This talk will give an introduction to geometric programming. The associated duality theory and relevant applications will be highlighted. The survey by Boyd, Kim, Vandenberghe, and Hassibi titled "A Tutorial on Geometric Programming" provides a nice overview, and will form the basis of the talk.

A plane cubic is in honeycomb form if its tropicalization exhibits a hexagonal cycle. We present algorithms for computing such representations, and we revisit the tropical group law for elliptic curves.

We survey methods for the computation of three-point covers of the projective line, including Gröbner basis, complex analytic, and p-adic lifting techniques. We give several examples and open problems.

Chemical reaction networks taken with mass-action kinetics are dynamical systems governed by polynomial differential equations that arise in systems biology. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. This talk focuses on systems with this property, and we say such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to admit toric steady states. Furthermore, we analyze the capacity of such a system to exhibit multiple steady states. An important application concerns the biochemical reaction networks networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism. No prior knowledge of chemical reaction network theory or binomial ideals will be assumed. This is joint work with Carsten Conradi, Mercedes Perez Millan, and Alicia Dickenstein.

We discuss the geometry underlying the difference between non-negative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids. This lecture is based on work of Greg Blekherman, and on our joint paper with Jonathan Hauenstein, John Christian Ottem and Kristian Ranestad.

We give several characterizations of stable intersections of tropical varieties and locally balanced fans in general. For tropical hypersurfaces of polytopes, the multiplicity of stable intersection is the mixed volume. The stable intersection of tropical varieties is the tropical variety of a ``perturbed'' intersection of varieties. These facts together give a proof of the Bernstein's Theorem. We will also show that computing stable intersection is equivalent to computations in the polytope algebra of McMullen. While general intersection products are disconnected in codimension 1, stable intersections of hypersurfaces are connected. We discuss several notions of connectedness and attempt to give sufficient conditions for connectivity of stable intersections. This is based on joint work in progress with Anders Jensen.

In the talk we review known facts about the algebraic structure of the Orlik-Solomon algebra associated to an arrangement of hyperplanes or more generally a matroid. We define a deformation of the Orlik-Solomon algebra of a matroid as a quotient of the free associative algebra over a commutative ring R with 1. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as R-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.

A universal Gröbner basis (UGB) is a set of polynomials which is a Gröbner basis with respect to any term order. Every ideal possesses a UGB, and there are many known cases where it has a particularly nice form. In this talk I will discuss some nice classes of examples, and consider the following two questions: In what ways could one prove that a set of polynomials is a universal Gröbner basis? How can we use computational tools to gain insight in the world of Gröbner bases? Binomial ideals will serve as a springboard to illustrate some of the intricacies involved.

Algorithms for inferring causality are heavily based on the Faithfulness assumption. The unfaithful distributions have measure zero and can be seen as a collection of hypersurfaces in a hypercube. The Faithfulness condition alone is not sufficient to guarantee uniform consistency and Strong-Faithfulness has been proposed to overcome this problem. In contrast to the original Faithfulness assumption, the set of distributions satisfying Strong-Faithfulness does not have measure one. We study the (Strong) Faithfulness condition from the point of view of real algebraic geometry and give upper and lower bounds on the proportion of unfaithful distributions for various classes of graphs.

Parameter identifiability analysis for dynamic system ODE models asks which unknown parameters can be quantified from given input-output data. If all parameters of a model have a finite number of solutions, then the model is said to be identifiable. A model is unidentifiable if the parameters can take on an infinite number of values and yet result in identical input-output data. The differential algebra approach is an effective method for analyzing identifiability properties, in particular, for proving global identifiability. In this talk, we examine unidentifiable models and a method for finding globally identifiable parameter combinations using Gröbner bases. We show that a set of algebraically independent identifiable parameter combinations can always be found from the Gröbner bases and can be used to reparameterize the model's input-output equations. We also discuss computational difficulties in finding identifiable parameter combinations and explore possible improvements to our algorithm.

In Computer Vision the multi-view variety is constructed by considering several cameras in general position in space, all focused on the same point. Given lines in all camera planes, how can I tell if the cameras were all looking at the same point? This question can be answered by finding implicit defining equations for the multi-view variety. We may also be interested in the algebraic problem to find the minimal generators of the defining ideal of the multi-view variety. The case of 3 cameras, the trifocal variety, is already interesting. In this talk I will explain our use of symbolic and numerical computations aided by Representation Theory and Numerical Algebraic Geometry to study the ideal of the trifocal variety. Our work builds on the work of others (such as Hartley-Zisserman, Alzati-Tortora and Papadopoulo-Faugeras) who have already considered this problem set-theoretically. This is joint work with Chris Aholt (Washington).

One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has singularities or other distinctive features of interest). A classical example of such a problem, described by Cayley and Salmon in 1852, is to determine whether or not a given plane curve of degree r > 3 has undulation points -- the points where the tangent line meets the curve with multiplicity four. Cayley proved that there exists an invariant of degree (r - 3)(3 r - 2) that vanishes if and only if the curve has undulation points. We construct this invariant explicitly for quartics (r=4) as the determinant of a 21 times 21 matrix with polynomial entries, and we conjecture a generalization to higher r.