Georgia Institute of Technology College of Sciences

A line arrangement is a collection of lines in the projective plane.  The intersection lattice of the line arrangement is the set of all lines and their intersections, ordered with respect to reverse inclusion.  A line arrangement is called free if the Jacobian ideal of the line arrangement is saturated.  The underlying motivation for this talk is a conjecture of Terao which says that whether a line arrangement is free can be detected from its intersection lattice.  This raises a question - in what ways does the saturation of the Jacobian ideal depend on the geometry of the lines and not just the intersection lattice?  A main objective of the talk is to introduce planar rigidity theory and show that 'infinitesimal rigidity' is a property of line arrangements which is not detected by the intersection lattice, but contributes in a very precise way to the saturation of the Jacobian ideal.  This connection builds a theory around a well-known example of Ziegler.  This is joint work with Jessica Sidman (Mt. Holyoke College) and Will Traves (Naval Academy).

The extremal function of a class of matroids maps each positive integer n to the maximum number of elements of a simple matroid in the class with rank at most n. We will present a result concerning the role of finite groups in minor-closed classes of matroids, and then use it to determine the extremal function for several natural classes of representable matroids. We will assume no knowledge of matroid theory. This is joint work with Jim Geelen and Peter Nelson.

Marcus (1972) and de Oliveira (1982) conjectured  bounds on the determinantal range of the sum of a pair of normal matrices with prescribed eigenvalues.  We show that this determinantal range is a flattened solid twisted permutahedron, which is, in turn, a finite union of flattened solid twisted hypercubes with prescribed vertices.  This complete geometric description, in particular, proves the conjecture. Our techniques are based on classical Lie theory, geometry, and combinatorics. I will give a pre-seminar that will be accessible to 1st year graduate students who like matrices, and provides an easy introduction to the topic. This is joint work with Matt Speck.

 Around 1997, Shub and Smale proved that sufficiently good upper bounds
on the number of integer roots of polynomials in one variable --- as a function
of the input complexity --- imply a variant of P not equal to NP. Since then,
later work has tried to go half-way: Trying to prove that easier root counts
(over fields instead) still imply interesting separations of complexity
classes. Koiran, Portier, and Tavenas have found such statements over the real
numbers.

        We present an analogous implication involving p-adic valuations:    
If the integer roots of SPS polynomials (i.e., sums of products of sparse polynomials) of size s never yield more than s^{O(1)} distinct p-adic
valuations, then the permanents of n by n matrices cannot be computed by constant-free, division-free arithmetic circuits of size n^{O(1)}. (The
implication would be a new step toward separating VP from VNP.) We also show that this conjecture is often true, in a tropical geometric sense (paralleling a similar result over the real numbers by Briquel and Burgisser). Finally, we prove a special case of our conjectured valuation bound, providing a p-adic analogue of an earlier real root count for polynomial systems supported on circuits. This is joint work with Joshua Goldstein, Pascal Koiran, and Natacha Portier.