Professor of Mathematics, UC Berkeley
Seminar on Commutative Algebra and Algebraic Geometry
I taught at Brandeis University for twenty-seven years,
with sabbatical time spent in Paris, Bonn, and Berkeley, and
became Director of the Mathematical Sciences Research Institute in
Berkeley in 1997.
At the same time I joined the
faculty of UC Berkeley as Professor of Mathematics.
Since the fall of 2007 I've been back to full-time teaching
and research at Berkeley.
I enjoyed my work at MSRI greatly for 10 years, and
retired from it, to go back to full-time teaching and research,
on August 1, 2007. I had the satisfaction
of supporting a huge amount of mathematics and related activity
at the Institute, and in helping it develop two things I thought
it needed badly: a physical facility worthy of such a world-class
institution, and (the beginning of) an endowment.
From 2003 to 2005 I was also President of the American Mathematical
Society, an organization I came to admire a great deal.
Some Artifacts From My Years as Director (under construction)
Interests and Current Activities
I am currently Chair of the
Editorial Board of the
Journal of Algebra and
Number Theory , which I helped found in 2006, and serve
on the Boards of the Bulletin du Société
Mathématique de France and Springer-Verlag's book series Algorithms and
Computation in Mathematics.
My first paper was about permutation groups, and my thesis and
subsequent few papers on non-commutative ring theory (my thesis
advisors were Saunders MacLane and, unofficially, the English
ring-theorist J.C. Robson.) I turned to commutative algebra, and
subsequently to singularity theory, knot theory, and algebraic
geometry. My papers also
include one on a statistical application of algebraic geometry and one
Recently I've worked mostly in commutative
algebra, algebraic geometry and computational aspects of these fields.
Ever since the early 70s I've used computers to produce
examples in algebraic geometry and commutative algebra, and I've
developed algorithms to extend the power of computation
in this area. I recently joined Mike Stillman
and Dan Grayson in the project to (further) develop the
Macaulay2 system for symbolic computation.
My interests outside mathematics include hiking, juggling, and, above
all, music. Originally a flutist, I now spend most of my musical time
singing art-songs (Schubert, Schumann, Brahms, Debussy, ...) I broke
down and bought a digital camera in November 2001, and you can find
some of the results on my photo page.
Some pictures of the
New Hampshire fall colors from October 9-11, 2004.
are more recent examples.
From my CV
Saunders Mac Lane: In Memoriam
Saunders Mac Lane died on April 14, 2005. He was my thesis advisor---Irving
Kaplansky was his first student, I was nearly his last; perhaps John
Thompson is the most illustrious. His
will soon appear, published by
AK Peters. A few months ago I wrote a
for this book which contains
some of my favorite stories about him. He was a great figure,
and very important for me personally. I and many others will miss him.
Current Events Bulletin at the Winter AMS Meeting
Since 2004 I've been organizing a session at each of the Winter
AMS meetings on Current Events in Mathematics. The format is simple:
four accessible 50-minute lectures on some of the most interesting
pure and applied mathematics of the last few years, presented by people
who are speaking on the work of others. The inspiration is of course the
famous Bourbaki Seminar, but the aim is to be broader and represent a
wider range of mathematics, particularly on the applied side.
booklet with writeups of the talks is available at the meeting
and online. Almost all of them become articles in the
the American Mathematical Society afterwards.
Some Ongoing Work
During my years as Director I was able to do quite a bit of mathematics,
but time always had to be stolen for it. Having retired from that
very busy job I have more time---though never enough, of course! Here
are some of my current projects
Boij and Soederberg made some marvelously bold conjectures on the ``shape''
of free resolutions, which Frank Schreyer and I have proven. These
things are closely related to the work on resolutions over exterior
algebras that has interested us for years. There's lots more to do!
Here is the
I am grateful to the National Scienced Foundation for partial support
in my work on these projects!
The fibers of a generic projection of a curve into P^2 have length at most 2,
and the fibers of a generic projection of a surface into P^3 have length
at most 3. I always wondered whether the lengths of fibers of a projections
of a smooth complex projective variety of dimension n into P^(n+1)
would have length at most n+1. Many people proved special cases of this;
Mather has shown that it is true at least up to n=14,
and that in general there are at most n+1 distinct points
in any fiber. However, Lazarsfeld
showed that it fails for large n: the length of the worst
fibers can be exponential in the square root of n !
In recent work, Beheshti, Harris Huneke and I have noticed some
further positive results. Conjecture: the Castelnuovo-Mumford regularity
of any fiber is bounded by n+1.
Having finished "The Geometry of Schemes", Joe Harris and I have been
thinking about a new book. Intended as an advanced course, introducing
a number of topics such as Hodge Theory, Deformation Theory, and
Intersection Theory that are fundamental tools of the trade but don't
make it into the basic texts. So far all that exists is an outline and
With Jeremy Gray I'm working on a biographical paper about the life
and work of F.S. Macaulay. Though his work contains aspects that
look quite various, I think there's a very strong underlying thread
that can be traced from the early work on plane curves right
through his invention (in the graded case) of the notion of what we now call
a Gorentstein ring.
Commutative Algebra Book
My book, "Commutative
Algebra with a View Toward Algebraic Geometry", published in 1995 by
lots of typos and also some more serious errors, which many readers
have been kind enough to point out to me. Fortunately I was able to
insert corrections in the second (1996) printing ( TeX source,
pdf), and more in the third (1999) printing (TeX source,
pdf). Note that the page numbers changed a little between the
first and second printings.
I'm still gathering, so if you are aware of further corrections that
should be inserted, or have any comments, please send
them to me! The pages above are now rather out-of-date; a project
I'll get to sooner or later.
Professor of Mathematics
Dept of Mathematics,
Berkeley, CA 94720
UC Berkeley Mathematics
Created: August 2, 1995. Last updated: January 23, 2008.