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Session Overview
MS184, part 2: Algebraic geometry for kinematics, mechanism science, and rigidity
Wednesday, 10/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-113
53 seats, 70m^2

3:00pm - 5:00pm

Algebraic geometry for kinematics, mechanism science, and rigidity

Chair(s): Matteo Gallet (SISSA, Trieste, Italy), Josef Schicho (JKU University Linz, Austria), Hans-Peter Schröcker (University of Innsbruck, Austria)

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

Tiago Guerreiro
Loughborough University, United Kingdom

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

Gaiane Panina1, Ilia Nekrasov2
1St. Petersburg Department of Steklov Mathematical Institute, Russia, 2University of Michigan, St. Petersburg State University

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

Senja Barthel

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

Johannes Siegele, Daniel Scharler, Hans-Peter Schröcker
University of Innsbruck

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.