SIAGA: Visualization of Algebraic Varieties

Seven pictures from Applied Algebra and Geometry: Picture #7

The Society for Industrial and Applied Mathematics, SIAM, has recently released a journal of Applied Algebra and Geometry called SIAGA. See here for more information on the new journal, which starts taking online submissions on March 23rd – in six days time.

The poster for the journal features seven pictures. In this final blog post I will talk about the seventh picture, on the subject of ‘Visualization of Algebraic Varieties’. This concluding blog post is short, and the picture is especially nice.

Thank you for reading this series of seven ‘SIAGA’ blog posts, and for following Rachael and my blog “Picture this Maths”. Check back here soon for future mathematical posts centered around pictures!

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The Context

There is a vast mathematical toolbox of techniques that can be used to understand algebraic varieties. We have encountered some such tools in this series of blog posts, for example polyhedral geometry. It’s great when we are actually able to draw the algebraic variety in question, using visualization software. When possible, this facilitates the most direct of observations to be made.

Although it poses an obvious restriction on the number of dimensions we can work in, even visualizing particular slices through our variety of interest is structurally revealing. Large polynomials with many terms can be very hard to get a handle on, and it makes sense to use modern-day computer tools to convert these equations into helpful pictures.

The Picture

This picture shows a Kummer Surface. It was made by Oliver Labs using the visualization software Surfex. Many beautiful pictures have been created in this way: for more, see the picture galleries from the Imaginary: Open Mathematics website.

It is an example of an irreducible surface in three-dimensional space of degree four. In general, these have at most 16 singular points. Kummer surfaces are those that attain this upper bound. The 16 singular points represent the 2-torsion points on the Jacobian of the underlying genus 2 curve.

This picture also represents the problem-solving areas of coding theory and cryptography, in which there can be found a broad range of applied algebra and geometry. The group law on an elliptic curve is fundamental for cryptography. Similarly, the group law on the Jacobian of hyperelliptic curves has been used for cryptographic purposes, see “Fast Cryptography in Genus 2” by Bos, Costello, Hisil and Lauter (2013), and “Applied Cryptography and Network Security” by Bao, Samarati and Zhou (2014). One of the authors of the first article is Kristin Lauter from Microsoft Research who is president of the Association for Women in Mathematics (AWM).

 

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