Georgia Institute of Technology College of Sciences

Hilbert’s 17th problem asked whether every nonnegative polynomial is a sum of squares of rational functions. This problem was solved affirmatively by Artin in the 1920’s, but very little is known about degree bounds (on the degrees of numerators and denominators) in such a representation. Artin’s original proof does not yield any upper bounds, and making such techniques quantitative results in bounds that are likely to be far from optimal, and very far away from currently known lower bounds. Before stating the 17th problem Hilbert was able to prove that any globally nonnegative polynomial in two variables is a sum of squares of rational functions, and the degree bounds in his proof have been best known for that two variable case since 1893. Taking inspiration from Hilbert’s proof we study degree bounds for nonnegative polynomials on surfaces. We are able to improve Hilbert’s bounds and also give degree bounds for some non-rational surfaces. I will present the history of the problem and outline our approach. Joint work with Rainer Sinn, Greg Smith and Mauricio Velasco.

We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of k linear functions. For networks with a single layer of maxout units, the linear regions correspond to the regions of an arrangement of tropical hypersurfaces and to the (upper) vertices of a Minkowski sum of polytopes. This is joint work with Guido Montufar and Leon Zhang.

If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.

For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is semistable -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities.  When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the cluster data associated to $f$.  When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated.  I will give an overview of how to construct "nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.