Holy Cross Mathematics and Computer Science

My graduate training, Ph.D. thesis, and my more recent work
have all dealt with aspects of the subject known as
*algebraic geometry*. This area of mathematics
grew out of the study the geometry of loci defined by
polynomial equations, starting with the
conic sections,
the curves defined by equations
*0 = Ax ^{2} + Bxy + Cy^{2} + Dx + Ey + F. *
Higher-dimensional objects can also be defined in this way.
For instance, here is a

Algebraic geometry uses abstract algebra (especially the theory of commutative rings, modules, and so forth) as its primary tool, though parts of the subject also draw on complex analysis, differential geometry, combinatorics, and other parts of mathematics.

Much of my earlier research was devoted to questions about algebraic curves and their Jacobian varieties related to one approach to the Schottky problem of characterizing the Jacobian varieties of curves among all principally polarized abelian varieties.

More recently, I have also become very interested in the
new areas of *computational algebraic geometry*,
computer algebra
and their applications.
The introduction of powerful tools such as *Groebner bases*
and various new versions of resultants for systems of polynomial
equations, coupled with the explosive growth in the speed and
availability of computational resources, has started to change
the face of algebraic geometry. One of the first steps
in this theory is a
*division algorithm* for polynomials in several
variables.

Starting from the time of my most recent sabbatical leave (spent
at Cornell University), I have also been especially interested
in another area where algebraic geometry and its new computational
methods are finding some exciting new applications. This area
is the field of *algebraic coding theory* -- specifically
the theory of error-correcting codes for reliable information
transmission. My interests here are primarily related to the
geometric Goppa codes and the construction of
algebraic encoding and decoding algorithms for them.

I also enjoy thinking about lots of different areas of mathematics. For example, I enjoy teaching and working on aspects of numerical analysis (as an "amateur"), and I have recently been learning more about Fourier analysis, wavelets, and signal processing.

Here is some information about my recent publications.