Barry Dayton's Space

This web space is primarily for my ongoing research and writing, although some of my older material is also included.

Other Stuff
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Personal Note: I am now living in Ridgefield Connecticut.

Google Scholar Page
MathSciNet Author

My Northeastern homepage

Theory of Equations Book
2002 edition

Witt vectors and Necklaces
Short version
Long version

More on Witt Vectors:
The Picard Group of a reduced G-Algebra
See also B.Singh On Subintegrality and M.Vitulli Weak Normality and Seminormality

Papers with C.A. Weibel:
Naturality of Pic, SK0 and SK1
Module Structures on the Hochschild and cyclic homology of graded rings


(with Z.Zeng and T.Y. Li) Multiple Zeros of Non-linear Systems

Algebraic Foundation of Local Multiplicity

Recent Talks and Papers

Approximate Local Rings 2006

Configurations of Lines
See Also Analysis of Bertini Experiments and Appendix
For a new take on these ideas see Section 1.9 of my new Book about Surfaces

FCRC11-SNC11 Duality Talk

AG11 Numerical Algebraic Geometry via Numerical Polynomial Algebra

AN12 Numerical Algebraic Geometry via Macaulay's Perspective

Quadratic Surface Intersection Curves For latest version see Explicit Regular QSIC.pdf

"Harnessing Natural Mosaics: Antibody-Instructed, Multi-Envelope HIV-1 Vaccine Design"
See also Mathematica Software for article

Original material in this website is covered by

except for material in the sub-directory curvebook which is copyrighted by Wolfram Media and Barry H Dayton.

This graphic shows, in black, the intersection curve of an ellipsoid with a paraboloid with non-vertical axis. For the latest version of my QSIC paper see Explicit Regular QSIC.pdf

My goal is an exposition of classical algebraic geometry (eg. 19th century) using Mathematica code. Abstract 20th century algebraic geometry, while interesting, is meaningless without a good classical background. Most current expositions of algorithms feature pseudo-code with an obsession in complexity, a totally useless concept. By using actual code the reader with access to Mathematica can directly verify my work and the speed of the algorithms.

Plane Curve Book

The main point of this book is to show that appropriate computer software can make this subject accessible to those who do not have the patience to master the big abstract theorems that usually define the field of abstract algebraic geometry. That doesn't mean we can ignore those theorems but does mean that we can concentrate on the application of these theorems rather than their development. To some extent this book was motivated by Shreeram Abhyankar's text Algebraic Geometry for Scientists and Engineers which, in my mind, missed the target audience by failing to illustrate the abstract ideas developed. I hope this book proves more interesting and readable to this audience.

The bulk of the book should be accessible to individuals who have had a standard pre-calculus course and have a working knowledge of the Wolfram Language. Numerical linear algebra is extensively used in the book but is hidden in the algorithms. An appendix is provided for those with a basic knowledge of linear algebra who want to learn more about this aspect.

Space Curve Book

Space curves in n-dimensional affine or projective space present a major challenge that we did not need to deal with for plane curves, instead of a single equation a system of n-1 or more equations are needed. This system is far from unique and, in many cases, may be over-determined. Since I allow numerical coefficients and, in general, numerical over-determined systems are to be avoided, specical methods must be used.

I first discuss naive space curves in 3-space, that is, curves defined by a system of two equations in 3 variables. Not only is this case familiar to what we learned in multivariable calculus but many methods in the plane curve case such as critical points and path tracing are still available.

Then I develop the methods from numerical linear algebra that will be needed: Macaulay and Sylvester matrices and duality. The basic definitions and techniques for the general case will then be established.

The last part covers, as examples, various situations I have recently presented in papers and talks. These are listed at the end of the left sidebar of this webpage.

Surface Story

This story covers naive implicit surfaces and parametric surfaces in reall projective three space. The second edition is now available. It is divided into 3 parts, the first part gives basic definitions and a preview of the whole book. The second part gives details on quadric surfaces, cubic surfaces and some higher dimensional surfaces, especialy the torus. The third part concerns the topology of hyperboloids and tori with an application to the topology of complex projective plane conic and cubic curves.

Surface Story Part I (95pp)   Surface StoryPart II (128pp)   Surface Story Part III (55pp)

Additional notes on Rational Curves

Here are several recent papers of mine. Note on Intersection of Rational Curves (PDF) and Interpolating Rational Normal Curves.

See also my earlier paper Degree vs Dimension for rational parametric curves in the Mathematica Journal.

NEW I have recently posted several self contained summaries of some of the material in my books in the Wolfram Community. These are available in web readable formats as well as Mathematica Notebooks. Barry H Dayton Wolfram Community Home