Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
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Location: Unitobler, F-113 53 seats, 70m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | Room free |
| Unitobler, F-113 | |
| 3:00pm - 5:00pm | MS184, part 1: Algebraic geometry for kinematics, mechanism science, and rigidity |
| Unitobler, F-113 | |
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3:00pm - 5:00pm
Algebraic geometry for kinematics, mechanism science, and rigidity Mathematicians became interested in problems concerning mobility and rigidity of mechanisms as soon as study of the subject began. Algebraists and geometers among them, notably Clifford and Study, developed tools still used today to investigate pertinent questions in the field. Recent renewed interest in techniques of algebraic geometry applied to kinematics and rigidity led to a modern classification of mechanisms, discovery of new families, development of algorithms for path planning and overall better understanding of rigid structures and configurations. A wide variety of techniques has been used in this regard and it is reasonable to expect that further influence of algebraic geometry upon kinematics and rigidity will produce deeper understanding leading to useful advancement of technology. We will focus on topics in algebraic geometry motivated by kinematics and rigidity or algebraic geometry methodology with potential application in kinematics and rigidity. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) On four-bar linkages, elliptic functions, and flexible polyhedra Darboux discovered that the (complexified) configuration space of a four-bar linkage is an elliptic curve. We present an explicit parametrization of the configuration space (in terms of the angles between the bars) by Jacobi elliptic functions and some geometric applications of this parametrization: an interpretation of Bottema's zigzag theorem, a derivation of the Dixon angle condition in the Burmester linkage, and examples of flexible quad-surfaces. Singularity distance computation for parallel manipulators of Stewart Gough Type The number of applications of parallel robots, ranging from medical surgery to astronomy, has increased enormously during the last decades due to their advantages of high speed, stiffness, accuracy, load/ weight ratio, etc. One of the drawbacks of these parallel robots are their singular configurations, where the manipulator has at least one uncontrollable instantaneous degree of freedom. Furthermore, the actuator forces can become very large, which may result in a breakdown of the mechanism. Therefore singularities have to be avoided. As a consequence the kinematic/robotic community is highly interested in evaluating the singularity closeness, but geometric a meaningful distance measure between a given manipulator configuration and the next singular configuration is still missing. We close this gap for parallel manipulators of Stewart Gough type by introducing such measures. Moreover the favored metric has a clear physical meaning, which is very important for the acceptance of this index by mechanical/constructional engineers. Every proposed singularity distance results from the solution of an algebraic system of equations, whose computational aspects are discussed on the basis of examples. Analysis of kinematic singularities through roadmap computations The analysis of kinematic singularities can be modeled as the problem of counting the connected components of a semi-algebraic set or finding a path joining two points in this set whenever such a path exists. These algorithmic problems are known to be difficult and usually tackled through the computation of a roadmap. This is an algebraic curve which will capture the connectivity of the semi-algebraic set under study. In this talk, I will review some recent progress on the state-of-the algorithms for computing roadmaps and report on their implementations which were used to analyze the kinematic singularities of some robots. Computing cognates of mechanisms A coupler cognate of a planar linkage is a different mechanism that has the same coupler curve. Roberts showed that there are 3 four-bar mechanisms that generate the same coupler curve. Dijksman provided a list of cognates for six-bar mechanisms but without proof the list was complete. This talk will describe a geometric approach to easily understand cognates which yields a simple method to generate cognates. We combine this with numerical algebraic geometry to give a method to produce a complete list of all coupler cognates. Examples on six - bar mechanisms will be shown to demonstrate the method. This is joint work with Jon Hauenstein and Charles Wampler. |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS166, part 1: Computational aspects of finite groups and their representations |
| Unitobler, F-113 | |
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10:00am - 12:00pm
Computational aspects of finite groups and their representations The theory of finite groups and their representations is not only an interesting topic for mathematicians but also provides powerful tools in solving problems in science. New computational tools are making this even more feasible. To name a few, one may find applications in physics, coding theory and cryptography. On the other hand representation theory is useful in different areas of mathematics such as algebraic geometry and algebraic topology. Due to this wide range of applications, new algorithmic methods are being developed to study finite groups and their representations from a computational perspective. Recent developments in computer algebra systems and more specifically computational linear algebra, provide tools for developments in computational aspects of finite groups and their representations. The aim of this minisymposium is to gather experts in the area to discuss the recent achievements and potential new directions. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Construction and enumeration of finite groups The talk gives a short survey on the state-of-the-art of computational methods to construct or to enumerate finite groups of a given order. Linear Time Fourier Transforms of $S_{n-k}$-invariant Functions on the Symmetric Group $S_n$ Motivated by Kondor's spectral approach to the NP-complete quadratic assignment problem, this talk discusses new techniques for the efficient computation of discrete Fourier transforms (DFTs) of Sn-k-invariant functions on the symmetric group Sn. We uncover diamond- and leaf-rake-like structures in Young's seminormal and orthogonal representations leading to relatively expensive diamond and cheaper leaf-rake computations. These local computations constitute the basis of a reduction/induction process. We introduce a new anticipation technique that avoids diamond computations at the expense of only a small arithmetic overhead for leaf-rake computations. This results in local fast Fourier transforms (FFTs). Combining these local FFTs with a multiresolution scheme closely related to the inductive version of Young's branching rule we obtain a global FFT algorithm that computes the DFT of Sn-k-invariant functions on Sn in linear time. More precisely, we show that for fixed k and all n ≥ 2k DFTs of Sn-k-invariant functions on Sn can be computed in at most ck·[Sn:Sn-k] scalar multiplications and additions, where ck denotes a positive constant depending only on k. This run-time is order-optimal and improves Maslen's algorithm. Quadratic Probabilistic Algorithms for Normal Bases It is well known that for any finite Galois extension field K/F, with Galois group G = Gal(K/F), there exists an element α in K whose orbit G·α forms an F-basis of K. Such an element α is called normal and G·α is called a normal basis. In this talk we introduce a probabilistic algorithm for finding a normal element when G is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether a random element α in K is normal can be reduced to deciding whether Σσ in G σ(α)σ in K[G] is invertible. In an algebraic model, the cost of our algorithm is quadratic in the size of G for metacyclic G and slightly subquadratic for abelian G. This is a joint work with Mark Giesbrecht (UWaterloo) and Eric Schost (UWaterloo). |
| 3:00pm - 5:00pm | MS184, part 2: Algebraic geometry for kinematics, mechanism science, and rigidity |
| Unitobler, F-113 | |
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3:00pm - 5:00pm
Algebraic geometry for kinematics, mechanism science, and rigidity Mathematicians became interested in problems concerning mobility and rigidity of mechanisms as soon as study of the subject began. Algebraists and geometers among them, notably Clifford and Study, developed tools still used today to investigate pertinent questions in the field. Recent renewed interest in techniques of algebraic geometry applied to kinematics and rigidity led to a modern classification of mechanisms, discovery of new families, development of algorithms for path planning and overall better understanding of rigid structures and configurations. A wide variety of techniques has been used in this regard and it is reasonable to expect that further influence of algebraic geometry upon kinematics and rigidity will produce deeper understanding leading to useful advancement of technology. We will focus on topics in algebraic geometry motivated by kinematics and rigidity or algebraic geometry methodology with potential application in kinematics and rigidity. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Bond theory and linkages with joints of helical type Linkages are rigid bodies assembled together by mechanical joints that allow for movement between the bodies when there is no physical constraint between them. These are arranged in 3-dimensional Euclidean space forming a closed loop. When the number of joints is not high enough, linkages are not mobile in general and are called overconstrained. However, mobile overconstrained linkages do exist and usually present very special geometric arrangements. A very recent algebraic tool used in order to retrieve what these arrangements might be is called bond theory and it has been applied in the quest for understanding and classifying such linkages. In this talk we explore how one can deal with the particular class of linkages containing helical joints following an algebraic point of view. Polygon spaces and other compactifications of M_{0,n}: Chow ring, \psi-classes and intersection numbers The moduli space of n-punctured rational curves M_{0,n} and its compactifications is a classical object, bringing together algebraic geometry, combinatorics, and topological robotics. Recently, D.I.Smyth classified all modular compactifications of M_{0,n}. In particular, an Alexander self-dual complex gives rise to a compactification of M_{0,n}, called ASD compactification. ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons. We make use of an interplay between different compactifications, and: describe the Chow rings of the ASD compactifications; compute for ASD compactifications the associated Kontsevich's psi-classes, their top monomials, and give a recurrence relation for the top monomials. Oversimplifying, the main approach is as follows. Some (but not all) ASD compactifications are the well-studied polygon spaces. A polygon space corresponds to a threshold Alexander self-dual complex. Its cohomology ring (which equals the Chow ring) is known due to J.-C. Hausmann and A. Knutson, and A. Klyachko. We shall use a computation-friendly presentation of the ring. Due to Smyth, all the modular compactifications correspond to preASD complexes, that is, to those complexes that are contained in an ASD complex. A removal of a facet of a preASD complex amounts to a blow up of the associated compactification. Each ASD compactification is achievable from a threshold ASD compactification by a sequence of blow ups and blow downs. Since the changes in the Chow ring are controllable, one can start with a polygon space, and then (by elementary steps) reach any of the ASD compactifications and describe its Chow ring. M. Kontsevich's psi-classes arise here in a standard way. Their computation of is a mere modification of the Chern number count for the tangent bundle over S^2 (a classical exercise in a topology course). The recursion and the top monomial counts follow. Distinguishing metal-organic frameworks Metal-organic frameworks are nanoporous crystalline materials that consist of metal centres that are connected by organic linkers. We consider two metal-organic frameworks as identical if they share the same bond network respecting the atom types. An algorithm is presented that decides whether two metal-organic frameworks are the same. It is based on distinguishing structures by comparing a set of invariants that is obtained from the bond network. We demonstrate our algorithm by analyzing the CoRe MOF database of DFT optimized structures with DDEC partial atomic charges using the program package ToposPro. This work is joint work with Zhenia Alexandrov, Davide Proserpio, and Berend Smit Degree Reduction of Rational Motions A rational motion can be represented by a polynomial in one indeterminate with coefficients in SE(3). In the matrix model of SE(3), the degree of the trajectories and the motion itself coincide. This is not the case for the dual quaternion model. A rational motion represented by a polynomial p in DH[t] has in general trajectories of degree 2 deg(p). However, polynomials where the degree of the trajectories is less than 2 deg p exist. In this case we speak of degree reduction. A necessary condition for degree reduction is existence of real polynomial factors in the primal part of p. In general each such factor decreases the trajectory degree by the amount of it’s own degree. There are motions with trajectories of even lower degree. We call this phenomenon exceptional degree reduction . An example of such a motion is the Darboux motion where deg p = 3, the primal part of p has a real polynomial factor of degree 2 but the degree of the trajectories is only 2. The Darboux motion also exhibits the rather strange property that the trajectory degree of the inverse motion, given by the conjugate polynomial bar{p}, has trajectory degree 4. Exceptional degree reduction can be explained in terms of one family of rulings on a certain quadric in the kinematic image space - a geometric entity which is not invariant with respect to conjugation. Moreover, our considerations yield a method to systematically construct rational motions with exceptional degree reduction. So far, the Darboux motion and its planar version were the only examples known to us. Further, we give a condition for rational motions to have a representation of lower degree in the extended kinematic image space. |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS180, part 1: Network coding and subspace designs |
| Unitobler, F-113 | |
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10:00am - 12:00pm
Network coding and subspace designs This symposium collects presentations about results on codes for linear network coding, either in the rank metric or in the subspace metric. Codes in the rank metric are usually subsets of the matrix space F_q^{m x n}, where F_q is a finite field; codes in the subspace metric are usually subsets of a finite Grassmann variety. Many interesting questions arise in this topic, e.g., about good packings in these two spaces, as well as fast encoding and decoding algorithms. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) More on exceptional scattered polynomials Let f be an F_q-linear function over F_(q^n). If the F_q-subspace U= { (x^(q^t), f(x)) : x in F_(q^n) } defines a maximum scattered linear set, then we call f a scattered polynomial of index t. We say a function f is an exceptional scattered polynomial of index t if the subspace U associated with f defines a maximum scattered linear set in PG(1, q^(mn)) for infinitely many m. There is a very interesting link between maximum scattered linear sets and the so called maximum rank distance (MRD for short) codes. In particular, a scattered polynomial over F_(q^n) defines an MRD code in (F_q)^(nxn) of minimum distance n-1. Exceptional scattered monic polynomials of index 0 (for q>5) and of index 1 have been already classified. In this work, we investigate the case t>1. The size of linear sets on a finite projective line A linear set in a finite projective space, i.e. a finite dimensional projective space over a finite field, is a set of points whose defining vectors belong to an additive subgroup of the underlying vector space of the projective space. Linear sets were introduced by G. Lunardon in the context of the construction of small blocking sets of finite Desarguesian projective planes. Meanwhile, linear sets have been used to construct and/or characterize many substructures of finite projective spaces. Recently, J. Sheekey described a correspondence between certain MRD codes and scattered linear sets on the finite projective line. After giving briefly some properties of linear sets, and connections with other objects like blocking sets and MRD codes, we will report on joint work with Geertrui Van de Voorde in which we showed a lower bound on the number of points of a linear set on a finite projective line. Rank Metric Codes and Subspace Codes in a Convolutional Setting Subspace codes have been introduced by Koetter and Kschischang in order to tackle coding problems in the area of random linear network coding. Subspace codes are subsets of a fixed Grassmannian defined over a finite field. This class of codes is also closely related to the class of ``rank metric codes''. In a first part we will show how rank metric codes induce in a natural way so called rank metric convolutional codes. We will then report about some basic properties of rank metric convolutional codes. In a second part we will show how rank metric convolutional codes can be lifted to subspace convolutional codes. Partitions of Matrix Spaces and q-Rook Polynomials I will describe the row-space and the pivot partition on the space of n x m matrices over GF(q). Both these partitions are Fourier-reflexive and yield invertible MacWilliams identities for matrix codes endowed with the row-space and the pivot enumerators, respectively. Moreover, they naturally give rise to notions of extremality. Codes that are extremal with respect to any of these notions satisfy strong rigidity properties, analogous to those of MRD codes. The Krawtchouk coefficients of both the row-space and the pivot partition can be explicitly computed using combinatorial methods. For the pivot partition, the computation relies on the properties of the q-rook polynomials associated with Ferrers diagrams, introduced by Garsia/Remmel and Haglund in the 80's. I will describe this connection between codes and rook theory, and present a closed formula for the q-rook polynomial (of any degree) associated to an arbitrary Ferrers board. The new results in this talk are joint work with Heide Gluesing-Luerssen. |
| 3:00pm - 5:00pm | MS155, part 1: Massively parallel computations in algebraic geometry |
| Unitobler, F-113 | |
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3:00pm - 5:00pm
Massively parallel computations in algebraic geometry Massively parallel methods have been a success story in high performance numerical simulation, but so far have rarely been used in computational algebraic geometry. Recent developments like the combination of the parallelization framework GPI-Space with the computer algebra system Singular have made such approaches accessible to the mathematician without the need to deal with a multitude of technical details. The minisymposium aims at bringing together researchers pioneering this approach, discussing the current state of the art and possible future developments. We plan to address applications in classical algebraic geometry, tropical geometry, geometric invariant theory and links to high energy physics. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) GPI-Space - Fraunhofer’s integrated solution to solve big problems on ultra scale machines High Performance Computing is an essential prerequisite for numerous modern scientific discoveries, which often rely on carefully tuned software stacks to harness computing power across thousands of resources and to handle massive amounts of data. One of the main challenges in this field is to enable domain scientists to take advantage of this huge computing power, while hiding the complexity of the efficient management of such resources. This talk will introduce GPI-Space, a workflow-management system for parallel applications, designed to automatically coordinate scalable, parallel executions in large, complex environments. A key advantage of GPI-Space is the separation it provides between the automatic management of parallel executions and the description of the problem-specific computational tasks and inter-dependencies. The talk will highlight the features that make GPI-Space a suitable run-time environment for algebraic geometry. More specifically, it will be discussed how parallel patterns in algorithms can be exploited by GPI-Space, with a focus on its description language (based on Petri nets), which allows scientists to model domain-specific applications independently of the execution environment. Finally, it will be presented how GPI-Space was used on top of the Singular computer algebra system to speed up the execution of algorithms for topics such Using Petri nets for parallelizing algorithms in algebraic geometry In theory, smoothness of an algebraic variety is checked by the classical Jacobian criterion. In many practical contexts, however, a direct application of this criterion is infeasible, in particular, if the codimension of the variety in its ambient space is large. A new hybrid smoothness test was recently suggested by Böhm and Frühbis-Krüger, which is based on the termination criterion from Hironaka's proof of resolution of singularities. This algorithm creates a sufficiently fine covering with affine charts, such that a relative version of the Jacobian criterion can be applied in each chart. In this talk, a massively parallel version of the algorithm is presented which has been implemented using Singular and GPI-Space. This is joint work with Janko Böhm, Wolfram Decker, Anne Frühbis-Krüger, Franz-Josef Pfreundt and Mirko Rahn. Parallel enumeration of triangulations Triangulations of point configurations are a key tool in many areas of mathematics. In tropical geometry, enumeration of all regular triangulations of certain point configurations gives rise to classification results. We will present the software mptopcom for computing triangulations of point configurations. Its core algorithm reverse search allows it to overcome the memory restraint that prevented previous software to succeed. Furthermore it is able to run in parallel. If there is a group acting on our point configuration, then it is often sufficient to enumerate orbits of triangulations rather than all triangulations. To this end we need an effective method of comparing triangulations up to symmetry. The method introduced in mptopcom avoids computing full orbits and hence is able to exploit the group action to improve performance. Module intersection method for multi-loop Feynman integral reduction We aim at a bottleneck problem of high energy physics, the linear reduction of multi-loop Feynman integrals. Our idea is to use module intersection computations in algebraic geometry to greatly simplify the Feynman integral linear relations, and then to apply a sparse linear algebra method for the reduction. With our method, the open problem of complete linear reduction of the 2-loop 5-point nonplanar hexagon-box Feynman integral was solved. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS180, part 2: Network coding and subspace designs |
| Unitobler, F-113 | |
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10:00am - 12:00pm
Network coding and subspace designs This symposium collects presentations about results on codes for linear network coding, either in the rank metric or in the subspace metric. Codes in the rank metric are usually subsets of the matrix space F_q^{m x n}, where F_q is a finite field; codes in the subspace metric are usually subsets of a finite Grassmann variety. Many interesting questions arise in this topic, e.g., about good packings in these two spaces, as well as fast encoding and decoding algorithms. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Sum-Rank Codes and Linearized Reed-Solomon Codes The sum-rank metric naturally extends both the Hamming and rank metrics in Coding Theory. In this talk, we will present some of their applications and general properties. We will also present linearized Reed-Solomon codes, which constitute the first general family of maximum sum-rank distance (MSRD) linear codes whose field sizes are subexponential in the code length. These codes are tightly connected to skew Reed-Solomon codes, and are natural hybrids between generalized Reed-Solomon codes and Gabidulin codes. On some automorphisms of polynomial rings and their applications in rank metric codes Recently, there is a growing interest in the study of rank metric codes. These codes have applications in network coding and cryptography. In this talk, I investigate some automorpshisms on polynomial rings over finite fields. We will show how the linear operators from these automorphisms can be used to construct some maximum rank distance (MRD) codes. First we will work on rank metric codes over arbitrary extension and then we will reduce these to finite fields extension. Some particular constructions give MRD codes which are not equivalent to twisted Gabidulin codes. Another application is to use these linear operators to construct some optimal rank metric codes from some Ferrers diagram. In fact we will give some examples of Ferrers diagrams for which there was no known construction of optimal rank metric codes. Invariants of rank-metric codes via Galois group action Codes in the rank metric have been introduced in 1978, but only in the last ten years they significantly gained interest due to their many applications in communications and security. These codes are linear subspaces of the space of matrices over a finite field, but they can be also seen as vectors over an extension field. There are not many explicit constructions of families of rank-metric codes with good parameters, and finding new ones hase become an important ongoing research question. However, when one considers new rank-metric codes constructions, it is important to check whether the new codes are equivalent to any other known construction. For this purpose, one wants to develop some criteria to check code equivalence. In this talk we introduce a new series of invariants of rank-metric codes obtained via the action of the Galois group of the underlying field extension. In particular, we consider the subspaces generated by the code and the application of several automorphisms to itself and show that their dimensions are invariant under code equivalence. This tool provides an easy checkable criterion for determining code inequivalence. We derive lower bounds on the number of equivalence classes of Gabidulin and twisted Gabidulin codes using this new invariant. In some special cases, the exact number of such equivalence classes is provided. |
| 3:00pm - 5:00pm | MS155, part 2: Massively parallel computations in algebraic geometry |
| Unitobler, F-113 | |
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3:00pm - 5:00pm
Massively parallel computations in algebraic geometry Massively parallel methods have been a success story in high performance numerical simulation, but so far have rarely been used in computational algebraic geometry. Recent developments like the combination of the parallelization framework GPI-Space with the computer algebra system Singular have made such approaches accessible to the mathematician without the need to deal with a multitude of technical details. The minisymposium aims at bringing together researchers pioneering this approach, discussing the current state of the art and possible future developments. We plan to address applications in classical algebraic geometry, tropical geometry, geometric invariant theory and links to high energy physics. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Tools for perturbative calculations from algebraic geometry In the last few years perturbative methods in scattering amplitudes have incorporated a lot of tools and methods from computational algebraic geometry. In this talk I will present some ideas that have been developed for reducing and calculating Feynman Integrals. A massively parallel fan traversal with applications to geometric invariant theory The GIT fan of an algebraic group action on an algebraic variety describes all GIT quotients arising from Mumford’s construction in Geometric Invariant Theory. Its computation poses a challenging hurdle due to Buchberger's algorithm with double exponential worst case complexity. We present an optimized and scalable approach for computing the GIT fan by means of computational convex geometry. By factoring out symmetries and utilizing an ultra scale computing center, we were able to traverse the GIT fan of 6-pointed stable curves of genus 0 in approximately 13 minutes, yielding 249.604 chamber orbits. Parallel algorithms for computing tropical varieties with symmetry In this talk I will report on the massively parallel computation of tropical varieties, taking their symmetry into account. For this purpose we build upon a massively parallel fan traversal method implemented by Christian Reinbold using Singular in conjunction with GPI-Space. To pass between neighboring tropical cones of the Gröbner fan we use recently developed methods by Hofmann and Ren. This is a joint work with Janko Boehm, Mirko Rahn and Yue Ren. Space sextics and their tritangents In this talk, we will briefly review the well-known algebro-geometric oddity that complex space sextic curves admit exactly 120 tritangent planes. We discuss recent works which shows that all tritangents can be totally real, as well as the current state on the question whether the 120 tritangents determine the curve uniquely. The latter context gives rise to computational algebro-combinatorial challenges in which parallelization would be of great help. This is joint work with Turku Celik, Avinash Kulkarni, Mahsa Sayyary, and Bernd Sturmfels. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS176: Algebraic geometry for kinematics and dynamics in robotics |
| Unitobler, F-113 | |
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10:00am - 12:00pm
Algebraic geometry for kinematics and dynamics in robotics A fundamental problem in robotics is to characterize the kinematics of the robotic mechanism, i.e. to infer the relationship between the joint configuration and the position of the end-effector of the robot, typically the gripper. Motions of robotics mechanisms, essentially composed by rigid links connected by joints, are often characterized using the group of rigid body motions SE(3). Exploiting Lie algebra properties, kinematics problems can be formulated as systems of polynomial equations that can be solved using algebraic geometry tools. Algebraic geometry can further be used to study the dynamics properties of robotics mechanisms, i.e. the effect of forces and torques on the robot motions. The goal of this minisymposium is to show the practical interest of algebraic geometry to analyze and control kinematic and dynamic motions of robotic systems in various applications such as solving inverse kinematic and dynamic problems, tracking manipulability ellipsoids or analyzing robots workspace. Furthermore, this minisymposium aims at bringing together mathematicians and roboticists to discuss further challenges in robotics involving application and development of algebraic geometry tools. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Some Applications of Classical Algebraic Geometry in Robotics The purpose of this talk is to give a brief overview of some problems in Robotics and how they can be viewed in terms of classical algebraic geometry. The group of proper rigid-body displacements plays a fundamental role in Robotics and a lot of Mechanical Engineering. Although this is not an algebraic group, using dual quaternions, it can be modelled as an open set in a six-dimensional non-singular quadric. Many of the linear subspaces of this quadric have important interpretations in terms of Robotics. In particular, lines through the identity element correspond to either rotational or translational one-parameter subgroups. In turn these correspond to mechanical joints in a robot or mechanism. Some other linear subspaces will be discussed. Series composition of joints then give rise to Segre varieties in the quadric and intersections of these varieties solve some important enumerative problems in Robotics. For some serial chains the displacements of the final link are the solution to a purely geometrical problem. For example, the displacements that maintain the contact between a point and a fixed plane. In these cases the solution lies in the intersection of the quadric representing the group of all rigid-body displacements and another non-singular quadric. This can be used to study parallel robots. Leading us to consider other realisations of the group. The standard 4-by-4 representation of the group gives a variety of degree 8 in P12. The rotation group SO(3) here is mapped to a Veronese variety. To look at the Gough-Stewart platform, a particular type of parallel robot, it is useful to look at an old idea, pentaspherical coodinates. This is equivalent to a realisation of the group of rigid-body displacements as a subgroup of the conformal group of R3. Finally some other application to robot motion and dynamics will be briefly discussed. A modular approach for kinematic and dynamic modeling of complex robotic systems using algebraic geometry Parallel mechanisms are increasingly being used as a modular subsystem units in the design of modern robotic systems for their superior stiffness and payload to weight ration. This leads to series-parallel hybrid robots which combine the advantages of both serial and parallel topologies but also inherit their kinematic complexity. One of the main challenges in modeling and simulation of these complex robotic systems is the existence of kinematic loops. Standard approaches in multi-body kinematics and dynamics adopt numerical resolution of loop closure constraints which leads to accuracy and inefficiency problems. These approaches give you a limited understanding of the geometry of the system. Recently, approaches from computational algebraic geometry have enabled a global description of the kinematic behavior of these complex systems. In this talk, we present a modular and analytical approach towards exploiting these algebraic methods for kinematics and dynamics modeling. This approach forms the basis of a software workbench called Hybrid Robot Dynamics (HyRoDyn). Further, we demonstrate its application in multi-body simulation and control of a complex series-parallel humanoid. Kinematics Analysis of Serial Manipulators via Computational Algebraic Geometry Kinematic singularities of a redundant serial manipulator with 7 rotational joints are analyzed and their effects on the possible self-motion are studied. We obtain the numerical kinematic singularities through algebraic varieties and demonstrate this on the kinematically redundant serial manipulator KUKA LBR iiwa. The algebraic equations for determining the variety are derived by taking the determinant of the 6-by-6 submatrix of the Jacobian matrix of the forward kinematics. By the primary decomposition, the singularities can be classified. Further analysis of the kinematic singularities including the inverse kinematics of the redundant manipulator provides us with valuable insights. Firstly, there are kinematic singularities where the inverse kinematics has no effect on the self-motion and cannot be used to avoid obstacles. Secondly, there are kinematic singularities, which lead to a single closed-loop connection with the serial redundant manipulator, so that a kinematotropic mechanism is achieved. Then we show the result of kinematic singularities of several industry robots which are obtained similarly. A special inverse kinematics analysis of a (2n+1)R serial manipulator is also presented in the end. Robot manipulability tracking and transfer Body posture influences human and robots performance in manipulation tasks, as appropriate poses facilitate motion or force exertion along different axes. In robotics, manipulability ellipsoids are used to analyze, control and design the robot dexterity as a function of the articulatory joints configuration. These ellipsoids can be designed according to different task requirements, such as tracking a desired position or applying a specific force. In the first part of this talk, we present a manipulability tracking formulation inspired by the classical inverse kinematics problem in robotics. Our formulation uses the Jacobian of the map from the joint space to the manipulability space. This relationship demands to consider that manipulability ellipsoids lie on the manifold of symmetric positive definite matrices, which is here tackled by exploiting tensor-based representations and Riemannian geometry. In the second part of the talk, we show how this tracking formulation can be combined with learning from demonstration techniques to transfer manipulability ellipsoids between robots. The presented approaches are illustrated with various robotic systems, including robotic hands, humanoids and dual-arm manipulators. |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F-113 | |
