Invited Speakers and Special Events

• Fall Workshop:
  • Greg Chirikjian: Computational Structural Biology and the Kinematics of Macromolecular Machines
    In this talk it is shown how methods originally developed to address problems in the field of robot kinematics have been applied to biophysical problems such as the analysis of conformational entropy of biopolymers and the animation of conformational transitions in multidomain proteins. In addition, it will be shown how techniques from differential geometry and noncommutative harmonic analysis can be used to analyze statistical patterns in the three-dimensional structure of proteins in the protein data bank. Open problems and future directions of the speaker's work will also be discussed.
  • Branko Grunbaum: Configurations -- combinatorial, topological, and geometric.
    This would be a survey covering, on the one hand, the history of the topic (starting about 130 years ago) with its abundant errors, as well as the recent (say last twenty years) revival and the very active developments and challenging open problems at the present time.
    
  • Igor Pak: Inflating polyhedral surfaces:
    Can you have a convex polyhedron of a smaller volume than a non-convex polyhedron with isometric surface? Imagine you start blowing air into a cube with a bendable but non-stretchable surface. What can be said about the limiting shape? These are the key questions we will consider. I will start the talk by discussing the history of the problem, present the examples and survey the earlier work on realization of surfaces. In the second part of the talk I will discuss my recent results on volume-increasing deformations of polyhedral surfaces.
    
  • Marjorie Senechal A point set puzzle revisited
    Two discrete point sets S1 and S2 are said to be homometric if, relative to some arbitrarily chosen origin, their difference sets S1' and S2' coincide, where S'={x-y : x,y in S}. Determining S' from S is entirely trivial, but the converse is not. Homometric point sets were first defined and studied in the 1930s, in connection with the interpretation of X-ray diffraction patterns. Later they were characterized algebraically. Recently, surprising examples have been found in aperiodic sequences. I hope that studying these different approaches together will shed light on their underlying geometry.
    
  • Special event: Joe O'Rourke 3D printer demo
    I will demonstrate the operation of a 3D printer, the Z-Printer 310, and a laser cutter, the Versa Laser 200. Both act as printers in the sense that, within a software application, the user selects File/Print, and they produce an object. The 3D printer is sent a file in .STL format, a special stereolithography representation of triangles in space. It "prints" the object on thin layers of powder, which it binds during the printing process. The result (after several hours!) is a solid 3D object. The laser cutter is something like a pen plotter, except with a 50 watt laser beam in place of the pen. One prepares a drawing in, say, Adobe Illustrator, and it is cut out of material (paper, plastic, wood) inside the cutter---quickly, accurately, and amidst flame and smoke.
• Rigidity Theory Day
  • Bob Connelly: Rigidity from Cauchy to granular material
    
    In 1813 A. L. Cauchy proved that convex polyhedra are rigid, even among convex polyhedra with given facets. Since then, this result has been looked at infinitesimally, lemmas extended globally, in higher dimensions, metrically, at second-order, and even extended to nearby non-convex polyhedra. The infinitesimal theory of rigidity has a wide range of applications, and there are some interesting difficulties in modeling problems concerning how it is used for packings and granular materials. I will give an overview of these questions from my point of view.
    
    
  • Henry Crapo: Rigidity as Homology
  • Bob Norton: Kinematics in Engineering
    Kinematics is fundamental to the design of all mechanisms and machinery. Engineers create linkages, cams, and gear trains to perform a multitude of tasks. We all use their mechanisms every day without realizing so. Our automobiles are full of them, as are our households. Most of the consumer products we buy are manufactured with mechanisms of this type. This talk will describe a few of the mathematical techniques (both geometric and algebraic) that mechanical engineers typically use to synthesize linkages and create mechanisms for practical applications. While the mathematical basis of linkage synthesis has been fairly well developed, there still remain unsolved problems in kinematics. Some of the mathematical techniques recently developed require cpu-hours of computer time to calculate. This makes them impractical for any real engineering problem solutions and useful only as academic exercises. There is still a great deal of experiential art and intuition needed to successfully design kinematic devices. Some of the fundamental precepts of linkage analysis will be briefly reviewed. A few geometrical synthesis techniques will be described and their algebraic analogues introduced. Very little engineering design is now done without extensive use of computers and commercial software, and these typically require algebraic methodology. A few Computer-Aided-Design (CAD) tools will be described in respect to their use in this context. Finally, a case study of the redesign of a typical mechanism using modern CAD techniques will be presented.
    
  • Rudi Penne: Instant centers in planar mechanisms
  • Brigitte Servatius: The molecular conjecture in 2-d
    Given a graph whose vertices represent rigid bodies in 2-space and whose edges represent pins holding their endpoints together, polarity translates the graph into a system of straight line segments pinned appropriately. The straight line segments can be interpreted as the rigid bodies with collinear pins. We will discuss the Jackson-Jordan proof that the collinearity of the pins does not decrease the rank.
  • Herman Servatius: Constrained Position Vector Configurations
    We examine general systems of vectors constrained by mutual inner products, and show how such a system is related to various geometric and combinatorial constructions.
  • Meera Sitharam: Rigidity and Geometric Constraint Decomposition
    Geometric Constraint Systems (GCSs) arise in numerous applications from mechanical computer aided design to molecular modeling. ``Good'' decompositions of GCSs are crucial not just for efficient solving and updates, but also for describing and sampling the solution space, simulating motion, autoconstraining of under-constrained systems, constraint system reformulation, appropriately dealing with over-constraints, etc. This talk will show the close connection between obtaining a ``good'' decomposition and obtaining a combinatorial characterization of generic rigidity.
• Special event: There will be an additional talk by Branko Grunbaum on Thursday, 5:00pm - joint event of the Math and CS departments at Smith College. If you plan to arrive on Thursday, you may catch it, too.
  The unexplored world of nonconvex polyhedra
  An examination of the two main misdirections taken in the theory of polyhedra (courtesy of Poinsot and Bruckner), and a presentation of some of the many exciting and unexpected classes of "new" polyhedra and their properties.