No abstract available.
Monday, November 17th, 2014
We begin by reviewing some useful M2 commands to work with tensors, starting from Sylvester algorithm to write a binary form as a sum of powers. We discuss two algorithms to check the identifiability of a specific tensor. The first one is deterministic (joint with Chiantini and Vannieuwenhoven) and explores the geometry of the decomposition, by computing the tangent space of the contact variety. The second one is probabilistic and uses software Bertini and homotopic continuation methods (joint with Hauenstein, Oeding, Sommese).
Tuesday, November 18th, 2014
I will present results on the geometry of sets of bivariate symmetric tensors of fixed typical rank. These sets are semialgebraic, andI will discuss algebraic boundary of these sets, number of connected components as well as the the geometry of the set of possible decomposition, which is also semialgebraic.
This is joint work in progress with Salvadore Barone.
The concept of E-eigenvector (or just eigenvector) of a tensor was introduced by L. Qi in 2007. This talk is about schemes associated with tensor eigenvectors (called eigenschemes for tensors). The focus of this talk is on eigenschemes for matrices. I will show that information about the Jordan canonical form of a matrix is encoded in the primary decomposition of the ideal of the eigenscheme for the matrix. If time permits, schemes associated with eigenvectors of more general tensors will be discussed.
A statistical model is a family of probability distributions where joint probabilities are often specified parametrically. If the joint probabilities are specified parametrically by polynomials, then the closure of the model is an algebraic variety. An underlying idea in algebraic statistics is that information about the variety yields statistical information about the model. Mixtures of independence models form a class of models that falls under this paradigm, indeed, their closures are sets of tensors with bounded border rank, i.e. secant varieties. In this talk, we will introduce mixture models, their closure, and their applications to phylogentics and evolutionary biology.
Wednesday, November 19th, 2014
In this talk, we will describe our computational and theoretical interest in toric vector bundles. After reviewing the basic features of toric vector bundles, we will introduce a collection of rational convex polytopes associated to a toric vector bundle. Lattice points in these polytopes correspond to generators for the space of global sections and edges are related to jets. These polyhedral tools also lead to new bundles with an intriguing mix of positivity properties.
No Abstract Available
Secant varieties of Segre and Segre-Veronese varieties, as well as linear systems interpolating multiple points, provide a bridge between tensor algebra and algebraic geometry and are of interest in applications.
In this talk I will introduce these objects and their dimensionality problem, and will give an overview about known results and open conjectures. I will tell about how the use of computer algebra systems such as Macaulay2 is fundamental for these questions, because computer based computations not only cover the initial cases (most of the proofs are by induction), but also help understand the general behaviour.
Thursday, November 20th, 2014
The BGG correspondence relates complexes over a polynomial ring and complexes over a corresponding exterior algebra. It also leads to exterior algebra methods for computing sheaf cohomology of a sheaf on projective space. I will discuss generalizations of these ideas to products of projective spaces, including a new exterior algebra approach for computing sheaf cohomology on a product of projective spaces. This is joint work with Eisenbud and Schreyer.
Generalized counting constraint satisfaction problems include Holant problems with planarity restrictions; polynomial-time algorithms for such problems include matchgates and matchcircuits, which are based on Pfaffians. In particular, they use gates which are expressible in terms of a vector of sub-Pfaffians of a skew-symmetric matrix. We introduce a new type of circuit based instead on determinants, with seemingly different expressive power. In these determinantal circuits, a gate is represented by the vector of all minors of an arbitrary matrix. Determinantal circuits permit a different class of gates. Applications of these circuits include proofs of theorems from algebraic graph theory including the Chung-Langlands formula for the number of rooted spanning forests of a graph and computing Tutte Polynomials of certain matroids. They also give a strategy for simulating quantum circuits with closed timelike curves. Monoidal category theory provides a useful language for discussing such counting problems, turning combinatorial restrictions into categorical properties. We introduce the counting problem in monoidal categories and count-preserving functors as a way to study FP subclasses of problems in settings which are generally #P-hard. Using this machinery we show that, surprisingly, determinantal circuits can be simulated by Pfaffian circuits at quadratic cost.