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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.

Series: Algebra Seminar

Duality is an important feature in convexity and in projective algebraic
geometry. We will discuss the interplay of these two dualities for the
cone of sums of squares of ternary forms and its dual cone, the Hankel
spectrahedron.

Series: Algebra Seminar

We will introduce, through examples, the philosophy of Delignethat "in characteristic zero, a deformation problem is controlled by adifferential graded (or "dg-") Lie algebra." Focusing on the deformationtheory of representations of a group, we will give an extension of thisphilosophy to positive characteristic. This will be justified by thepresence of a dg-algebra controlling the deformations, and the fact thatthe cohomology of the dg-algebra has an A-infinity algebra structureexplicitly presenting the deformation problem. This structure can bethought of as "higher cup products" on group cohomology, extending theusual cup product and often computable as Massey products. We will writedown concrete, representation-theoretic questions that are answered bythese higher cup products. To conclude, we will show that the cup productstructure on Galois cohomology, which is the subject of e.g. the motivicBloch-Kato conjecture and its proofs, is enriched by these higher cupproducts, and that this enrichment reflects properties of the Galois group.Familiarity with dg-algebras and infinity-algebras will not be presumed.

Series: Algebra Seminar

After reminding everyone why the symmetric powers Sym^n X of a scheme arise and are interesting from the point of view of the Weil conjectures, I'll recall the Dold-Thom theorem of algebraic topology, which governs the behavior of symmetric powers of a topological space. I'll then explain how the notion of étale homotopy allows us to compare these two realms of arithmetic geometry and algebraic topology, providing a homotopical refinement of a small part of the Weil conjectures.

Series: Algebra Seminar

A reciprocal linear space is the image of a linear space under
coordinate-wise inversion. This nice algebraic variety appears in many
contexts and its structure is governed by the combinatorics of the
underlying hyperplane arrangement. A reciprocal linear space is also an
example of a hyperbolic variety, meaning that there is a family of
linear spaces all of whose intersections with it are real. This special
real structure is witnessed by a determinantal representation of its
Chow form in the Grassmannian. In this talk, I will introduce reciprocal
linear spaces and discuss the relation of their algebraic properties to
their combinatorial and real structure.