A characteristic feature of current developments in mathematics - and defi-
nitely a trend for the future - is the emergence of various "non-classical" geomet-
ric and algebraic structures. It is impossible to list them all: from one direction
come "spaces" and "groups" that are not spaces nor groups in the usual sense but
are rather super or quantum objects; from another direction come multiple cate-
gories (with various levels of strictness) and multiple groupoids; thirdly, there come
"homotopy relatives" of Lie and associative algebras; and there are many other
examples. Some of these structures actually came into being decades ago, but have
only now started to receive due appreciation (supermanifolds and supergroups; Lie
groupoids and algebroids); others have gotten a new impetus due to new powerful
examples (Hopf algebras from quantum groups), or have undergone an amazing re-
birth (operads and various homotopy structures). Classical objects like Lie groups
and Lie algebras of course remain important, but they are no longer felt sufficient.
The motivation for new structures comes, to a great extent, from mathematical
and theoretical physics. Experts debate whether some of the "new physics" could
actually be related to physical reality, but no one can doubt that it has supplied a
plentitude of new mathematical problems, results, methods and inspirations. Apart
from that, it seems completely certain to the writer that the new geometric and
algebraic structures are not merely a product of fashion. They possess a vitality
of their own, and, as is becoming more and more evident, they have natural moti-
vations from and firm links with the "classical" differential geometry and algebraic
topology of the twentieth century.
Quantization theory and related mathematical areas (like deformation theory)
are in the focus of rapid development. As landmarks one can see the works by
Boris Fedosov on the quantization of general symplectic manifolds and by Maxim
Kontsevich on the quantization of arbitrary Poisson manifolds. The latter work
manifested the rebirth (or rethinking) of the deformation theory of various mathe-
matical structures (cf. "Nijenhuis's program" of the 1960s) and attracted general
attention to such "non-classical" tools as odd brackets, homotopy algebras, oper-
ads, etc., which had already been in use in the mathematical physics of recent years.
These results gave an enormous boost to the field, which is now extensively devel-
oping. Independently, a "cohomological physics", to use Jim Stasheff's word, grew
out of gauge-invariant field theory. An example is the renowned Batalin-Vilkovisky
quantization procedure. Since about 1990, there have also been rapid developments
in the theory of Lie algebroids, Lie bialgebroids and Poisson-Lie groupoids. These
theories often provide a semi-classical form of quantization, which mediates between
classical mathematics and fully quantum theories.
Previous Page Next Page