classifi
cation of manifolds).
Algebra
is not beyond the limits of analysis either. In number theory p-adic functional
analysis provides us with results on the Weyl conjectures, while one cannot
overlook the work of Hormander and its consequences in commutative algebra and
complex algebraic geometry (the effective Nullstellensatz" easily comes to
mind).
On
the other hand, all these require concrete estimates. Norms have to be
calculated, operators must be inverted and various types of constants should be
explicitly constructed. Furthermore, for some time now, computers have become important
not only as computational tools, but as valuable \coworkers" as well.
Computer assisted proofs are here to stay, allowing us to draw de
finite conclusions on problems dicult to attack from a more \old fashioned" viewpoint.
The
four color map theorem and the existence of a strange attractor in the Lorenz
system were proved by mathematicians using computers to perform calculations
unattainable by hand.
All these are but a few examples of the effectiveness
of analysis, along with reliable calculations, on various problems arising in
all areas of mathematics. We hope that this journal will serve as a widely
respected forum for communicating techniques and results that will further
broaden our understanding of the mathematical and physical universe.