Preface
Algebraic geometry is a classical discipline which for many years sat at the
intersection of algebra, number theory, several complex variables, and geometry
in all its incarnations. The advent of personal computing, and more so the de-
velopment of software for symbolic computations, introduced a new facet of the
discipline in the 1980’s. Software packages such as Singular, Macaulay, and CoCoA
have partially changed the scope of problems that can be considered. Nonetheless,
the mainstream approach of algebraic geometry remained distinctly separate from
the computational (particularly numerical) flavor found in many other branches of
mathematics.
In the 1990’s, a numerical approach to algebraic geometry was pioneered, due
largely to the work of Andrew Sommese and his collaborators. In the early days,
the fundamental technique, homotopy continuation, was used to compute solu-
tions to polynomial systems arising mostly from kinematics and, more generally,
engineering. These new numerical techniques opened the door for researchers to
attack polynomial systems from a different angle, in many cases expanding the
classes of polynomial systems that could be solved in practice. In the last few
years, the numerical approach has grown quickly and extended its reach into areas
often considered to be strictly the domain of exact symbolic computation. It is
now being realized that numerical techniques and symbolic, classical methods need
not compete and can harmoniously complement each other, as multiple tools in
the practitioner’s toolbox. Active sharing of ideas, progress, and problems in both
directions between the classical and numerical algebraic geometry communities is
vital.
Accordingly, with the help of Juan Migliore, we organized a conference aimed at
enhancing this interaction. The conference was titled Interactions of Classical and
Numerical Algebraic Geometry and took place at the University of Notre Dame over
a period of three days in May 2008. There were 11 talks by world leaders in these
two fields, with a lively discussion period at the end of each day. Believing that the
intersection of the two fields was (and is) ripe for rapid growth fueled by joint work,
we hoped that bringing these communities together in an interactive forum would
help spark important advances. The strong participation in the meeting indicates
that the communities are indeed receptive to interaction, and the contributions to
this volume show that valuable results have already begun to accrue.
The inspiration of this meeting was to honor the career, past and present, of
a pioneer of both classical and particularly numerical algebraic geometry, Profes-
sor Andrew J. Sommese, the Vincent J. and Annamarie Micus Duncan Professor of
Mathematics at the University of Notre Dame, in the year of his 60th birthday. An-
drew spent roughly the first 15-20 years of his career primarily focused on complex
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