BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michael Farber (School of Mathematical Sciences Queen Mary\, Unive rsity of London) DTSTART:20200917T140000Z DTEND:20200917T144500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/1 DESCRIPTION:Title: Topology of parametrised motion planning algorithms\nby Mic hael Farber (School of Mathematical Sciences Queen Mary\, University of Lo ndon) as part of CMO workshop: Topological Complexity and Motion Planning\ n\n\nAbstract\nWe introduce and study a new concept of parameterised topol ogical complexity\, a topological invariant motivated by the motion planni ng problem of robotics. In the parametrised setting\, a motion planning al gorithm has high degree of universality and flexibility\, it can function under a variety of external conditions (such as positions of the obstacles etc). We explicitly compute the parameterised topological complexity of o bstacle-avoiding collision-free motion of many particles (robots) in 3-dim ensional space. Our results show that the parameterised topological comple xity can be significantly higher than the standard (non-parametrised) inva riant. Joint work with Daniel Cohen and Shmuel Weinberger.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Ayse Borat (Bursa Technical University) DTSTART:20200917T150000Z DTEND:20200917T154500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/2 DESCRIPTION:Title: A simplicial analog of homotopic distance\nby Ayse Borat (B ursa Technical University) as part of CMO workshop: Topological Complexity and Motion Planning\n\n\nAbstract\nHomotopic distance as introduced by Ma cias-Virgos and Mosquera-Lois in [2] can be realized as a generalization o f topological complexity (TC) and Lusternik Schnirelmann category (cat). I n this talk\, we will introduce a simplicial analog (in the sense of Gonza lez in [1]) of homotopic distance and show that it has a relation with sim plicial complexity (SC) as homotopic distance has with TC. We will also in troduce some basic properties of simplicial distance.\n\n[1] J. Gonzalez\, Simplicial Complexity: Piecewise Linear Motion Planning in Robotics\, New York Journal of Mathematics 24 (2018)\, 279-292.\n\n[2] E. Macias-Virgos\ , D. Mosquera-Lois\, Homotopic Distance between Maps\, preprint. arXiv: 18 10.12591v2.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Koditschek (University of Pennsylvania) DTSTART:20200917T161500Z DTEND:20200917T163000Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/3 DESCRIPTION:Title: Vector Field Methods of Motion Planning\nby Daniel Koditsch ek (University of Pennsylvania) as part of CMO workshop: Topological Compl exity and Motion Planning\n\n\nAbstract\nA long tradition in robotics has deployed dynamical systems as “reactive” motion planners by encoding g oals as attracting sets and obstacles as repelling sets of vector fields a rising from suitably constructed feedback laws [1] . This raises the prosp ects for a topologically informed notion of “closed loop” planning com plexity [2]\, holding substantial interest for robotics\, and whose contra st with the original “open loop” notion [3] may be of mathematical int erest as well. This talk will briefly review the history of such ideas and provide context for the next three talks which discuss some recent advanc es in the closed loop tradition\, reviewing the implications for practical robotics as well as associated mathematical questions.\n\n[1] D. E. Kodit schek and E. Rimon\, “Robot navigation functions on manifolds with bound ary\,” Adv. Appl. Math.\, vol. 11\, no. 4\, pp. 412–442\, 1990\, doi: doi:10.1016/0196-8858(90)90017-S.\n\n[2] Y. Baryshnikov and B. Shapiro\, “How to run a centipede: a topological perspective\,” in Geometric Con trol Theory and Sub-Riemannian Geometry\, Springer International Publishin g\, 2014\, pp. 37–51.\n\n[3] M. Farber\, “Topological complexity of mo tion planning\,” Discrete Comput. Geom.\, vol. 29\, no. 2\, pp. 211–22 1\, 2003.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Vasileios Vasilopoulos (University of Pennsylvania) DTSTART:20200917T163000Z DTEND:20200917T164500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/4 DESCRIPTION:Title: Doubly Reactive Methods of Task Planning for Robotics\nby V asileios Vasilopoulos (University of Pennsylvania) as part of CMO workshop : Topological Complexity and Motion Planning\n\n\nAbstract\nA recent advan ce in vector field methods of motion planning for robotics replaced the ne ed for perfect a priori information about the environment’s geometry wit h a real-time\, “doubly reactive” construction that generates the vect or field as well as its flow at execution time – directly from sensory i nputs – but at the cost of assuming a geometrically simple environment [ 5] . Still more recent developments have adapted to this doubly reactive o nline setting the original offline deformation of detailed obstacles into their geometrically simple topological models. Consequent upon these new i nsights and algorithms\, empirical navigation can now be achieved in parti ally unknown unstructured physical environments by legged robots\, with fo rmal guarantees that ensure safe convergence for simpler\, wheeled mechani cal platforms. These ideas can be extended to cover a far broader domain o f robot task planning wherein the robot has the job of rearranging objects in the world by visiting\, grasping\, moving them [10] and then repeating as necessary until the rearrangement task is complete.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul Gustafson (Wright State University) DTSTART:20200917T164500Z DTEND:20200917T170000Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/5 DESCRIPTION:Title: A Category Theoretic Treatment of Robot Hybrid Dynamics with Ap plications to Reactive Motion Planning and Beyond\nby Paul Gustafson ( Wright State University) as part of CMO workshop: Topological Complexity a nd Motion Planning\n\n\nAbstract\nHybrid dynamical systems have emerged fr om the engineering literature as an interesting new class of mathematical objects that intermingle features of both discrete time and continuous tim e systems. In a typical engineering setting\, a hybrid system describes th e evolution of states driven into different physical modes by events that may be instigated by an external controller or simply imposed by the natur al world. Extending the formal convergence and safety guarantees of the or iginal omniscient reactive systems introduced in the first talk of this se ries to the new imperfectly known environments negotiated by their doubly reactive siblings introduced in the second talk requires reasoning about h ybrid dynamical systems wherein each new encounter with a different obstac le triggers a reset of the continuous model space [11]. A recent categoric al treatment [12] of robot hybrid dynamical systems [13] affords a method of hierarchical composition\, raising the prospect of further formal exten sions that might cover as well the more broadly useful class of mobile man ipulation tasks assigned to dynamically dexterous (e.g.\, legged) robots.\ n\n[11] V. Vasilopoulos\, G. Pavlakos\, K. Schmeckpeper\, K. Daniilidis\, and D. E. Koditschek\, “Reactive Navigation in Partially Familiar Non-Co nvex Environments Using Semantic Perceptual Feedback\,” Rev.\, p. (under review)\, 2019\, [Online]. Available: https://arxiv.org/abs/2002.08946.\n \n[12] J. Culbertson\, P. Gustafson\, D. E. Koditschek\, and P. F. Stiller \, “Formal composition of hybrid systems\,” Theory Appl. Categ.\, no. arXiv:1911.01267 [cs\, math]\, p. (under review)\, Nov. 2019\, Accessed: N ov. 24\, 2019. [Online]. Available: http://arxiv.org/abs/1911.01267.\n\n[1 3] A. M. Johnson\, S. A. Burden\, and D. E. Koditschek\, “A hybrid syste ms model for simple manipulation and self-manipulation systems\,” Int. J . Robot. Res.\, vol. 35\, no. 11\, pp. 1354--1392\, Sep. 2016\, doi: 10.11 77/0278364916639380.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthew Kvalheim (University of Pennsylvania) DTSTART:20200917T170000Z DTEND:20200917T171500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/6 DESCRIPTION:Title: Toward a Task Planning Theory for Robot Hybrid Dynamics\nby Matthew Kvalheim (University of Pennsylvania) as part of CMO workshop: To pological Complexity and Motion Planning\n\n\nAbstract\nA theory of topolo gical dynamics for hybrid systems has recently begun to emerge [14]. This talk will discuss this theory and\, in particular\, explain how suitably r estricted objects in the formal category introduced in the third talk of t his series can be shown to admit a version of Conley’s Fundamental Theor em of Dynamical Systems. This raises the hope for a more general theory of dynamical planning complexity that might bring mathematical insights from both the open loop [3] and closed loop [2] tradition to the physically in eluctable but mathematically under-developed class of robot hybrid dynamic s [13].\n\n[2] Y. Baryshnikov and B. Shapiro\, “How to run a centipede: a topological perspective\,” in Geometric Control Theory and Sub-Riemann ian Geometry\, Springer International Publishing\, 2014\, pp. 37–51.\n\n [3] M. Farber\, “Topological complexity of motion planning\,” Discrete Comput. Geom.\, vol. 29\, no. 2\, pp. 211–221\, 2003.\n\n[13] A. M. Joh nson\, S. A. Burden\, and D. E. Koditschek\, “A hybrid systems model for simple manipulation and self-manipulation systems\,” Int. J. Robot. Res .\, vol. 35\, no. 11\, pp. 1354--1392\, Sep. 2016\, doi: 10.1177/027836491 6639380.\n\n[14] M. D. Kvalheim\, P. Gustafson\, and D. E. Koditschek\, “Conley’s fundamental theorem for a class of hybrid systems\,” ArXiv 200503217 Cs Math\, p. (under review)\, May 2020\, Accessed: May 31\, 2020 . [Online]. Available: http://arxiv.org/abs/2005.03217.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Jie Wu (Hebei Normal University and National University of Singapo re) DTSTART:20200918T140000Z DTEND:20200918T144500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/7 DESCRIPTION:Title: Topological complexity of the work map\nby Jie Wu (Hebei No rmal University and National University of Singapore) as part of CMO works hop: Topological Complexity and Motion Planning\n\n\nAbstract\nWe introduc e the topological complexity of the work map associated to a robot system. In broad terms\, this measures the complexity of any algorithm controllin g\, not just the motion of the configuration space of the given system\, b ut the task for which the system has been designed. From a purely topologi cal point of view\, this is a homotopy invariant of a map which generalize s the classical topological complexity of a space. Joint work with Aniceto Murillo.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Petar Pavesic (University of Ljubljana) DTSTART:20200918T150000Z DTEND:20200918T154500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/8 DESCRIPTION:Title: Two questions on TC\nby Petar Pavesic (University of Ljublj ana) as part of CMO workshop: Topological Complexity and Motion Planning\n \n\nAbstract\n1. What is the $TC$ of a wedge?\n\nIn the literature one can find two relatively coarse estimates of $TC(X\\vee Y)$:\nFarber states th at\n$$\\max\\{TC(X)\,TC(Y)\\} \\le TC(X\\vee Y)\\le \\max\\{TC(X)\,TC(Y)\, cat(X)+cat(Y)-1\\}$$\n(where the proof of the upper bound is only sketche d)\, while\nDranishnikov gives \n$$\\max\\{TC(X)\,TC(Y)\, cat(X\\times Y) \\} \\le TC(X\\vee Y)\\le TC(X)+TC(Y)+1.$$\nAt first sight the two estimat es almost contradict each other\, because the overlap of the two \ninterva ls is very small. Nevertheless\, all known examples satisfy both estimates . We will show \nthat under suitable assumptions Dranishnikov's method yie lds a proof of Farber's upper bound.\n\n2. What can be said about closed m anifolds with small TC?\n\nIf $M$ is a closed manifold with $TC(M)=2$\, th en by Grant\, Lupton and Oprea $M$ is homeomorphic to an odd-dimensional s phere. We will make another step and study closed manifolds whose topologi cal complexity is equal to 3.\n\nOf course\, all spaces considered are CW- complexes and $TC(\\mathbf{\\cdot})=1$.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Hellen Colman (Wright College) DTSTART:20200918T161500Z DTEND:20200918T170000Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/9 DESCRIPTION:Title: Morita Invariance of Invariant Topological Complexity\nby H ellen Colman (Wright College) as part of CMO workshop: Topological Complex ity and Motion Planning\n\n\nAbstract\nWe show that the invariant topologi cal complexity defines a new numerical invariant for orbifolds.\n\nOrbifol ds may be described as global quotients of spaces by compact group actions with finite isotropy groups. The same orbifold may have descriptions invo lving different spaces and different groups. We say that two actions are M orita equivalent if they define the same orbifold. Therefore\, any notion defined for group actions should be Morita invariant to be well defined fo r orbifolds.\n\nWe use the homotopy invariance of equivariant principal bu ndles to prove that the equivariant A-category of Clapp and Puppe is invar iant under Morita equivalence. As a corollary\, we obtain that both the eq uivariant Lusternik-Schnirelmann category of a group action and the invari ant topological complexity are invariant under Morita equivalence. This al lows a definition of topological complexity for orbifolds.\n\nThis is join t work with Andres Angel\, Mark Grant and John Oprea\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Dranishnikov (University of Florida) DTSTART:20200919T161500Z DTEND:20200919T170000Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/12 DESCRIPTION:Title: On topological complexity of hyperbolic groups\nby Alexand er Dranishnikov (University of Florida) as part of CMO workshop: Topologic al Complexity and Motion Planning\n\n\nAbstract\nWe will discuss the proof of the equality TC(G)=2cd(G) for nonabelian hyperbolic groups\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/12/ END:VEVENT BEGIN:VEVENT SUMMARY:David Recio-Mitter (Lehigh University) DTSTART:20200920T140000Z DTEND:20200920T144500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/13 DESCRIPTION:Title: Geodesic complexity and motion planning on graphs\nby Davi d Recio-Mitter (Lehigh University) as part of CMO workshop: Topological Co mplexity and Motion Planning\n\n\nAbstract\nThe topological complexity TC( X) of a space X was introduced in 2003 by Farber to measure the instabilit y of robot motion planning in X. The motion is not required to be along sh ortest paths in that setting. We define a new version of topological compl exity in which we require the robot to move along shortest paths (more spe cifically geodesics)\, which we call the geodesic complexity GC(X). In ord er to study GC(X) we introduce the total cut locus.\n\nWe show that the ge odesic complexity is sensitive to the metric and in general differs from t he topological complexity\, which only depends on the homotopy type of the space. We also show that in some cases both numbers agree. In particular\ , we construct the first optimal motion planners on configuration spaces o f graphs along shortest paths (joint work with Donald Davis and Michael Ha rrison).\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/13/ END:VEVENT BEGIN:VEVENT SUMMARY:John Oprea (Cleveland State University) DTSTART:20200920T150000Z DTEND:20200920T154500Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/14 DESCRIPTION:Title: Logarithmicity\, the TC-generating function and right-angled A rtin groups\nby John Oprea (Cleveland State University) as part of CMO workshop: Topological Complexity and Motion Planning\n\n\nAbstract\nThe - generating function associated to a space is the formal power series For m any examples \, it is known that where is a polynomial with . Is this true in general? I shall discuss recent developments concerning this question\ , including observing that the answer is related to satisfying logarithmic ity of LS-category. Also\, in the examples mentioned above\, it is always the case that has degree less than or equal to . Is this true in general? I shall discuss this question in the context of right-angled Artin (RAA) g roups and along the way see how RAA groups yield some interesting byproduc ts for the study of .\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Don Davis (Lehigh University) DTSTART:20200920T161500Z DTEND:20200920T170000Z DTSTAMP:20260404T041200Z UID:CMO_20w5194/15 DESCRIPTION:Title: Geodesic complexity of non-geodesic spaces\nby Don Davis ( Lehigh University) as part of CMO workshop: Topological Complexity and Mot ion Planning\n\n\nAbstract\nWe define the notion of near geodesic between points where no geodesic exists\, and use this to define geodesic complexi ty for non-geodesic spaces. We determine explicit near geodesics and geode sic complexity in a variety of cases.\n LOCATION:https://stable.researchseminars.org/talk/CMO_20w5194/15/ END:VEVENT END:VCALENDAR