Hello! Welcome to my home page.

If you are looking for information about my father, you can read about his funeral and my eulogy.

You can write to me at volcheck@acm.org.

I am a mathematician working as a cryptographic vulnerability analyst for the (US) National Security Agency. I work in the Cryptographic Components Evaluation Division of NSA's Cryptographic Evaluations Center. I did my graduate work at the UCLA Mathematics Department and held a post-doctoral position at the Research Institute for Symbolic Computation.

You can read my CV.

There was a special session on Computational Algebraic Geometry for Curves and Surfaces at the 2001 AMS/MAA Joint Meetings in New Orleans, Louisiana. Professor Mika Seppälä of Florida State University and I jointly organized this event. We plan to do this again in Baltimore in January 2003. At the 1999 Joint Meetings in San Antonio, Texas, we held the first special session on this topic.

Research Interests

• Algebraic Curves
• Computer Algebra

My research centers on the algorithmic questions raised by the theory of algebraic curves. Some fundamental questions are

• How do you parametrize a rational curve?
• How do you compute the integral closure of the function ring of a curve?
• How do you add divisors in the divisor class group of a curve? (or equivalently, how do you add points on the Jacobian variety of a curve?)
• How do you resolve the singularities of a plane algebraic curve without extending the ground field?
• How do you find the smallest possible degree (with respect to a single variable) of a plane model of a curve? For hyperelliptic curves, this value is two. (This number is the gonality of the curve.)
• How do you efficiently compute the dual of a plane algebraic curve?

Publications

Here is a manuscript on addition in the Jacobian of an algebraic curve. I presented this work at the June 1995 Oberwolfach conference on number theory. An earlier version of this work appeared in the Proceedings of the first Algorithmic Number Theory Symposium (ANTS-1) at Cornell University in 1994. (download PS or dvi)

Here is a paper that appeared in the Proceedings of ISSAC 1997 on computing the dual of a plane algebraic curve.

Service

Read about my service activities.

Links

Here are the home pages of people whose work I like and find rather interesting:

• Mark Encarnación
• Dexter Kozen
• Josef Schicho
• Franz Winkler

Thanks for visiting my page. Let me leave you with some words by George Kempf from the introduction to his textbook ``Algebraic Varieties'':

Algebraic geometry is a mixture of the ideas of two Mediterranean cultures. It is the superposition of the Arab science of the lightning calculation of the solutions of equations over the Greek art of position and shape. This tapestry was originally woven on European soil and is still being refined under the influence of international fashion. Algebraic geometry studies the delicate balance between the geometrically plausible and the algebraically possible. Whenever one side of this mathematical teeter-totter outweighs the other, one immediately loses interest and runs off in search of a more exciting amusement.

(George Kempf was a long-time member of the Johns Hopkins University Department of Mathematics. He passed away in the summer of 2002.)