Algebraic Geometry Seminar

Computing, counting, or even deciding on the existence of real solutions to a system of polynomial equations is a very challenging problem that is important in many applications of mathematics. There is an emerging landscape of structure in the possible numbers of real solutions to systems of polynomial equations. These include fewnomial upper bounds, gaps or congruences, and lower bounds. My talk will survey what is known about these bounds, focussing on lower bounds---which are existence proofs of solutions---and open problems, including some concrete challenges.
Abstract

In this talk, I will give a brief introduction to quantum cohomology of flag varieties first, and then introduce a nice Z^2 grading on the quantum cohomology of a complete flag variety. As an application, I will show a quantum Pieri rule for the tautological subbundle over a Grassmannian of type C. This is my joint work with Naichung Conan Leung.

We introduce an object with many properties and applications: the vector partition function. This can be seen as the number of integer points in a variable polytope, and it is better understood by considering a toric arrangement, i.e. a suitable familiy of hypersurfaces in torus. Such an arrangement has a topological model and a wonderful model, that can be used to compute several invariants. The combinatorial counterpart of a toric arrangement is an "arithmetic matroid": we study this object by associating an arithmetic Tutte polynomial, with applications to toric arrangements, partition functions, polytopes, and graphs.

Kinematicians - engineers who study mechanisms such as robotic arms - are interested in finding examples of mechanisms that have unexpected mobility, called "exceptional mechanisms." The range of motion for a particular mechanism is governed by a set of polynomial equations. The solution set of this polynomial system then consists of all ways the mechanism can be constructed and, accordingly, all ways in which it can move. The problem of finding exceptional mechanisms can thus be translated into the problem of finding points in some parameter space (the coordinates of which correspond to various attributes of the mechanism, such as linkage lengths) for which the associated polynomial system has a solution set of a higher dimension than the dimension of the solution set at a general point. Andrew Sommese and Charles Wampler developed a numeical method for finding such parameter values, based on the use of fiber products. However, the computational cost needed for their initial approach was beyond what could be handled by the software available at that time. Eric Hanson, Jon Hauenstein, Wampler, and I are now working on a new approach, making use of both more recent techniques (e.g., diagonal homotopies) and more efficient software (Bertini). In this talk, I will describe both the original fiber product approach of Sommese and Wampler and some of our improvements, illustrated with a very simple (non-kinematics) toy problem. I will not assume any prior knowledge about kinematics, fiber products, or numerical algebraic geometry.

One common question regarding a subspace arrangement is to determine the integral homology of the complement. In many cases, one can relate the homology to the homology of the intersection lattice, and then use combinatorial techniques, such as shellability. However, for certain subspace arrangements, this approach fails to work. In this talk, we will demonstrate how discrete Morse theory can be used to prove results about integral homology for r-disjoint k-equal arrangements, which are a generalization of the k-equal arrangement that has been well-studied in the literature. This is joint work with Helene Barcelo and Christopher Severs.

In this joint work with K. Schwede and W. Zhang, we describe combinatorially the ideals on a toric variety that are fixed by a given Cartier algebra. We also show that geometrically such ideals can be view as certain generalization of the defining ideals of unions of log canonical centers in the minimal model program.