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:20260422T185219Z UID:CMO_20w5194/1 DESCRIPTION:Title: Topology of parametrised motion planning algorithms\nby Michael Fa rber (School of Mathematical Sciences Queen Mary\, University of London) a s part of CMO workshop: Topological Complexity and Motion Planning\n\n\nAb stract\nWe introduce and study a new concept of parameterised topological complexity\, a topological invariant motivated by the motion planning prob lem of robotics. In the parametrised setting\, a motion planning algorithm 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 obstacle -avoiding collision-free motion of many particles (robots) in 3-dimensiona l space. Our results show that the parameterised topological complexity ca n be significantly higher than the standard (non-parametrised) invariant. Joint work with Daniel Cohen and Shmuel Weinberger.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Ayse Borat (Bursa Technical University) DTSTART:20200917T150000Z DTEND:20200917T154500Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/2 DESCRIPTION:Title: A simplicial analog of homotopic distance\nby Ayse Borat (Bursa Te chnical University) as part of CMO workshop: Topological Complexity and Mo tion Planning\n\n\nAbstract\nHomotopic distance as introduced by Macias-Vi rgos and Mosquera-Lois in [2] can be realized as a generalization of topol ogical complexity (TC) and Lusternik Schnirelmann category (cat). In this talk\, we will introduce a simplicial analog (in the sense of Gonzalez in [1]) of homotopic distance and show that it has a relation with simplicial complexity (SC) as homotopic distance has with TC. We will also introduce some basic properties of simplicial distance.\n\n[1] J. Gonzalez\, Simpli cial Complexity: Piecewise Linear Motion Planning in Robotics\, New York J ournal of Mathematics 24 (2018)\, 279-292.\n\n[2] E. Macias-Virgos\, D. Mo squera-Lois\, Homotopic Distance between Maps\, preprint. arXiv: 1810.1259 1v2.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Koditschek (University of Pennsylvania) DTSTART:20200917T161500Z DTEND:20200917T163000Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/3 DESCRIPTION:Title: Vector Field Methods of Motion Planning\nby Daniel Koditschek (Uni versity of Pennsylvania) as part of CMO workshop: Topological Complexity a nd Motion Planning\n\n\nAbstract\nA long tradition in robotics has deploye d dynamical systems as “reactive” motion planners by encoding goals as attracting sets and obstacles as repelling sets of vector fields arising from suitably constructed feedback laws [1] . This raises the prospects fo r a topologically informed notion of “closed loop” planning complexity [2]\, holding substantial interest for robotics\, and whose contrast with the original “open loop” notion [3] may be of mathematical interest a s well. This talk will briefly review the history of such ideas and provid e context for the next three talks which discuss some recent advances in t he closed loop tradition\, reviewing the implications for practical roboti cs as well as associated mathematical questions.\n\n[1] D. E. Koditschek a nd E. Rimon\, “Robot navigation functions on manifolds with boundary\, ” Adv. Appl. Math.\, vol. 11\, no. 4\, pp. 412–442\, 1990\, doi: doi:1 0.1016/0196-8858(90)90017-S.\n\n[2] Y. Baryshnikov and B. Shapiro\, “How to run a centipede: a topological perspective\,” in Geometric Control T heory and Sub-Riemannian Geometry\, Springer International Publishing\, 20 14\, pp. 37–51.\n\n[3] M. Farber\, “Topological complexity of motion p lanning\,” Discrete Comput. Geom.\, vol. 29\, no. 2\, pp. 211–221\, 20 03.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Vasileios Vasilopoulos (University of Pennsylvania) DTSTART:20200917T163000Z DTEND:20200917T164500Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/4 DESCRIPTION:Title: Doubly Reactive Methods of Task Planning for Robotics\nby Vasileio s Vasilopoulos (University of Pennsylvania) as part of CMO workshop: Topol ogical Complexity and Motion Planning\n\n\nAbstract\nA recent advance in v ector field methods of motion planning for robotics replaced the need for perfect a priori information about the environment’s geometry with a rea l-time\, “doubly reactive” construction that generates the vector fiel d as well as its flow at execution time – directly from sensory inputs – but at the cost of assuming a geometrically simple environment [5] . S till more recent developments have adapted to this doubly reactive online setting the original offline deformation of detailed obstacles into their geometrically simple topological models. Consequent upon these new insight s and algorithms\, empirical navigation can now be achieved in partially u nknown unstructured physical environments by legged robots\, with formal g uarantees that ensure safe convergence for simpler\, wheeled mechanical pl atforms. These ideas can be extended to cover a far broader domain of robo t task planning wherein the robot has the job of rearranging objects in th e world by visiting\, grasping\, moving them [10] and then repeating as ne cessary until the rearrangement task is complete.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul Gustafson (Wright State University) DTSTART:20200917T164500Z DTEND:20200917T170000Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/5 DESCRIPTION:Title: A Category Theoretic Treatment of Robot Hybrid Dynamics with Applicati ons to Reactive Motion Planning and Beyond\nby Paul Gustafson (Wright State University) as part of CMO workshop: Topological Complexity and Moti on Planning\n\n\nAbstract\nHybrid dynamical systems have emerged from the engineering literature as an interesting new class of mathematical objects that intermingle features of both discrete time and continuous time syste ms. In a typical engineering setting\, a hybrid system describes the evolu tion of states driven into different physical modes by events that may be instigated by an external controller or simply imposed by the natural worl d. Extending the formal convergence and safety guarantees of the original omniscient reactive systems introduced in the first talk of this series to the new imperfectly known environments negotiated by their doubly reactiv e siblings introduced in the second talk requires reasoning about hybrid d ynamical systems wherein each new encounter with a different obstacle trig gers a reset of the continuous model space [11]. A recent categorical trea tment [12] of robot hybrid dynamical systems [13] affords a method of hier archical composition\, raising the prospect of further formal extensions t hat might cover as well the more broadly useful class of mobile manipulati on 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-Convex En vironments 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\, “F ormal composition of hybrid systems\,” Theory Appl. Categ.\, no. arXiv:1 911.01267 [cs\, math]\, p. (under review)\, Nov. 2019\, Accessed: Nov. 24\ , 2019. [Online]. Available: http://arxiv.org/abs/1911.01267.\n\n[13] A. M . Johnson\, S. A. Burden\, and D. E. Koditschek\, “A hybrid systems mode l for simple manipulation and self-manipulation systems\,” Int. J. Robot . Res.\, vol. 35\, no. 11\, pp. 1354--1392\, Sep. 2016\, doi: 10.1177/0278 364916639380.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthew Kvalheim (University of Pennsylvania) DTSTART:20200917T170000Z DTEND:20200917T171500Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/6 DESCRIPTION:Title: Toward a Task Planning Theory for Robot Hybrid Dynamics\nby Matthe w Kvalheim (University of Pennsylvania) as part of CMO workshop: Topologic al Complexity and Motion Planning\n\n\nAbstract\nA theory of topological d ynamics for hybrid systems has recently begun to emerge [14]. This talk wi ll discuss this theory and\, in particular\, explain how suitably restrict ed objects in the formal category introduced in the third talk of this ser ies can be shown to admit a version of Conley’s Fundamental Theorem of D ynamical Systems. This raises the hope for a more general theory of dynami cal planning complexity that might bring mathematical insights from both t he open loop [3] and closed loop [2] tradition to the physically ineluctab le but mathematically under-developed class of robot hybrid dynamics [13]. \n\n[2] Y. Baryshnikov and B. Shapiro\, “How to run a centipede: a topol ogical perspective\,” in Geometric Control Theory and Sub-Riemannian Geo metry\, 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. Johnson\, 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: 10.1177/0278364916639380 .\n\n[14] M. D. Kvalheim\, P. Gustafson\, and D. E. Koditschek\, “Conley ’s fundamental theorem for a class of hybrid systems\,” ArXiv200503217 Cs Math\, p. (under review)\, May 2020\, Accessed: May 31\, 2020. [Online ]. Available: http://arxiv.org/abs/2005.03217.\n LOCATION:https://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:20260422T185219Z UID:CMO_20w5194/7 DESCRIPTION:Title: Topological complexity of the work map\nby Jie Wu (Hebei Normal Un iversity and National University of Singapore) as part of CMO workshop: To pological Complexity and Motion Planning\n\n\nAbstract\nWe introduce the t opological complexity of the work map associated to a robot system. In bro ad terms\, this measures the complexity of any algorithm controlling\, not just the motion of the configuration space of the given system\, but the task for which the system has been designed. From a purely topological poi nt of view\, this is a homotopy invariant of a map which generalizes the c lassical topological complexity of a space. Joint work with Aniceto Murill o.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Petar Pavesic (University of Ljubljana) DTSTART:20200918T150000Z DTEND:20200918T154500Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/8 DESCRIPTION:Title: Two questions on TC\nby Petar Pavesic (University of Ljubljana) as part of CMO workshop: Topological Complexity and Motion Planning\n\n\nAbs tract\n1. What is the $TC$ of a wedge?\n\nIn the literature one can find t wo relatively coarse estimates of $TC(X\\vee Y)$:\nFarber states that\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 sketched)\, wh ile\nDranishnikov gives \n$$\\max\\{TC(X)\,TC(Y)\, cat(X\\times Y)\\} \\l e TC(X\\vee Y)\\le TC(X)+TC(Y)+1.$$\nAt first sight the two estimates almo st contradict each other\, because the overlap of the two \nintervals is v ery small. Nevertheless\, all known examples satisfy both estimates. We wi ll show \nthat under suitable assumptions Dranishnikov's method yields a p roof of Farber's upper bound.\n\n2. What can be said about closed manifold s with small TC?\n\nIf $M$ is a closed manifold with $TC(M)=2$\, then by G rant\, Lupton and Oprea $M$ is homeomorphic to an odd-dimensional sphere. We will make another step and study closed manifolds whose topological com plexity is equal to 3.\n\nOf course\, all spaces considered are CW-complex es and $TC(\\mathbf{\\cdot})=1$.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Hellen Colman (Wright College) DTSTART:20200918T161500Z DTEND:20200918T170000Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/9 DESCRIPTION:Title: Morita Invariance of Invariant Topological Complexity\nby Hellen C olman (Wright College) as part of CMO workshop: Topological Complexity and Motion Planning\n\n\nAbstract\nWe show that the invariant topological com plexity defines a new numerical invariant for orbifolds.\n\nOrbifolds may be described as global quotients of spaces by compact group actions with f inite isotropy groups. The same orbifold may have descriptions involving d ifferent spaces and different groups. We say that two actions are Morita e quivalent if they define the same orbifold. Therefore\, any notion defined for group actions should be Morita invariant to be well defined for orbif olds.\n\nWe use the homotopy invariance of equivariant principal bundles t o prove that the equivariant A-category of Clapp and Puppe is invariant un der Morita equivalence. As a corollary\, we obtain that both the equivaria nt Lusternik-Schnirelmann category of a group action and the invariant top ological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.\n\nThis is joint work with Andres Angel\, Mark Grant and John Oprea\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Dranishnikov (University of Florida) DTSTART:20200919T161500Z DTEND:20200919T170000Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/12 DESCRIPTION:Title: On topological complexity of hyperbolic groups\nby Alexander Dran ishnikov (University of Florida) as part of CMO workshop: Topological Comp lexity 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://researchseminars.org/talk/CMO_20w5194/12/ END:VEVENT BEGIN:VEVENT SUMMARY:David Recio-Mitter (Lehigh University) DTSTART:20200920T140000Z DTEND:20200920T144500Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/13 DESCRIPTION:Title: Geodesic complexity and motion planning on graphs\nby David Recio -Mitter (Lehigh University) as part of CMO workshop: Topological Complexit y and Motion Planning\n\n\nAbstract\nThe topological complexity TC(X) of a space X was introduced in 2003 by Farber to measure the instability of ro bot motion planning in X. The motion is not required to be along shortest paths in that setting. We define a new version of topological complexity i n which we require the robot to move along shortest paths (more specifical ly geodesics)\, which we call the geodesic complexity GC(X). In order to s tudy GC(X) we introduce the total cut locus.\n\nWe show that the geodesic complexity is sensitive to the metric and in general differs from the topo logical complexity\, which only depends on the homotopy type of the space. We also show that in some cases both numbers agree. In particular\, we co nstruct the first optimal motion planners on configuration spaces of graph s along shortest paths (joint work with Donald Davis and Michael Harrison) .\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/13/ END:VEVENT BEGIN:VEVENT SUMMARY:John Oprea (Cleveland State University) DTSTART:20200920T150000Z DTEND:20200920T154500Z DTSTAMP:20260422T185219Z UID:CMO_20w5194/14 DESCRIPTION:Title: Logarithmicity\, the TC-generating function and right-angled Artin gr oups\nby John Oprea (Cleveland State University) as part of CMO worksh op: Topological Complexity and Motion Planning\n\n\nAbstract\nThe -generat ing function associated to a space is the formal power series For many exa mples \, it is known that where is a polynomial with . Is this true in gen eral? I shall discuss recent developments concerning this question\, inclu ding observing that the answer is related to satisfying logarithmicity of LS-category. Also\, in the examples mentioned above\, it is always the cas e 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) groups a nd along the way see how RAA groups yield some interesting byproducts for the study of .\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Don Davis (Lehigh University) DTSTART:20200920T161500Z DTEND:20200920T170000Z DTSTAMP:20260422T185219Z 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 Motion Pla nning\n\n\nAbstract\nWe define the notion of near geodesic between points where no geodesic exists\, and use this to define geodesic complexity for non-geodesic spaces. We determine explicit near geodesics and geodesic com plexity in a variety of cases.\n LOCATION:https://researchseminars.org/talk/CMO_20w5194/15/ END:VEVENT END:VCALENDAR