Here's a list of topics related to our course, aside from the basics of homotopy continuation and numerical irreducible decomposition. If you want to dig into something else, then, by all means, be my guest! The goal is to have you think about something related (somehow) to numerical algebraic geometry that is of interest to you.

If you are an expert (or not), whether you are in the class or not, please feel free to send me comments, corrections, or other new ideas....

One way to find papers is to check out the citations of the papers or software (or people) discussed in class.

Not so open, but still interesting

• Theory underlying homotopy methods (theory): Why do you find the same number of points over a Zariski open subset of the parameter space? Why is it that you necessarily have a path going to every solution of the target system if you build a total degree (or other) start system? The appendix of the Sommese-Wampler book is the best place to start reading up on these ideas.
• Traces (theory): Similar to the previous; can you cook up a good explanation of the trace test?
• Puiseux series (theory): What are they? Why do they come up near singularities (think of the fractional power series endgame)? The power series paper (Morgan, Sommese, and Wampler) is a good place to start this, and you should also definitely check out Jan Verschelde's new paper on tropisms! (What is a tropism??)
• General nonlinear path-tracking (theory, computation): How does homotopy continuation work with non-polynomials systems of equations? Allgower-Georg is the standard reference for this.
• Polyhedral homotopies (theory, computation): These are very powerful methods for small or sparse problems. They rely heavily on convex geometry and (somewhat) optimization. TY Li's 2003 Acta Numerica article is a good source.
• Multihomogeneous or linear product homotopies: What are they? How do you go from a root count to a homotopy? (See root count topics in the next section.) The Sommese-Wampler book is a good reference for this.

Kind of open

• Simple heuristics for homotopy continuation (computation): We halve steplength upon failure of Newton's method, then double upon K successful steps in a row. Is 1/2 the optimal fraction? Do different choices of K affect run times much? You can play with this stuff with Bertini. A thorough study could make a decent conference paper and would be appreciated by the community, but don't count on it being adequate for a PhD thesis (MS maybe).
• Root counts - complex (theory): What are the various root counts you can get for polynomial systems? Given some system, how do you choose the best root count? What's the BKK bound? (Sommese-Wampler is good for the first; TY Li's Acta Numerica article is good for the last, as is Frank Sottile's new AMS book.)
• Root counts - real (theory): How can you bound the number of real isolated solutions of a polynomial system? Check out Frank Sottile's new AMS book or the string of papers by Bihan and Sottile (see Sottile's website).
• Monodromy groups (theory, computation): Recall the monodromy portion of equidimensional decomposition. Can you find monodromy loops (for some fixed problem) that generate the full permutation group? Must loops necessarily act transitively?
• Deflation: This is a process for regularizing singular solution sets. The main results in this line are from Leykin-Verschelde-Zhao and the recent Hauenstein-Wampler isosingular paper. The Sommese-Wampler book also has a write-up. The process is known but could probably be made more efficient.... One question: If you deflate at a nonsingular point, what happens?
• Existing applications: There are applications in kinematics (see the preprint page of any of the Bertini authors), control theory (Bates preprints), string theory (Hauenstein preprints), PDEs (Sommese or Hauenstein preprints), and others.

Way open

• p-adic homotopy continuation (theory and computation): Some people really want to find solutions of polynomial systems over a finite field. p-adic analysis (analogous in some ways to numerical analysis) might provide some foundation for this. For example, there is a p-adic version of Newton's method. Can you cook up and implement p-adic homotopy continuation? (This is likely to be very hard, i.e., PhD thesis level.)
• Kinematics software (computation): Build some nice code for solving kinematics problems in kinematics language, probably using Bertini as an engine. This will be time-consuming and should only be pursued by a person who has some kinematics training (say from Tony Maciejewski's course in ECE). The Sommese-Wampler book and their new Acta Numerica survey article are nice starting points for thinking about this. A really nice job (probably with influence from Dan Brake, Wampler, Maciejewski, or others) might result in a paper, though the code itself (maybe in Matlab?) would be a nice deliverable that could go on a cv.
• Monomials and geometry (theory or computation): What happens to the geometry when you remove a monomial from a polynomial (or polynomial system)? Of course the degree of the monomial removed will matter, but what happens to the hypersurface(s)? This is worth considering as there could be nice algorithms coming from the initial removal (to create a simpler system), then re-introduction of some monomials.
• Syzygies and free resolutions (theory or computation): Is there a way to compute these numerically? What are the right data types? Can you at least get the Betti numbers?
• Applications (computation, application): There are all manner of polynomial systems out there in the literature (economic equilibria, dynamical systems, chemical equilibria, etc.). Go find them and try some problems in Bertini. There are also some applications for which some work has been done; see the previous section.
• Comparisons: Benchmark Bertini against other numerical code (PHCpack, HOM4PS-2.0) or symbolic code (M2, Singular, etc.) and consider complexity differences. I've done some of this (Bertini/Singular -- see the paper), but much more could be done.
• Complexity: Work out the complexity of the methods used in practice (not general homotopy continuation with 0th order predictors, etc., as studied by Smale, et al.).