Mathematics is a tool for thought. A highly necessary  tooll  in  a world where both feedbaclt  and  non-
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other paJts  and  for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics  ..  .'; 'One service logic has rendered com-
puter science  ..  .'; 'One service category theory has rendered mathematics  ..  .'. All arguably true. And
all statements obtainable this way form  part  of the raison d'etre of this series.
This series,  Mathematics  and  Its Applications,  started in  1977.  Now that over  one:  hundred
volumes have appeared  it  seems opportune to reexamine its scope.  At  the time I wrote
"Growing specialization  and  diversification have brought a host  of  monographs  and
textbooks  on  increasingly specialized topics. H()wever, the  'tree'  of  knowledge  of
mathematics and related fields does not grow only by putting forth new branches.  It
also happens, quite often  in  fact, that branches which were thought to be compI:tely
disparate are suddenly seen to be related. Further, the kind  and  level of sophisticntion
of  mathematics applied  in  various sciences  has  changed drastically in recent  y ~ : a r s :
measure theory  is  used (non-trivially)  in  regional  and  theoretical economics; algehraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the strue:ture
of  water meet one another  in  packing  and  covering theory; quantum fields,  c r ~ s t a l
defects and mathematical programming profit from homotopy theory; Lie algebral, are
relevant  to  filtering;  and  prediction and electrical engineering can use Stein spaces. And
in addition to  this  there are such new emerging subdisciplines as 'experimt:Iltal
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-:;cale
order', which are almost impossible to  fit  into the  ~ : x i s t i n g  classification schemes. They
draw upon widely different sections of mathematics."
By  and large, all this still applies today.  It  is still true that  at  first sight mathematics  sec:ms  rather
fragmented  and  that to find, see, and exploit the deeper underlying interrelations more effort is
needed  and  so are books that can help mathematicians and scientists  do  so. Accordingly MIA will
continue  to  try to make such books available  ..
If  anything, the description I gave in  1977  is now  an  understatement. To the examples  ·:>f  interac-
tion areas one should add string theory where Riemann surfaces, algebraic geometry, moe.ular func-
tions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and morc) all come
together. And to the examples of  things  which can  be  usefully applied let me add the topic 'finite
geometry'; a combination of words which sounds like  it  might not even exist, let alone be applica-
ble. And yet  it  is  being applied: to statistics via designs, to radar/sonar detection arrays (via finite
projective planes), and to bus connections  of  VLSI chips (via difference sets). There seems  to  be  no
part of (so-called pure) mathematics that is not  in  immediate danger of being applied.  And,  accord-
ingly, the applied mathematician needs to  be:  aware of much more. Besides analysis and numerics,
the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, a:ld so on.

Classical and New Inequalities in Analysis PDF

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