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Series: Algebra Seminar

This talk will be about polynomial decompositions that are relevant in machine learning. I will start with the well-known low-rank symmetric tensor decomposition, and present a simple new algorithm with local convergence guarantees, which seems to handily outperform the state-of-the-art in experiments. Next I will consider a particular generalization of symmetric tensor decomposition, and apply this to estimate subspace arrangements from very many, very noisy samples (a regime in which current subspace clustering algorithms break down). Finally I will switch gears and discuss representability of polynomials by deep neural networks with polynomial activations. The various polynomial decompositions in this talk motivate questions in commutative algebra, computational algebraic geometry and optimization. The first part of this talk is joint with Emmanuel Abbe, Tamir Bendory, Joao Pereira and Amit Singer, while the latter part is joint with Matthew Trager.

Series: Algebra Seminar

Wachspress defined barycentric coordinates on polygons in 1975. Warren generalized his construction to higher dimensional polytopes in 1996. In contrast to the classical case of simplices, barycentric coordinates on other polytopes are not unique. So the coordinates defined by Warren are now commonly known as Wachspress coordinates. They are used in a variety of applications, such as geometric modeling.

We connect the constructions by Warren and Wachspress by proving the conjecture that there is a unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. This is the adjoint polynomial introduced by Warren. Our formulation is the natural generalization of Wachspress' original idea.

The algebraic geometry of the map defined by the Wachspress coordinates was studied in the case of polygons by Irving and Schenk in 2014. We extend their results to higher dimensional polytopes. In particular, we show that the image of this Wachspress map is the projection from the image of the adjoint. For three-dimensional polytopes, we show that their adjoints are adjoints of K3- or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

Series: Algebra Seminar

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime (tropical) ideal is either empty or consists of a single point. This is joint work with D. Joó.

Series: Algebra Seminar

In this talk we discuss the following problem due to Peskine and Kollar: Let E be a flat family of rank two bundles on P^n parametrized by a smooth variety T. If E_t is isomorphic to O(a)+O(b) for general t in T, does it mean E_0 is isomorphic to O(a)+O(b) for a special point 0 in T? We construct counter-examples in over P^1 and P^2, and discuss the problem in P^3 and higher P^n.

Series: Algebra Seminar

The classical statement that there are 27 lines on every smooth cubic surface in $\mathbb{P}^3$ fails to hold under tropicalization: a tropical cubic surface in $\mathbb{TP}^3$ often contains infinitely many tropical lines. This pathology can be corrected by reembedding the cubic surface in $\mathbb{P}^{44}$ via the anticanonical bundle.

Under this tropicalization, the 27 classical lines become an arrangement of metric trees in the boundary of the tropical cubic surface in $\mathbb{TP}^{44}$. A remarkable fact is that this arrangement completely determines the combinatorial structure of the corresponding tropical cubic surface. In this talk, we will describe their metric and topological type as we move along the moduli space of tropical cubic surfaces. Time permitting, we will discuss the matroid that emerges from their tropical convex hull.

This is joint work with Anand Deopurkar.

Series: Algebra Seminar

This is a two day conference (March 30-31) to be held at Georgia Tech on algebraic geometry and related areas. We will have talks by Sam Payne, Eric Larson, Angelica Cueto, Rohini Ramadas, and Jennifer Balakrishnan. See https://sites.google.com/view/gattaca/home for more information.

Series: Algebra Seminar

If a finite group $G$ acts on a Cohen-Macaulay ring $A$, and the order of $G$ is a unit in $A$, then the invariant ring $A^G$ is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of $G$ is not a unit in $A$ then the Cohen-Macaulayness of $A^G$ is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that $A^G$ is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G$ on $A$ acting on strict henselizations of appropriate localizations of $A$. In a case of long-standing interest—a permutation group acting on a polynomial ring—we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.

Series: Algebra Seminar

Problems from enumerative geometry have Galois groups. Like those from field extensions, these Galois groups reflect the internal structure of the original problem. The Schubert calculus is a class of problems in enumerative geometry that is very well understood, and may be used as a laboratory to study new phenomena in enumerative geometry.I will discuss this background, and sketch a picture that is emerging from a sustained study of Schubert problems from the perspective of Galois theory. This includes a conjecture concerning the possible Schubert Galois groups, a partial solution of the inverse Galois problem, as well as glimpses of the outline of a possible classification of Schubert problems for their Galois groups.

Series: Algebra Seminar

Let X be a degree d curve in the projective space P^r.

A general hyperplane H intersects X at d distinct points; varying H defines a monodromy action on X∩H. The resulting permutation group G is the sectional monodromy group of X. When the ground field has characteristic zero the group G is known to be the full symmetric group.

By work of Harris, if G contains the alternating group, then X satisfies a strengthened Castelnuovo's inequality (relating the degree and the genus of X).

The talk is concerned with sectional monodromy groups in positive characteristic. I will describe all non-strange non-degenerate curves in projective spaces of dimension r>2 for which G is not symmetric or alternating. For a particular family of plane curves, I will compute the sectional monodromy groups and thus answer an old question on Galois groups of generic trinomials.

Series: Algebra Seminar

We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on the log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Connor Simpson and Spencer Backman.