Anthony Nixon
Title: Global rigidity of linearly constrained frameworks.
Abstract: A bar-joint framework (G,p) in d-dimensional Euclidean space is the combination of a graph G and a map p assigning positions to the vertices of G. The framework is rigid if the only edge-length-preserving continuous motions of the vertices arise from isometries of d-space. The framework is globally rigid if every other framework with the same edge lengths arises from isometries of d-space. I will survey some important results about rigid and globally rigid frameworks and then introduce linearly constrained frameworks. Linearly constrained frameworks are a generalisation of frameworks in which some vertices are constrained to lie on one or more given hyperplanes. Streinu and Theran characterised rigid linearly constrained generic frameworks in 2-space in 2010. I will describe an analogous result for the global rigidity of linearly constrained generic frameworks in 2-space. Extensions of results for rigidity and global rigidity to higher dimensions will also be discussed.
This is joint work with Hakan Guler and Bill Jackson.
Alexander Kasprzyk
Title: Mutations: towards a classification of terminal Q-Fano 3-folds?
Abstract: Recent work by Coates, Corti, Kasprzyk, and others has focused on the idea of classifying Fano manifolds via mirror symmetry. Crucial to this idea is the concept of a mutation of a toric variety: a Q-Gorenstein deformation that has a particularly nice combinatorial interpretation. This leads naturally to the study of mutation-equivalence classes of Fano polytope, and the classification of the so-called rigid maximally mutable Laurent polynomials (rigid MMLPs). We conjecture that rigid MMLPs correspond to (certain) Q-factorial terminal Fano varieties, something that was hidden in low dimensions. In this talk I will explain how we can used mutations to study Fano varieties in this way.
Marc Diesse
Title: On local real algebraic geometry and applications to kinematics.
Abstract: We address the problem of identifying non-smooth points of real algebraic sets, which is important in kinematics for analyzing configuration spaces of linkages. In contrast to complex varieties, a real algebraic variety X can still be smooth at singular points in the sense that X is locally an analytic submanifold of real euclidean space. This happens because analytic branches of the variety might not be visible in real space. By analyzing the formal completion (of the integral closure) of the local ring at such points we can work out conditions to (dis)prove smoothness at singular points. For an application we will demonstrate the method on some algebraic curves and the configuration spaces of a class of planar linkages.
Fatemeh Mohammadi
Title: Generalized Permutohedra from Probabilistic Graphical Models
Abstract: Graphical models (Bayesian networks) based on directed acyclic graphs (DAGs) are used to model complex cause-and-effect systems. A graphical model is a family of joint probability distributions over the nodes of a graph which encodes conditional independence relations via the Markov properties. One of the fundamental problems in causality is to learn an unknown graph based on a set of observed conditional independence relations. In this talk, I will describe a greedy algorithm for DAG model selection that operates via edge walks on so-called DAG associahedra. For an undirected graph, the set of conditional independence relations are represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. For any regular Gaussian model, and its associated set of conditional independence relations we construct the analogous polytope DAG associahedron which can be defined using relative entropy. For DAGs we construct this polytope as a Minkowski sum of matroid polytopes corresponding to Bayes-ball paths in a graph. This is joint work with Caroline Uhler, Charles Wang, and Josephine Yu.
Hans-Peter Schröcker
Title: A Plea for Non-Euclidean Geometry in Mechanism Science
Abstract: The year 2013 saw the advent of two fundamentally new methods for the synthesis and for the analysis of mechanisms: Motion factorization and bond theory. The former decomposes rational motions into a sequence of coupled elementary motions, usually rotations. The latter exploits combinatorial and geometric properties related to points with “degenerate kinematic behaviour” in a mechanism’s configuration curve. Ever since, both theories proved their theoretical value and their suitability for applications. In spite of considerable progress of both theories, some mysteries still remain.
In this talk we summarize recent findings in motion factorization and bond theory. We also aim at a better understanding by looking at non-Euclidean interpretations. While motion factorization and bond theory in their original formulation are based on the dual quaternion model of rigid body displacements, non-Euclidean formulations use different quaternion algebras and unveil properties that are not easily accessible in the Euclidean setup.
Josef Schicho
Title: Algebraic Methods and Questions for Linkages with Helical Joints
Abstract: The stabilizers of the helices are the only closed subgroups of the group of Euclidean displacements that are not algebraic. Consequently, the configuration spaces of linkages with helical joints are, in general, only analytic and not algebraic. But by [ArXiV:1312.1060], every linkage with helical joints can be related to a linkage without helical joints by a method that is able to explain the mobility of the former ("screw carving"). After briefly explaining the method, we will discuss some implications and pose more algebraic questions that would have implications for linkages with helical joints.