Polynomial systems show up all over the place, and numerical algebraic geometry is the field dedicated to solving them numerically, accurately, and efficiently. This is a very young field, and there are still many exciting open problems to be tackled!

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:

- Expose non-expert students to NAG methods;
- Give students already on the road to expertise an opportunity to fill in gaps and focus as a group on the core techniques of the field;
- Investigate numerous applicaitons and open problems, hopefully leading down the road to original research; and
- Experiment with a book currently being written by Bates, Hauenstein, Sommese, and Wampler, spotting errors and providing feedback to the authors.

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.