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BEGIN:VEVENT
SUMMARY:Michael Farber (School of Mathematical Sciences Queen Mary\, Unive
rsity of London)
DTSTART;VALUE=DATE-TIME:20200917T140000Z
DTEND;VALUE=DATE-TIME:20200917T144500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/1
DESCRIPTION:Title: Topology of parametrised motion planning algorithms\nby
Michael Farber (School of Mathematical Sciences Queen Mary\, University o
f London) as part of CMO workshop: Topological Complexity and Motion Plann
ing\n\n\nAbstract\nWe introduce and study a new concept of parameterised t
opological complexity\, a topological invariant motivated by the motion pl
anning problem of robotics. In the parametrised setting\, a motion plannin
g algorithm has high degree of universality and flexibility\, it can funct
ion under a variety of external conditions (such as positions of the obsta
cles etc). We explicitly compute the parameterised topological complexity
of obstacle-avoiding collision-free motion of many particles (robots) in 3
-dimensional space. Our results show that the parameterised topological co
mplexity can be significantly higher than the standard (non-parametrised)
invariant. Joint work with Daniel Cohen and Shmuel Weinberger.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayse Borat (Bursa Technical University)
DTSTART;VALUE=DATE-TIME:20200917T150000Z
DTEND;VALUE=DATE-TIME:20200917T154500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/2
DESCRIPTION:Title: A simplicial analog of homotopic distance\nby Ayse Bora
t (Bursa Technical University) as part of CMO workshop: Topological Comple
xity and Motion Planning\n\n\nAbstract\nHomotopic distance as introduced b
y Macias-Virgos and Mosquera-Lois in [2] can be realized as a generalizati
on of topological complexity (TC) and Lusternik Schnirelmann category (cat
). In this talk\, we will introduce a simplicial analog (in the sense of G
onzalez in [1]) of homotopic distance and show that it has a relation with
simplicial complexity (SC) as homotopic distance has with TC. We will als
o introduce some basic properties of simplicial distance.\n\n[1] J. Gonzal
ez\, Simplicial Complexity: Piecewise Linear Motion Planning in Robotics\,
New York Journal of Mathematics 24 (2018)\, 279-292.\n\n[2] E. Macias-Vir
gos\, D. Mosquera-Lois\, Homotopic Distance between Maps\, preprint. arXiv
: 1810.12591v2.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Koditschek (University of Pennsylvania)
DTSTART;VALUE=DATE-TIME:20200917T161500Z
DTEND;VALUE=DATE-TIME:20200917T163000Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/3
DESCRIPTION:Title: Vector Field Methods of Motion Planning\nby Daniel Kodi
tschek (University of Pennsylvania) as part of CMO workshop: Topological C
omplexity and Motion Planning\n\n\nAbstract\nA long tradition in robotics
has deployed dynamical systems as “reactive” motion planners by encodi
ng goals as attracting sets and obstacles as repelling sets of vector fiel
ds arising from suitably constructed feedback laws [1] . This raises the p
rospects for a topologically informed notion of “closed loop” planning
complexity [2]\, holding substantial interest for robotics\, and whose co
ntrast with the original “open loop” notion [3] may be of mathematical
interest as well. This talk will briefly review the history of such ideas
and provide context for the next three talks which discuss some recent ad
vances in the closed loop tradition\, reviewing the implications for pract
ical robotics as well as associated mathematical questions.\n\n[1] D. E. K
oditschek and E. Rimon\, “Robot navigation functions on manifolds with b
oundary\,” Adv. Appl. Math.\, vol. 11\, no. 4\, pp. 412–442\, 1990\, d
oi: 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
Control Theory and Sub-Riemannian Geometry\, Springer International Publis
hing\, 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
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vasileios Vasilopoulos (University of Pennsylvania)
DTSTART;VALUE=DATE-TIME:20200917T163000Z
DTEND;VALUE=DATE-TIME:20200917T164500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/4
DESCRIPTION:Title: Doubly Reactive Methods of Task Planning for Robotics\n
by Vasileios Vasilopoulos (University of Pennsylvania) as part of CMO work
shop: Topological Complexity and Motion Planning\n\n\nAbstract\nA recent a
dvance in vector field methods of motion planning for robotics replaced th
e need for perfect a priori information about the environment’s geometry
with a real-time\, “doubly reactive” construction that generates the
vector field as well as its flow at execution time – directly from senso
ry inputs – but at the cost of assuming a geometrically simple environme
nt [5] . Still more recent developments have adapted to this doubly reacti
ve online setting the original offline deformation of detailed obstacles i
nto their geometrically simple topological models. Consequent upon these n
ew insights and algorithms\, empirical navigation can now be achieved in p
artially unknown unstructured physical environments by legged robots\, wit
h formal guarantees that ensure safe convergence for simpler\, wheeled mec
hanical platforms. These ideas can be extended to cover a far broader doma
in of robot task planning wherein the robot has the job of rearranging obj
ects in the world by visiting\, grasping\, moving them [10] and then repea
ting as necessary until the rearrangement task is complete.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Gustafson (Wright State University)
DTSTART;VALUE=DATE-TIME:20200917T164500Z
DTEND;VALUE=DATE-TIME:20200917T170000Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/5
DESCRIPTION:Title: A Category Theoretic Treatment of Robot Hybrid Dynamics
with Applications to Reactive Motion Planning and Beyond\nby Paul Gustafs
on (Wright State University) as part of CMO workshop: Topological Complexi
ty and Motion Planning\n\n\nAbstract\nHybrid dynamical systems have emerge
d from the engineering literature as an interesting new class of mathemati
cal objects that intermingle features of both discrete time and continuous
time systems. In a typical engineering setting\, a hybrid system describe
s the evolution of states driven into different physical modes by events t
hat may be instigated by an external controller or simply imposed by the n
atural world. Extending the formal convergence and safety guarantees of th
e original omniscient reactive systems introduced in the first talk of thi
s series to the new imperfectly known environments negotiated by their dou
bly reactive siblings introduced in the second talk requires reasoning abo
ut hybrid dynamical systems wherein each new encounter with a different ob
stacle triggers a reset of the continuous model space [11]. A recent categ
orical treatment [12] of robot hybrid dynamical systems [13] affords a met
hod of hierarchical composition\, raising the prospect of further formal e
xtensions that might cover as well the more broadly useful class of mobile
manipulation tasks assigned to dynamically dexterous (e.g.\, legged) robo
ts.\n\n[11] V. Vasilopoulos\, G. Pavlakos\, K. Schmeckpeper\, K. Daniilidi
s\, and D. E. Koditschek\, “Reactive Navigation in Partially Familiar No
n-Convex Environments Using Semantic Perceptual Feedback\,” Rev.\, p. (u
nder review)\, 2019\, [Online]. Available: https://arxiv.org/abs/2002.0894
6.\n\n[12] J. Culbertson\, P. Gustafson\, D. E. Koditschek\, and P. F. Sti
ller\, “Formal composition of hybrid systems\,” Theory Appl. Categ.\,
no. arXiv:1911.01267 [cs\, math]\, p. (under review)\, Nov. 2019\, Accesse
d: 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 s
ystems model for simple manipulation and self-manipulation systems\,” In
t. J. Robot. Res.\, vol. 35\, no. 11\, pp. 1354--1392\, Sep. 2016\, doi: 1
0.1177/0278364916639380.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Kvalheim (University of Pennsylvania)
DTSTART;VALUE=DATE-TIME:20200917T170000Z
DTEND;VALUE=DATE-TIME:20200917T171500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
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
: Topological Complexity and Motion Planning\n\n\nAbstract\nA theory of to
pological dynamics for hybrid systems has recently begun to emerge [14]. T
his talk will discuss this theory and\, in particular\, explain how suitab
ly restricted objects in the formal category introduced in the third talk
of this series can be shown to admit a version of Conley’s Fundamental T
heorem of Dynamical Systems. This raises the hope for a more general theor
y of dynamical planning complexity that might bring mathematical insights
from both the open loop [3] and closed loop [2] tradition to the physicall
y ineluctable but mathematically under-developed class of robot hybrid dyn
amics [13].\n\n[2] Y. Baryshnikov and B. Shapiro\, “How to run a centipe
de: a topological perspective\,” in Geometric Control Theory and Sub-Rie
mannian Geometry\, Springer International Publishing\, 2014\, pp. 37–51.
\n\n[3] M. Farber\, “Topological complexity of motion planning\,” Disc
rete 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: 10.1177/02783
64916639380.\n\n[14] M. D. Kvalheim\, P. Gustafson\, and D. E. Koditschek\
, “Conley’s fundamental theorem for a class of hybrid systems\,” ArX
iv200503217 Cs Math\, p. (under review)\, May 2020\, Accessed: May 31\, 20
20. [Online]. Available: http://arxiv.org/abs/2005.03217.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Wu (Hebei Normal University and National University of Singapo
re)
DTSTART;VALUE=DATE-TIME:20200918T140000Z
DTEND;VALUE=DATE-TIME:20200918T144500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/7
DESCRIPTION:Title: Topological complexity of the work map\nby Jie Wu (Hebe
i Normal University and National University of Singapore) as part of CMO w
orkshop: Topological Complexity and Motion Planning\n\n\nAbstract\nWe intr
oduce the topological complexity of the work map associated to a robot sys
tem. In broad terms\, this measures the complexity of any algorithm contro
lling\, 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 topo
logical point of view\, this is a homotopy invariant of a map which genera
lizes the classical topological complexity of a space. Joint work with Ani
ceto Murillo.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petar Pavesic (University of Ljubljana)
DTSTART;VALUE=DATE-TIME:20200918T150000Z
DTEND;VALUE=DATE-TIME:20200918T154500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/8
DESCRIPTION:Title: Two questions on TC\nby Petar Pavesic (University of Lj
ubljana) as part of CMO workshop: Topological Complexity and Motion Planni
ng\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 state
s 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 ske
tched)\, while\nDranishnikov gives \n$$\\max\\{TC(X)\,TC(Y)\, cat(X\\time
s Y)\\} \\le TC(X\\vee Y)\\le TC(X)+TC(Y)+1.$$\nAt first sight the two est
imates almost contradict each other\, because the overlap of the two \nint
ervals is very small. Nevertheless\, all known examples satisfy both estim
ates. We will show \nthat under suitable assumptions Dranishnikov's method
yields a proof of Farber's upper bound.\n\n2. What can be said about clos
ed manifolds with small TC?\n\nIf $M$ is a closed manifold with $TC(M)=2$\
, then by Grant\, Lupton and Oprea $M$ is homeomorphic to an odd-dimension
al sphere. We will make another step and study closed manifolds whose topo
logical complexity is equal to 3.\n\nOf course\, all spaces considered are
CW-complexes and $TC(\\mathbf{\\cdot})=1$.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hellen Colman (Wright College)
DTSTART;VALUE=DATE-TIME:20200918T161500Z
DTEND;VALUE=DATE-TIME:20200918T170000Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/9
DESCRIPTION:Title: Morita Invariance of Invariant Topological Complexity\n
by Hellen Colman (Wright College) as part of CMO workshop: Topological Com
plexity and Motion Planning\n\n\nAbstract\nWe show that the invariant topo
logical complexity defines a new numerical invariant for orbifolds.\n\nOrb
ifolds may be described as global quotients of spaces by compact group act
ions with finite isotropy groups. The same orbifold may have descriptions
involving different spaces and different groups. We say that two actions a
re Morita equivalent if they define the same orbifold. Therefore\, any not
ion defined for group actions should be Morita invariant to be well define
d for orbifolds.\n\nWe use the homotopy invariance of equivariant principa
l bundles to prove that the equivariant A-category of Clapp and Puppe is i
nvariant under Morita equivalence. As a corollary\, we obtain that both th
e equivariant Lusternik-Schnirelmann category of a group action and the in
variant topological complexity are invariant under Morita equivalence. Thi
s allows a definition of topological complexity for orbifolds.\n\nThis is
joint work with Andres Angel\, Mark Grant and John Oprea\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dranishnikov (University of Florida)
DTSTART;VALUE=DATE-TIME:20200919T161500Z
DTEND;VALUE=DATE-TIME:20200919T170000Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/12
DESCRIPTION:Title: On topological complexity of hyperbolic groups\nby Alex
ander Dranishnikov (University of Florida) as part of CMO workshop: Topolo
gical Complexity and Motion Planning\n\n\nAbstract\nWe will discuss the pr
oof of the equality TC(G)=2cd(G) for nonabelian hyperbolic groups\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Recio-Mitter (Lehigh University)
DTSTART;VALUE=DATE-TIME:20200920T140000Z
DTEND;VALUE=DATE-TIME:20200920T144500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/13
DESCRIPTION:Title: Geodesic complexity and motion planning on graphs\nby D
avid Recio-Mitter (Lehigh University) as part of CMO workshop: Topological
Complexity and Motion Planning\n\n\nAbstract\nThe topological complexity
TC(X) of a space X was introduced in 2003 by Farber to measure the instabi
lity of robot motion planning in X. The motion is not required to be along
shortest paths in that setting. We define a new version of topological co
mplexity in which we require the robot to move along shortest paths (more
specifically geodesics)\, which we call the geodesic complexity GC(X). In
order to study GC(X) we introduce the total cut locus.\n\nWe show that the
geodesic complexity is sensitive to the metric and in general differs fro
m the topological complexity\, which only depends on the homotopy type of
the space. We also show that in some cases both numbers agree. In particul
ar\, we construct the first optimal motion planners on configuration space
s of graphs along shortest paths (joint work with Donald Davis and Michael
Harrison).\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Oprea (Cleveland State University)
DTSTART;VALUE=DATE-TIME:20200920T150000Z
DTEND;VALUE=DATE-TIME:20200920T154500Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/14
DESCRIPTION:Title: Logarithmicity\, the TC-generating function and right-a
ngled Artin groups\nby John Oprea (Cleveland State University) as part of
CMO workshop: Topological Complexity and Motion Planning\n\n\nAbstract\nTh
e -generating function associated to a space is the formal power series Fo
r many examples \, it is known that where is a polynomial with . Is this t
rue in general? I shall discuss recent developments concerning this questi
on\, including observing that the answer is related to satisfying logarith
micity of LS-category. Also\, in the examples mentioned above\, it is alwa
ys the case that has degree less than or equal to . Is this true in genera
l? I shall discuss this question in the context of right-angled Artin (RAA
) groups and along the way see how RAA groups yield some interesting bypro
ducts for the study of .\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Don Davis (Lehigh University)
DTSTART;VALUE=DATE-TIME:20200920T161500Z
DTEND;VALUE=DATE-TIME:20200920T170000Z
DTSTAMP;VALUE=DATE-TIME:20200921T071533Z
UID:CMO_20w5194/15
DESCRIPTION:Title: Geodesic complexity of non-geodesic spaces\nby Don Davi
s (Lehigh University) as part of CMO workshop: Topological Complexity and
Motion Planning\n\n\nAbstract\nWe define the notion of near geodesic betwe
en points where no geodesic exists\, and use this to define geodesic compl
exity for non-geodesic spaces. We determine explicit near geodesics and ge
odesic complexity in a variety of cases.\n
END:VEVENT
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