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Series: Algebra Seminar

We consider functions on finite abelian groups that are nonnegative and also sparse in the Fourier basis. We investigate conditions under which such functions admit sparse sum-of-certificates certificates of nonnegativity, i.e., certificates where the functions in the sum of squares decomposition have a small common sparsity pattern. Our conditions are purely combinatorial in nature, and are based on finding particularly nice chordal covers of a certain Cayley graph. These techniques allow us to show that any nonnegative quadratic function in binary variables is a sum of squares of functions of degree at mostceil(n/2), resolving a conjecture of Laurent. After discussing the connection with semidefinite programming lifts of polytopes, we also see how our techniques provide an example of separation between sizes ofsemidefinite programming lifts and linear programming lifts. This is joint work with James Saunderson and Pablo Parrilo.

Series: Algebra Seminar

Useful background:The paper I’m discussing: <a href="http://arxiv.org/abs/1502.03736" title="http://arxiv.org/abs/1502.03736">http://arxiv.org/abs/1502.03736</a>Terry Tao’s blog post on Dvir’s theorem: <a href="https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel... title="https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel... earlier paper with Terry and Richard Oberlin about Kakeya restriction over finite fields: <a href="http://arxiv.org/abs/0903.1879" title="http://arxiv.org/abs/0903.1879">http://arxiv.org/abs/0903.1879</a>

The study of extremal configurations of points and subspaces sits at the boundary between combinatorics, harmonic analysis, and number theory; since Dvir’s 2008 resolution of the Kakeya conjecture over finite fields, it has been clear that algebraic geometry is also part of the story.We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant non-reduced subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer. It will, however, try to convince everyone in the room that it can be useful to be an algebraic geometer.This is joint work with Daniel Erman.

Series: Algebra Seminar

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points for this optimization problem. We focus on projective varieties seen in engineering applications, and we discuss tools for exact computation. Our running example is the Eckart-Young Theorem which relates the nearest point map for low rank matrices with the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

Series: Algebra Seminar

Jacobians aren't particularly interesting from the point of view of the minimal model program, and neither are the moduli spaces of vector bundles on curves. But once we pass to vector bundles of higher rank (or torsion-free sheaves) on surfaces, then the birational geometry becomes very interesting. In this talk, I want to describe some recent results that rely on "tilting" the category of coherent sheaves on a surface to produce birational models of moduli that are themselves moduli spaces that come up naturally in the minimal model program.

Series: Algebra Seminar

Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix
inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. It is a convex basic closed semialgebraic subset of R^g. Given a spectrahedron S_L, the
matrix cube problem of Nemirovskii asks for the biggest cube [-r,r]^g included in S_L. We solve a relaxation of this problem based on``matricial’’ spectrahedra and estimate the error inherent in this
relaxation.
The talk is based on joint work with B. Helton, S. McCullough and M. Schweighofer.

Series: Algebra Seminar

In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves. I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform. These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.

Series: Algebra Seminar

Let K be a complete, algebraically closed, non-Archimedean field, and let $\phi$ be a rational function defined over K with degree at least 2. Recently, Robert Rumely introduced two objects that carry information about the arithmetic and the dynamics of $\phi$. The first is a function $\ord\Res_\phi$, which describes the behavior of the resultant of $\phi$ under coordinate changes on the projective line. The second is a discrete probability measure $\nu_\phi$ supported on the Berkovich half space that carries arithmetic information about $\phi$ and its action on the Berkovich line. In this talk, we will show that the functions $\ord\Res_\phi(x)$ converge locally uniformly to the Arakelov-Green's function attached to $\phi$, and that the family of measures $\nu_{\phi^n}$ attached to the iterates of $\phi$ converge to the equilibrium measure of $\phi$.

Series: Algebra Seminar

Using the ideas of Poonen and Stoll, we develop a modified version of Chabauty's method, which shows that a positive proportion of hyperelliptic curves have as few quadratic points as possible.

Series: Algebra Seminar

Faltings' theorem states that curves of genus g> 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry and tropical geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

Series: Algebra Seminar

Smooth curves in the tropical plane correspond to unimodulartriangulations of lattice polygons. The skeleton of such a curve is ametric graph whose genus is the number of lattice points in the interior ofthe polygon. In this talk we report on work concerning the followingrealizability problem: Characterize all metric graphs that admit a planarrepresentation as a smooth tropical curve. For instance, about 29.5 percentof metric graphs of genus 3 have this property. (Joint work with SarahBrodsky, Michael Joswig, and Bernd Sturmfels.)