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Series: Algebra Seminar

The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. More generally, one can associate a polyhedral dual complex to any toroidal degeneration. It is natural to ask for connections between the geometry of an algebraic variety and the combinatorial properties of its dual complex. In this talk, I will explain one such result: The dual complex of an n-dimensional uniruled variety has the homotopy type of an (n-1)-dimensional simplicial complex. The key technical tool is a specialization map to dual complexes and a balancing condition for these specialization.

Series: Algebra Seminar

A Linear system on metric graphs is a set of effective divisors. It has the structure of a cell complex. We introduce the anchor divisors in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We can also compute the extremal generators of the tropical convex hull using the landmarks. We apply these methods to some examples - $K_{4}$ and $K_{3,3}$..

Series: Algebra Seminar

Tropicalization involves passing to an ordered group M, usually taken to be
(R, +) or (Q, +), and viewed as a semifield. Although there is a rich theory arising from
this viewpoint, idempotent semirings possess a restricted algebraic structure theory, and
also do not reflect important valuation-theoretic properties, thereby forcing researchers to
rely often on combinatoric techniques.
Our research in tropical algebra has focused on coping with the fact that the max-plus’
algebra lacks negation, which is used throughout the classical structure theory of modules.
At the outset one is confronted with the obstacle that different cosets need not be disjoint,
which plays havoc with the traditional structure theory.
Building on an idea of Gaubert and his group (including work of Akian and Guterman),
we introduce a general way of artificially providing negation, in a manner similar to the
construction of Z from N but with one crucial difference necessitated by the fact that the
max-plus algebra is not additively cancelative! This leads to the possibility of defining many
auxiliary tropical structures, such as Lie algebras and exterior algebras, and also providing
a key ingredient for a module theory that could enable one to use standard tools such as
homology.

Series: Algebra Seminar

Given a graph G on p vertices we consider the cone of concentration matrices associated to G; that is, the cone of all (p x p) positive semidefinite matrices with zeros in entries corresponding to the nonedges of G. Due to its applications in PSD-completion problems and maximum-likelihood estimation, the geometry of this cone is of general interest. A natural pursuit in this geometric investigation is to characterize the possible ranks of the extremal rays of this cone. We will investigate this problem combinatorially using the cut polytope of G and its semidefinite relaxation, known as the elliptope of G. For the graphs without K_5 minors, we will see that the facet-normals of the cut polytope identify a distinguished set of extremal rays for which we can recover the ranks. In the case that these graphs are also series-parallel we will see that all extremal ranks are given in this fashion. Time permitting, we will investigate the potential for generalizing these results. This talk is based on joint work with Caroline Uhler and Ruriko Yoshida.

Series: Algebra Seminar

Let det_n be the homogeneous
polynomial obtained by taking the determinant of an n x n matrix of
indeterminates. In this presentation linear maps called Young
flattenings will be defined and will be used to show new lower bounds on
the symmetric border rank of det_n.

Series: Algebra Seminar

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We develop a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. Joint work with Mateusz Michałek, Bernd Sturmfels, and Piotr Zwiernik.

Series: Algebra Seminar

Given a non-isotrivial elliptic curve E over K=Fq(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=C(t), the rank E(L) is zero for any rational function field L=C(u). The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

Series: Algebra Seminar

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. In this talk, I will advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input one. Our motivating examples will be plane elliptic cubics and genus two hyperelliptic curves. Based on joint work with Hannah Markwig (arXiv:1409.7430) and ongoing work in progress with Hannah Markwig and Ralph Morrison.

Series: Algebra Seminar

Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. The standard certificates are sumsof squares (SOS), which trace back to Hilbert (see Hilbert’s 17th problem).In this talk we completely characterize sections of the cones of nonnegativepolynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial. Based on these results, we obtain a completely new class of nonnegativity certificates independent from SOS certificates. Furthermore, nonnegativity of such circuit polynomials f coincides with solidness of the amoeba of f , i.e., the Log-absolute-value image of the algebraic variety V(f) in C^n of f. These results establish a first direct connection between amoeba theory and nonnegativity of polynomials.These results generalize earlier works by Fidalgo, Ghasemi, Kovacec, Marshall and Reznick. The talk is based on joint work with Sadik Iliman.

Series: Algebra Seminar

In numerical algebraic geometry the key idea is to solve systems of polynomial equations via homotopy continuation. By this is meant, that the solutions of a system are tracked as the coefficients change continuously toward the system of interest. We study the tropicalisation of this process. Namely, we combinatorially keep track of the solutions of a tropical polynomial system as its coefficients change. Tropicalising the entire regeneration process of numerical algebraic geometry, we obtain a combinatorial algorithm for finding all tropical solutions. In particular, we obtain the mixed cells of the system in a mixed volume computation. Experiments suggest that the method is not only competitive but also asymptotically performs better than conventional methods for mixed cell enumeration. The method shares many of the properties of a recent tropical method proposed by Malajovich. However, using symbolic perturbations, reverse search and exact arithmetic our method becomes reliable, memory-less and well-suited for parallelisation.