Algebraic geometry of quantum graphical models

The conventional wisdom expressed at several recent joint conferences and workshops on algebraic geometry and geometric modeling – Algebraic Geometry and Geometric Modeling 2002, Vilnius, Lithuania, The MSRI Workshop on Real Algebraic Geometry in Applications, Berkeley, 2004, and Algebraic Geometry and Geometric Modeling 2004, Nice, France – is that algebraic geometry has much to offer geometric modeling. Algebraic geometry and geometric modeling both deal with polynomials: Algebraic geometry investigates the algebraic and geometric properties of polynomials; geometric modeling uses polynomials to build computer models for industrial design and manufacture. Since algebraic geometry is by far the older, more mature discipline, geometric modeling should have a lot to learn from algebraic geometry.

Mathematical algorithms are a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. This book provides a bridge between algebraic geometry and geometric modelling algorithms, formulated within a computer science framework.Apart from the algebraic geometry topics covered, the entire book is based on the unifying concept of using algebraic techniques properly specialized to solve geometric problems to seriously improve accuracy, robustness and efficiency of CAD-systems. It provides new approaches as well as industrial applications to deform surfaces when animating virtual characters, to automatically compare images of handwritten signatures and to improve control of NC machines.This book further introduces a noteworthy representation based on 2D contours, which is essential to model the metal sheet in industrial processes. It additionally reviews applications of numerical algebraic geometry to differential equations systems with multiple solutions and bifurcations.Future Vision and Trends on Shapes, Geometry and Algebra is aimed specialists in the area of mathematics and computer science on the one hand and on the other hand at those who want to become familiar with the practical application of algebraic geometry and geometric modelling such as students, researchers and doctorates.

The two fields of Geometric Modeling and Algebraic Geometry, though closely related, are traditionally represented by two almost disjoint scientific communities. Both fields deal with objects defined by algebraic equations, but the objects are studied in different ways. This contributed book presents, in 12 chapters written by leading experts, recent results which rely on the interaction of both fields. Some of these results have been obtained in the frame of the European GAIA II project (IST 2001-35512) entitled Intersection algorithms for geometry-based IT applications using approximate algebraic methods.

Urban scene recognition by graphical model and 3D geometry

We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

This paper provides an overview of the tools used to conduct independent modeling and simulation of Artemis 1 separation events and highlights challenges and lessons learned across a decade of analysis. An overview of each of the tools along with the process flow and key components of maintaining independence from the Program’s baseline tool set will be provided. In addition, each of the separation analyses conducted will be discussed, making note of key models incorporated for each event and unique challenges that were addressed. The technical work described in this paper is part of an on-going technical assessment conducted by the NASA Engineering and Safety Center to develop and maintain an independent modeling and simulation capability in support of Space Launch System (SLS) and Multi-Purpose Crew Vehicle (MPCV) throughout design, verification, and flight readiness cycles. For Artemis 1, an end-to-end analysis approach was taken with events simulated from launch to re-entry. This paper will focus on the separation events, including liftoff clearance analysis, solid rocket booster (SRB) separation from Core Stage (CS), Service Module (SM) panel jettison, ICPS / Core Stage Separation, and MPCV/ ICPS separation. To assess separation clearances, the NESC has developed a high fidelity, independent 6-degree of freedom (DOF) simulation of the separation events. The simulation was developed using the Program to Optimize Simulated Trajectories II (POST2). POST2 is an industry standard flight trajectory optimization and simulation tool that is Class D compliant and follows NASA NPR 7150.2, which includes regression testing, unit testing, and configuration control of the simulation. The primary objective of the SLS separation and clearance analyses was to estimate the clearances between different vehicle components and/or ground support equipment (GSE) during separation events. The clearance is defined as the minimum distance from any point on one 3D geometry model associated with one vehicle component (or GSE) to any point on another 3D geometry model associated with a vehicle component separating from the main body. The vehicle geometry models used by the NESC to compute the separation clearances were derived from CAD models. POST2 simulation results were post processed in the Exploration Visualization Environment (EVE) tool to compute clearances, and detect recontact occurrences, if any, between the separating stages. EVE is a simulation, visualization, and analysis system designed to integrate time-based dynamics data with detailed graphical models in a full-scale virtual environment. EVE was used to simultaneously calculate more than 30 different separation clearances by “driving” detailed vehicle geometry models with position and attitude time history data obtained from the POST2 simulation. EVE displays each clearance distance visually, at any given instant of time, with a straight line connecting the two closest points on each object and records the time history of the clearance distance in a file. If the clearance reduces to zero, then EVE reports it as a recontact. To perform separation analyses, POST2 Monte Carlo analyses were performed by running 2000 dispersed cases that included over hundreds of uncertainties and dispersions for a range of simulation models including aerodynamics, propulsion, navigation sensors, control actuators, slosh and flexible body dynamics, and winds and atmosphere. Clearances were computed for each of the dispersed Monte Carlo trajectories and overall separation performance was statistically assessed. Over the past decade the POST2 simulation and EVE visualization tools have been used to perform detailed assessments of the separation events that occur during the Artemis 1 ascent. To support these assessments, numerous unique simulation models were developed and integrated into the standard suite of POST2 simulation models to address and resolve key concerns specific to each of the Artemis 1 separation events. In addition to presenting an overview of the multi-body 6-DOF simulation and separation analysis capability, the paper will discuss the integration of key simulation models such as flexible body dynamics, slosh dynamics, separation mechanisms, multi-body aerodynamics, and environmental models. Additional details are provided about other challenges that were addressed, including the use of a convex hull to reduce computational time for clearance calculations, the modeling of incidental recontact during separation events, and the capability to simulate failure scenarios.

This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. Here, we address the question of how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. The Newton polytope of a statistical model plays a key role. Our results are applied to the hidden Markov model and the general Markov model on a binary tree.

We study submodels of Gaussian directed acyclic graph (DAG) models defined by partial homogeneity constraints imposed on the model error variances and structural coefficients. We represent these models with coloured DAGs and investigate their properties for use in statistical and causal inference. Local and global Markov properties are provided and shown to characterize the coloured DAG model. Additional properties relevant to causal discovery are studied, including the existence and nonexistence of faithful distributions and structural identifiability. Extending prior work of Peters and Bühlmann and Wu and Drton, we prove structural identifiability under the assumption of homogeneous structural coefficients, as well as for a family of models with partially homogeneous structural coefficients. The latter models, termed blocked properly edge-coloured DAGS (BPEC-DAGs), capture additional causal insights by clustering the direct causes of each node into communities according to their effect on their common target. An analogue of the greedy equivalence search algorithm for learning BPEC-DAGs is given and evaluated on real and synthetic data. Regarding model geometry, we provide a proof of a conjecture of Sullivant which generalizes to coloured DAG models, coloured undirected graphical models and directed ancestral graph models. The proof yields a tool for identification of Markov properties for any rationally parametrized model with globally, rationally identifiable parameters.

Robot manipulators generally rely on complete knowledge of object geometry in order to plan motions and compute successful grasps. However, manipulating real-world objects poses a substantial modelling challenge. New instances of known object classes may vary from learned models. Objects that are not perfectly rigid may appear in new configurations that do not match any of the known geometries. In this paper we describe an algorithm for learning generative probabilistic models of object geometry for the purposes of manipulation; these models capture both non-rigid deformations of known objects and variability of objects within a known class. Given a single image of partially occluded objects, the model can be used to recognize objects based on the visible portion of each object contour, and then estimate the complete geometry of the object to allow grasp planning. We provide two main contributions: a probabilistic model of shape geometry and a graphical model for performing correspondence between shape descriptions. We show examples of learned models from image data and demonstrate how the learned models can be used by a manipulation planner to grasp objects in cluttered visual scenes.

This article discusses the connection between the matrix models and algebraic geometry. In particular, it considers three specific applications of matrix models to algebraic geometry, namely: the Kontsevich matrix model that describes intersection indices on moduli spaces of curves with marked points; the Hermitian matrix model free energy at the leading expansion order as the prepotential of the Seiberg-Witten-Whitham-Krichever hierarchy; and the other orders of free energy and resolvent expansions as symplectic invariants and possibly amplitudes of open/closed strings. The article first describes the moduli space of algebraic curves and its parameterization via the Jenkins-Strebel differentials before analysing the relation between the so-called formal matrix models (solutions of the loop equation) and algebraic hierarchies of Dijkgraaf-Witten-Whitham-Krichever type. It also presents the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, along with higher expansion terms and symplectic invariants.

Algebraic Geometry provides an impressive theory targeting the understanding of geometric objects defined algebraically. Geometric Modeling uses every day, in order to solve practical and difficult problems, digital shapes based on algebraic models. In this book, we have collected articles bridging these two areas. The confrontation of the different points of view results in a better analysis of what the key challenges are and how they can be met. We focus on the following important classes of problems: implicitization, classification, and intersection. The combination of illustrative pictures, explicit computations and review articles will help the reader to handle these subjects

Let ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

This document contains notes from the lectures of Corti, Kollar, Lazarsfeld, and Mustaţa at the workshop ``Minimal and canonical models in algebraic geometry at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan.

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [24]. We expand on the recent work of Demailly and Kollar [14] and Johnson and Kollar [20] who give methods for constructing Kahler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors [9], circle V-bundles over log del Pezzo surfaces with Kahler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [36] together with [11] must be diffeomorphic to S5#l(S2×S3). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S2×S3 and infinite families of such structures on #l(S2×S3) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.

Algebraic geometry and stochastic complexity of hidden Markov models