In case the name (especially the algebraic geometry part) worries you, rest assured - I will not expect that you know any algebraic geometry and barely any will be presented. In fact, this field might more aptly be named numerical nonlinear algebra as it is really an extension of numerical linear algebra to the polynomial (nonlinear) case. Algebraic geometry is the field dedicated to polynomials and their solution sets, so it is natural that it should come into play occasionally.
This topics course has a few goals, listed here in no particular order:
I encourage you to experiment and investigate topics or problems that interest you and to bring questions or results to class. While I will typically have an idea of what to discuss each day, I can be very flexible with what we cover and when.
Please note that this course WILL count as a substitute APPLIED course for your breadth requirement but not a substitue for algebra.
Given that the principal work for this class will be independent investigation/experimentation/study, I expect that you will get out of this class approximately what you want. That might be increased expertise within the field, exposure to non-standard applied problems, practical experience with computations and software, or perhaps something else entirely.
I will also be passing you chapters (in pdf format) of the book that the Bertini Team is currently writing. It will be useful for your understanding of the material (hopefully!), and we'd very much appreciate any feedback you can provide.
This course will have three basic phases: Basic Ideas, Recent Advances, and Independent Studies.
Basic Ideas: Unless you are my student, this field is probably pretty new to you. These techniques don't show up in standard graduate courses (yet), so we'll spend several weeks on basic concepts. We'll cover the basic ideas around polynomial systems and their (complex and real) solution sets, the main computational engine of the field (homotopy continuation), and the more sophisticated methods that have been built from homotopy continuation to find so-called witness sets on each irreducible component of the solution set.
This material relies on some very basic algebraic geometry (which I'll cover), some complex analysis, some numerical linear algebra, and bits and pieces from a few other fields (ODEs, combinatorics, numerical analysis, several complex variables, etc.). I'll gladly provide background as needed.
At the conclusion of this component of the course, you'll be in good shape to read papers in the field and to begin experimenting with the software Bertini.
Recent Advances: Numerical algebraic geometry is a happening place, and there are some very exciting recent developments that we'll cover. We'll hit a variety of topics, including certification of numerical computations, real (rather than complex) computational techniques, dual space techniques, and a wealth of applications (likely including at least kinematics, control theory, string theory, magnetism, and tumor modeling). Some of these topics might require some special knowledge, which I'll provide as necessary.
During the several weeks we spend on this component, you should plan to read (at least skim) some technical papers and start working on your independent studies.
Independent Studies: My goal with this component is to get each student (or small team of students) working on some problem(s) of current interest. I am very flexible about the structure of this work, and I am making no formal requirements about deliverables or presentations. I would like everybody to dig into something that sounds cool, then report on it sometime during class (even for only a few minutes). We can discuss your progress one-on-one outside of class or during class, if the latter seems appropriate. Given that such investigations often lead to interesting problems (some of which I will have encountered previously), my hope is that this work will lead to some original research. While I don't plan to set aside a particular set of times for presentations, I am happy to work them into our regular class meetings as appropriate. Here is a list of topics for this. (My students are welcome to look into the theoretical side of NAG, as in Appendix A of Sommese-Wampler. This is unlikely to lead to original research, but it's an important part of your training.)
Maple worksheet for drawing some real univariate solution paths.
Maple worksheet for determining the rank of the Jacobian at various points on "illustrative example" (sphere, cubic, 3 lines, point).
Bertini input file for a simple example where squaring the system (going from 3 eqs in 2 vars to 2 eqs in 2 vars) results in "Bertini junk." Here is the main output file showing all 8 (instead of 4!) solutions.