Loop Topological Complexity
|
|
|
- Lionel Shaw
- 5 years ago
- Views:
Transcription
1 XXIST OPORTO MEETING Loop Topological Complexity LISBOA, PORTUGAL My Ismail Mamouni Agrégé-Doctor CRMEF Rabat, Morocco http ://myismail.net My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
2 Joint work with Derfoufi Younes Fac. Sc. Meknes, Morocco My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
3 Let s Go : Love and Peace My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
4 Content 1 Motion Planning Algorithms (MPA) 2 My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
5 Motion Planning Algorithms (MPA) Motion Planning Algorithms Farber (2003) Let X a path configuration space, a motion planning algorithm consists to produce continuously from a pair of point (A, B) X X (as an input), a path in X (as an output) from A to B. Roughly speaking, that is any continuous section s : X X PX of the projection π : PX X X γ (γ(0), γ(1)). My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
6 Motion Planning Algorithms (MPA) Motion Planning Algorithms Farber (2003) MPA exists iff X contractible My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
7 Motion Planning Algorithms (MPA) Motion Planning Algorithms On the sequel X contractible, and M(X) denotes the non empty set of all motion planning algorithms on X topologized (as a subset of map(x X, PX)) with the compact-open topology. Derfoufi, M. (2015) M(X) is also contractible My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
8 Motion Planning Algorithms (MPA) Motion Planning Algorithms Open Question 1 If X and Y are homeomorphic, M(X) and M(Y ) are also homeomorphic? My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
9 Motion Planning Algorithms (MPA) Motion Planning Algorithms Partial Answer (compact manifold context) From Urysohn metrization theorem X is metrizable, whenever X is a compact manifold ; Let d X be a such metric on X, it induces a metric δ X on M(X) defined for any s, s M(X) by δ X (s, s ) := sup (A,B) X 2 0 t 1 d X (s(a, B)(t), s (A, B)(t)); The open-compact topology of M(X) is the same than that induced by the metric δ X. My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
10 Motion Planning Algorithms (MPA) Motion Planning Algorithms Derfoufi, M. (Classification Theorem, 2015) (X, d X ) and (Y, d Y ) are isometric, then (M(X), δ X ) and (M(Y ), δ Y ) are also. Sketch of the proof To any isometry ϕ : (X, d X ) (Y, d Y ), we associate the isometry M(ϕ) : (M(Y ), δ Y ) (M(X), δ X ) s ϕ 1 s (ϕ ϕ) My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
11 Motion Planning Algorithms (MPA) Motion Planning Algorithms A contravariant functor X : M : ϕ : contactible path-connected compact mfld Isometrie M(X) : M(ϕ) : metrizable space Isometrie My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
12 Motion Planning Algorithms (MPA) Motion Planning Algorithms Derfoufi, M. (2015) (M(X), δ) is complete. Interpretation For a fixed pair of points (departure, target), then from any wider class of closed motion planning algorithms, emerges one (their limit). Example from Nature Ant colony optimization algorithm based on the behaviour of ants seeking for an optimal path nest-food My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
13 Motion Planning Algorithms (MPA) Motion Planning Algorithms Open Question 2 (M(X), δ) is compact? Interpretation The compactness of (M(X), δ) means that if one fix a pair of points (departure, target), then from any wider class of motion planning algorithms (not necessary closed), emerges one (their limit). My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
14 Loop Motion Planning Algorithms Derfoufi, M. (2015) Loop Motion Planning Algorithm (LMPA) : any continuous section s : X X X S1 of the loop evaluation ev : X S1 X X γ (γ(0), γ( 1 2 )). Interpretation Input : a pair of points (departure, target) ; Output : a goings and comings motion by requiring the robot a come-back to it departure point. My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
15 Loop Motion Planning Algorithms Application areas The motion of an drone like an unmanned warplane or a guided TV camera ; The famous vehicle routing problem with pick-up and delivery Derfoufi, M. (2015) LMPA exits iff X contractible. My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
16 Loop Motion Planning Algorithms On the sequel X contractible, and M LP (X) denotes the non empty set of all loop motion planning algorithms on X topologized (as a subset of map(x X, X S1 )) with the compact-open topology. Derfoufi, M. (2015) M LP (X) is also contractible. My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
17 Loop Topological Complexity Derfoufi, M. (2015) TC LP (X) : the minimal number k (or infinity) such that X X can be recovered by k open sets U 1,... U k such that on each of which there exists a continuous loop motion planning algorithm, si LP over U i verifying si LP (A i, B i )(1) = A i for any (A i, B i ) U i. Derfoufi, M. (2015) TC LP (X) = TC(X) My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
18 Loop Motion Planning Product Derfoufi, M. (2015) For any (s 1, s 2 ) M LP (X) M LP (X), we put : s 1 s 2 (a, b)(t) = s 1 (a, b)(t) if 0 t 1 2 = s 1 (a, b)(3t 1) if 1 2 t 2 3 = s 2 (a, b)(3t 2) if 2 3 t 1 My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
19 Loop Motion Planning Product Two loop motion planning algorithms are composable if and only if they have two common base points. B, t=1/2 B t=1/2 B t=1/2 * = A t=0,t=1 A t=0,t=1 A t=0,t=2/3,t=1 My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
20 Open Question 3 How one may use the loop motion product or interpret the equality TC = TC LP? My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
21 Research Project Key Idea loop motion product leads to a string motion product in the same spirit of that of Chas-Sullivan on M LP (X) Main Obstacle M LP (X) is contractible, its homology is trivial. My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
22 Research Project Global Question Is there any way to define global LMPA for a non contractible compact manifold My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
23 References Y. Derfoufi, M., Motion planning algorithms, topological proprieties and affine approximation, submitted on December 2014 Y. Derfoufi, M., On the motion of a mobile robot in a contractible, path connected and compact manifold, final touches My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
24 Acknowledgements XXIst Oporto Meeting Organizers My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
25 Acknowledgements Mark Grant, Aberdeen, UK Introduction to Topological Robotics and 2 Years Assistance My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
26 Acknowledgements MAAT, Moroccan Research Group My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
27 Acknowledgements MAAT, Moroccan Research Group MAAT : Moroccan Area of Algebraic Topology (Born : 2012) ; Logo : and Home Page : http ://algtop.net Members : 3 professors, 7 PhD Students, 30 Master Students ; Scientific production 1 1 published papers, 7 submitted since November 2013 ; 2 Monthly seminar 3 Bi-Annual Research School :Geometry, Topology in Physics and Mathematics 4 GeToPhyMa-2016 (July, http ://algtop.net/geto16) : On Rational Homotopty Theory and its Interactions, will be dedicated to D. Sullivan and J. Stasheff. My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
28 Acknowledgements Kind Portuguese People My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
29 Questions or Comments are accepted in slowly formulated My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
30 My Ismail Mamouni XXIst Oporto Meeting, Lisboa, Portugal 4th-7th February, / 30
Point-Set Topology II
Point-Set Topology II Charles Staats September 14, 2010 1 More on Quotients Universal Property of Quotients. Let X be a topological space with equivalence relation. Suppose that f : X Y is continuous and
Metrics on diagrams and persistent homology
Background Categorical ph Relative ph More structure Department of Mathematics Cleveland State University p.bubenik@csuohio.edu http://academic.csuohio.edu/bubenik_p/ July 18, 2013 joint work with Vin
Lecture 0: Reivew of some basic material
Lecture 0: Reivew of some basic material September 12, 2018 1 Background material on the homotopy category We begin with the topological category TOP, whose objects are topological spaces and whose morphisms
Computing the Betti Numbers of Arrangements. Saugata Basu School of Mathematics & College of Computing Georgia Institute of Technology.
1 Computing the Betti Numbers of Arrangements Saugata Basu School of Mathematics & College of Computing Georgia Institute of Technology. 2 Arrangements in Computational Geometry An arrangement in R k is
Net Tape Graphs. M. El-Ghoul Mathematics Department, Faculty of Science Tanta University, Tanta, Egypt
www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. ; February Net Tape Graphs M. El-Ghoul Mathematics Department, Faculty of Science Tanta University, Tanta, Egypt E-mail: m.elghoul@msn.com
Math 734 Aug 22, Differential Geometry Fall 2002, USC
Math 734 Aug 22, 2002 1 Differential Geometry Fall 2002, USC Lecture Notes 1 1 Topological Manifolds The basic objects of study in this class are manifolds. Roughly speaking, these are objects which locally
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY KARL L. STRATOS Abstract. The conventional method of describing a graph as a pair (V, E), where V and E repectively denote the sets of vertices and edges,
A new homology for infinite graphs and metric continua
A new homology for infinite graphs and metric continua Technische Universität Graz and Mathematisches Seminar Universität Hamburg Bremen, 10.11.09 Cycle Space The cycle space C(G) of a finite graph: A
Orientation of manifolds - definition*
Bulletin of the Manifold Atlas - definition (2013) Orientation of manifolds - definition* MATTHIAS KRECK 1. Zero dimensional manifolds For zero dimensional manifolds an orientation is a map from the manifold
Simplicial Hyperbolic Surfaces
Simplicial Hyperbolic Surfaces Talk by Ken Bromberg August 21, 2007 1-Lipschitz Surfaces- In this lecture we will discuss geometrically meaningful ways of mapping a surface S into a hyperbolic manifold
Simplicial Objects and Homotopy Groups
Simplicial Objects and Homotopy Groups Jie Wu Department of Mathematics National University of Singapore July 8, 2007 Simplicial Objects and Homotopy Groups -Objects and Homology Simplicial Sets and Homotopy
Cannon s conjecture, subdivision rules, and expansion complexes
Cannon s conjecture, subdivision rules, and expansion complexes W. Floyd (joint work with J. Cannon and W. Parry) Department of Mathematics Virginia Tech UNC Greensboro: November, 2014 Motivation from
Elementary Topology. Note: This problem list was written primarily by Phil Bowers and John Bryant. It has been edited by a few others along the way.
Elementary Topology Note: This problem list was written primarily by Phil Bowers and John Bryant. It has been edited by a few others along the way. Definition. properties: (i) T and X T, A topology on
Lecture 17: Continuous Functions
Lecture 17: Continuous Functions 1 Continuous Functions Let (X, T X ) and (Y, T Y ) be topological spaces. Definition 1.1 (Continuous Function). A function f : X Y is said to be continuous if the inverse
Definition A metric space is proper if all closed balls are compact. The length pseudo metric of a metric space X is given by.
Chapter 1 Geometry: Nuts and Bolts 1.1 Metric Spaces Definition 1.1.1. A metric space is proper if all closed balls are compact. The length pseudo metric of a metric space X is given by (x, y) inf p. p:x
A Tour of General Topology Chris Rogers June 29, 2010
A Tour of General Topology Chris Rogers June 29, 2010 1. The laundry list 1.1. Metric and topological spaces, open and closed sets. (1) metric space: open balls N ɛ (x), various metrics e.g. discrete metric,
Topological space - Wikipedia, the free encyclopedia
Page 1 of 6 Topological space From Wikipedia, the free encyclopedia Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity.
Topology 550A Homework 3, Week 3 (Corrections: February 22, 2012)
Topology 550A Homework 3, Week 3 (Corrections: February 22, 2012) Michael Tagare De Guzman January 31, 2012 4A. The Sorgenfrey Line The following material concerns the Sorgenfrey line, E, introduced in
Geodesic and curvature of piecewise flat Finsler surfaces
Geodesic and curvature of piecewise flat Finsler surfaces Ming Xu Capital Normal University (based on a joint work with S. Deng) in Southwest Jiaotong University, Emei, July 2018 Outline 1 Background Definition
Introduction to Immersion, Embedding, and the Whitney Embedding Theorems
Introduction to Immersion, Embedding, and the Whitney Embedding Theorems Paul Rapoport November 23, 2015 Abstract We give an overview of immersion in order to present the idea of embedding, then discuss
Continuous Selections of Multivalued Mappings
Continuous Selections of Multivalued Mappings by Dusan Repovs Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia and Pavel Vladimirovic Semenov Department of Mathematics, Moscow City
TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3.
TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 301. Definition. Let m be a positive integer, and let X be a set. An m-tuple of elements of X is a function x : {1,..., m} X. We sometimes use x i instead
Manifolds (Relates to text Sec. 36)
22M:132 Fall 07 J. Simon Manifolds (Relates to text Sec. 36) Introduction. Manifolds are one of the most important classes of topological spaces (the other is function spaces). Much of your work in subsequent
Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary?
Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case?
Surfaces Beyond Classification
Chapter XII Surfaces Beyond Classification In most of the textbooks which present topological classification of compact surfaces the classification is the top result. However the topology of 2- manifolds
Geometry of Spaces of Planar Quadrilaterals
Geometry of Spaces of Planar Quadrilaterals Jessica L. StClair Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements
Some fixed fuzzy point results using Hausdorff metric in fuzzy metric spaces
Annals of Fuzzy Mathematics and Informatics Volume 13, No 5, (May 017), pp 641 650 ISSN: 093 9310 (print version) ISSN: 87 635 (electronic version) http://wwwafmiorkr @FMI c Kyung Moon Sa Co http://wwwkyungmooncom
Homotopy type of the complement. complex lines arrangements
On the homotopy type of the complement of complex lines arrangements Department of Geometry-Topology Institute of Mathematics, VAST, Hanoi Singapore December 15, 2008 Introduction Let A be an l-arrangement,
Notes on Topology. Andrew Forrester January 28, Notation 1. 2 The Big Picture 1
Notes on Topology Andrew Forrester January 28, 2009 Contents 1 Notation 1 2 The Big Picture 1 3 Fundamental Concepts 2 4 Topological Spaces and Topologies 2 4.1 Topological Spaces.........................................
Topological Data Analysis - I. Afra Zomorodian Department of Computer Science Dartmouth College
Topological Data Analysis - I Afra Zomorodian Department of Computer Science Dartmouth College September 3, 2007 1 Acquisition Vision: Images (2D) GIS: Terrains (3D) Graphics: Surfaces (3D) Medicine: MRI
Topology problem set Integration workshop 2010
Topology problem set Integration workshop 2010 July 28, 2010 1 Topological spaces and Continuous functions 1.1 If T 1 and T 2 are two topologies on X, show that (X, T 1 T 2 ) is also a topological space.
I. An introduction to Boolean inverse semigroups
I. An introduction to Boolean inverse semigroups Mark V Lawson Heriot-Watt University, Edinburgh June 2016 1 0. In principio The monograph J. Renault, A groupoid approach to C - algebras, Lecture Notes
Continuous functions and homeomorphisms
Continuous functions and homeomorphisms 1 Motivation Up to now we have defined just a few topological properties, like the first three T -axioms and the countability properties (separable, ccc, first and
2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to
2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to be connected if it is not disconnected. A subset of
Mathematical Research Letters 1, (1994) MÖBIUS CONE STRUCTURES ON 3-DIMENSIONAL MANIFOLDS. Feng Luo
Mathematical Research Letters 1, 257 261 (1994) MÖBIUS CONE STRUCTURES ON 3-DIMENSIONAL MANIFOLDS Feng Luo Abstract. We show that for any given angle α (0, 2π), any closed 3- manifold has a Möbius cone
Proximal Manifold Learning via Descriptive Neighbourhood Selection
Applied Mathematical Sciences, Vol. 8, 2014, no. 71, 3513-3517 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.42111 Proximal Manifold Learning via Descriptive Neighbourhood Selection
By taking products of coordinate charts, we obtain charts for the Cartesian product of manifolds. Hence the Cartesian product is a manifold.
1 Manifolds A manifold is a space which looks like R n at small scales (i.e. locally ), but which may be very different from this at large scales (i.e. globally ). In other words, manifolds are made by
Detection and approximation of linear structures in metric spaces
Stanford - July 12, 2012 MMDS 2012 Detection and approximation of linear structures in metric spaces Frédéric Chazal Geometrica group INRIA Saclay Joint work with M. Aanjaneya, D. Chen, M. Glisse, L. Guibas,
Some new higher separation axioms via sets having non-empty interior
Bhat & Das, Cogent Mathematics (015), : 109695 http://dx.doi.org/10.1080/3311835.015.109695 PURE MATHEMATICS RESEARCH ARTICLE Some new higher separation axioms via sets having non-empty interior Pratibha
On normal embedding of complex algebraic surfaces
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 On normal embedding of complex algebraic surfaces Lev Birbrair, Alexandre Fernandes and Walter D. Neumann Dedicated to our
The Borsuk-Ulam theorem- A Combinatorial Proof
The Borsuk-Ulam theorem- A Combinatorial Proof Shreejit Bandyopadhyay April 14, 2015 1 Introduction The Borsuk-Ulam theorem is perhaps among the results in algebraic topology having the greatest number
Regarding Python level necessary for the course
Logistics First two recitations (next two weeks) Python basics (installation, basic syntax, basic programming), optional Making models for 3D printing w/ Blender Will announce details through Sakai Regarding
Bounded subsets of topological vector spaces
Chapter 2 Bounded subsets of topological vector spaces In this chapter we will study the notion of bounded set in any t.v.s. and analyzing some properties which will be useful in the following and especially
6.3 Poincare's Theorem
Figure 6.5: The second cut. for some g 0. 6.3 Poincare's Theorem Theorem 6.3.1 (Poincare). Let D be a polygon diagram drawn in the hyperbolic plane such that the lengths of its edges and the interior angles
Stable maps and branched shadows of 3-manifolds
Stable maps and branched shadows of 3-manifolds Yuya KODA Hiroshima University) joint work with Masaharu ISHIKAWA (Tohoku University) RIMS Seminar, Numazu, May 26, 2015 Topology, Geometry and Algebra of
4. Definition: topological space, open set, topology, trivial topology, discrete topology.
Topology Summary Note to the reader. If a statement is marked with [Not proved in the lecture], then the statement was stated but not proved in the lecture. Of course, you don t need to know the proof.
MA651 Topology. Lecture 4. Topological spaces 2
MA651 Topology. Lecture 4. Topological spaces 2 This text is based on the following books: Linear Algebra and Analysis by Marc Zamansky Topology by James Dugundgji Fundamental concepts of topology by Peter
Hierarchy of graph matchbox manifolds
Hierarchy of graph matchbox manifolds Olga Lukina University of Leicester, UK July 2th, 2012 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, 2012 1 / 31 Introduction I will
On Soft Topological Linear Spaces
Republic of Iraq Ministry of Higher Education and Scientific Research University of AL-Qadisiyah College of Computer Science and Formation Technology Department of Mathematics On Soft Topological Linear
Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS Definition 1.1. Let X be a set and T a subset of the power set P(X) of X. Then T is a topology on X if and only if all of the following
d(γ(a i 1 ), γ(a i )) i=1
Marli C. Wang Hyperbolic Geometry Hyperbolic surfaces A hyperbolic surface is a metric space with some additional properties: it has the shortest length property and every point has an open neighborhood
M3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements.
M3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements. Chapter 1: Metric spaces and convergence. (1.1) Recall the standard distance function
Homotopy theory of higher categorical structures
University of California, Riverside August 8, 2013 Higher categories In a category, we have morphisms, or functions, between objects. But what if you have functions between functions? This gives the idea
Reflection groups 4. Mike Davis. May 19, Sao Paulo
Reflection groups 4 Mike Davis Sao Paulo May 19, 2014 https://people.math.osu.edu/davis.12/slides.html 1 2 Exotic fundamental gps Nonsmoothable aspherical manifolds 3 Let (W, S) be a Coxeter system. S
An introduction to Topological Data Analysis through persistent homology: Intro and geometric inference
Sophia-Antipolis, January 2016 Winter School An introduction to Topological Data Analysis through persistent homology: Intro and geometric inference Frédéric Chazal INRIA Saclay - Ile-de-France frederic.chazal@inria.fr
Lecture notes for Topology MMA100
Lecture notes for Topology MMA100 J A S, S-11 1 Simplicial Complexes 1.1 Affine independence A collection of points v 0, v 1,..., v n in some Euclidean space R N are affinely independent if the (affine
On the duality of topological Boolean algebras
On the duality of topological Boolean algebras Matthew de Brecht 1 Graduate School of Human and Environmental Studies, Kyoto University Workshop on Mathematical Logic and its Applications 2016 1 This work
Simplicial Complexes: Second Lecture
Simplicial Complexes: Second Lecture 4 Nov, 2010 1 Overview Today we have two main goals: Prove that every continuous map between triangulable spaces can be approximated by a simplicial map. To do this,
Trees in Dynamics and Probability
Trees in Dynamics and Probability Mario Bonk (UCLA) joint work with Huy Tran (TU Berlin) and with Daniel Meyer (U. Liverpool) New Developments in Complex Analysis and Function Theory University of Crete,
On Fuzzy Topological Spaces Involving Boolean Algebraic Structures
Journal of mathematics and computer Science 15 (2015) 252-260 On Fuzzy Topological Spaces Involving Boolean Algebraic Structures P.K. Sharma Post Graduate Department of Mathematics, D.A.V. College, Jalandhar
Situation #1: Translating Functions Prepared at University of Georgia William Plummer EMAT 6500 Date last revised: July 28, 2013
Situation #1: Translating Functions Prepared at University of Georgia William Plummer EMAT 6500 Date last revised: July 28, 2013 Prompt An Algebra class is discussing the graphing of quadratic functions
Introduction to Matchbox Manifolds
University of Illinois at Chicago www.math.uic.edu/ hurder... Report on works with Alex Clark and Olga Lukina Matchbox manifolds Let M be a continuum. Suppose that each x M has an open neighborhood homeomorphic
Simplicial volume of non-compact manifolds
Simplicial volume of non-compact manifolds Clara Löh April 2008 Abstract. Degree theorems are statements bounding the mapping degree in terms of the volumes of the domain and target manifolds. A possible
Lower bounds on the barrier parameter of convex cones
of convex cones Université Grenoble 1 / CNRS June 20, 2012 / High Performance Optimization 2012, Delft Outline Logarithmically homogeneous barriers 1 Logarithmically homogeneous barriers Conic optimization
Generalized Moment-Angle Complexes
Generalized Moment-Angle Complexes Fred Cohen 1-5 June 2010 joint work with Tony Bahri, Martin Bendersky, and Sam Gitler The setting and the problems: This report addresses joint work with Tony Bahri,
On expansive chaotic maps in G-spaces
International Journal of Applied Mathematical Research, 3 (3) (2014) 225-232 Science Publishing Corporation wwwsciencepubcocom/indexphp/ijamr doi: 1014419/ijamrv3i32286 Research Paper On expansive chaotic
The Brownian map A continuous limit for large random planar maps
The Brownian map A continuous limit for large random planar maps Jean-François Le Gall Université Paris-Sud Orsay and Institut universitaire de France Seminar on Stochastic Processes 0 Kai Lai Chung lecture
Real Analysis, 2nd Edition, G.B.Folland
Real Analysis, 2nd Edition, G.B.Folland Chapter 4 Point Set Topology Yung-Hsiang Huang 4.1 Topological Spaces 1. If card(x) 2, there is a topology on X that is T 0 but not T 1. 2. If X is an infinite set,
Topology and its Applications
Topology and its Applications 159 (2012) 3137 3142 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol A topological dynamical system on the Cantor
Graphs associated to CAT(0) cube complexes
Graphs associated to CAT(0) cube complexes Mark Hagen McGill University Cornell Topology Seminar, 15 November 2011 Outline Background on CAT(0) cube complexes The contact graph: a combinatorial invariant
SPERNER S LEMMA MOOR XU
SPERNER S LEMMA MOOR XU Abstract. Is it possible to dissect a square into an odd number of triangles of equal area? This question was first answered by Paul Monsky in 970, and the solution requires elements
Metric and metrizable spaces
Metric and metrizable spaces These notes discuss the same topic as Sections 20 and 2 of Munkres book; some notions (Symmetric, -metric, Ψ-spaces...) are not discussed in Munkres book.. Symmetric, -metric,
An Introduction to Belyi Surfaces
An Introduction to Belyi Surfaces Matthew Stevenson December 16, 2013 We outline the basic theory of Belyi surfaces, up to Belyi s theorem (1979, [1]), which characterizes these spaces as precisely those
Lecture 7: Jan 31, Some definitions related to Simplical Complex. 7.2 Topological Equivalence and Homeomorphism
CS 6170 Computational Topology: Topological Data Analysis University of Utah Spring 2017 School of Computing Lecture 7: Jan 31, 2017 Lecturer: Prof. Bei Wang Scribe: Avani Sharma,
Coxeter Groups and CAT(0) metrics
Peking University June 25, 2008 http://www.math.ohio-state.edu/ mdavis/ The plan: First, explain Gromov s notion of a nonpositively curved metric on a polyhedral complex. Then give a simple combinatorial
Strata and stabilizers of trees
Vincent Guirardel Joint work with G. Levitt Institut de Mathématiques de Toulouse Goal of the talk Outer space CV N = { minimal free actions of F N on simplicial trees } /. Compactification CV N = { minimal
Polyhedra inscribed in a hyperboloid & AdS geometry. anti-de Sitter geometry.
Polyhedra inscribed in a hyperboloid and anti-de Sitter geometry. Jeffrey Danciger 1 Sara Maloni 2 Jean-Marc Schlenker 3 1 University of Texas at Austin 2 Brown University 3 University of Luxembourg AMS
Final Exam, F11PE Solutions, Topology, Autumn 2011
Final Exam, F11PE Solutions, Topology, Autumn 2011 Question 1 (i) Given a metric space (X, d), define what it means for a set to be open in the associated metric topology. Solution: A set U X is open if,
simply ordered sets. We ll state only the result here, since the proof is given in Munkres.
p. 1 Math 490 Notes 20 More About Compactness Recall that in Munkres it is proved that a simply (totally) ordered set X with the order topology is connected iff it satisfies: (1) Every subset bounded above
Reconstruction of Filament Structure
Reconstruction of Filament Structure Ruqi HUANG INRIA-Geometrica Joint work with Frédéric CHAZAL and Jian SUN 27/10/2014 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches
Rational maps, subdivision rules, and Kleinian groups
Rational maps, subdivision rules, and Kleinian groups J. Cannon 1 W. Floyd 2 W. Parry 3 1 Department of Mathematics Brigham Young University 2 Department of Mathematics Virginia Tech 3 Department of Mathematics
Two Connections between Combinatorial and Differential Geometry
Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut für Mathematik, Technische Universität Berlin Berlin Mathematical School DFG Research Group Polyhedral Surfaces
Persistence stability for geometric complexes
Lake Tahoe - December, 8 2012 NIPS workshop on Algebraic Topology and Machine Learning Persistence stability for geometric complexes Frédéric Chazal Joint work with Vin de Silva and Steve Oudot (+ on-going
TOPOLOGY CHECKLIST - SPRING 2010
TOPOLOGY CHECKLIST - SPRING 2010 The list below serves as an indication of what we have covered in our course on topology. (It was written in a hurry, so there is a high risk of some mistake being made
Sheaves and Stacks. November 5, Sheaves and Stacks
November 5, 2014 Grothendieck topologies Grothendieck topologies are an extra datum on a category. They allow us to make sense of something being locally defined. Give a formal framework for glueing problems
Notes on metric spaces and topology. Math 309: Topics in geometry. Dale Rolfsen. University of British Columbia
Notes on metric spaces and topology Math 309: Topics in geometry Dale Rolfsen University of British Columbia Let X be a set; we ll generally refer to its elements as points. A distance function, or metric
arxiv: v1 [math.at] 8 Jan 2015
![arxiv: v1 [math.at] 8 Jan 2015](/thumbs/74/70043747.jpg "arxiv: v1 [math.at] 8 Jan 2015")
HOMOLOGY GROUPS OF SIMPLICIAL COMPLEMENTS: A NEW PROOF OF HOCHSTER THEOREM arxiv:1501.01787v1 [math.at] 8 Jan 2015 JUN MA, FEIFEI FAN AND XIANGJUN WANG Abstract. In this paper, we consider homology groups
Using context and model categories to define directed homotopies
Using context and model categories to define directed homotopies p. 1/57 Using context and model categories to define directed homotopies Peter Bubenik Ecole Polytechnique Fédérale de Lausanne (EPFL) peter.bubenik@epfl.ch
Teichmüller Space and Fenchel-Nielsen Coordinates
Teichmüller Space and Fenchel-Nielsen Coordinates Nathan Lopez November 30, 2015 Abstract Here we give an overview of Teichmüller space and its realization as a smooth manifold through Fenchel- Nielsen
3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines
3.5 Day 1 Warm Up Graph each line. 1. y = 4x 2. y = 3x + 2 3. y = x 3 4. y = 4 x + 3 3 November 2, 2015 3.4 Proofs with Perpendicular Lines Geometry 3.5 Equations of Parallel and Perpendicular Lines Day
Lecture : Topological Space
Example of Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Example of 1 2 3 Example of 4 5 6 Example of I Topological spaces and continuous
Topological properties of convex sets
Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory Winter 2018 Topic 5: Topological properties of convex sets 5.1 Interior and closure of convex sets Let
Geometric structures on manifolds
CHAPTER 3 Geometric structures on manifolds In this chapter, we give our first examples of hyperbolic manifolds, combining ideas from the previous two chapters. 3.1. Geometric structures 3.1.1. Introductory
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 10
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 10 RAVI VAKIL Contents 1. Schemes 1 1.1. Affine schemes 2 1.2. Schemes 3 1.3. Morphisms of affine schemes 3 1.4. Morphisms of general schemes 4 1.5. Scheme-theoretic
LECTURE 8: SMOOTH SUBMANIFOLDS
LECTURE 8: SMOOTH SUBMANIFOLDS 1. Smooth submanifolds Let M be a smooth manifold of dimension n. What object can be called a smooth submanifold of M? (Recall: what is a vector subspace W of a vector space
PS Geometric Modeling Homework Assignment Sheet I (Due 20-Oct-2017)
Homework Assignment Sheet I (Due 20-Oct-2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist
Computer Vision I - Algorithms and Applications: Multi-View 3D reconstruction
Computer Vision I - Algorithms and Applications: Multi-View 3D reconstruction Carsten Rother 09/12/2013 Computer Vision I: Multi-View 3D reconstruction Roadmap this lecture Computer Vision I: Multi-View
Combinatorial Algebraic Topology and applications to Distributed Computing
Combinatorial Algebraic Topology and applications to Distributed Computing Dmitry Feichtner-Kozlov University of Bremen XXIst Oporto Meeting on Geometry, Topology and Physics LISSABON 2015 Lecture I: Simplicial
Geometric structures on 2-orbifolds
Geometric structures on 2-orbifolds Section 1: Manifolds and differentiable structures S. Choi Department of Mathematical Science KAIST, Daejeon, South Korea 2010 Fall, Lectures at KAIST S. Choi (KAIST)
Topology I Test 1 Solutions October 13, 2008
Topology I Test 1 Solutions October 13, 2008 1. Do FIVE of the following: (a) Give a careful definition of connected. A topological space X is connected if for any two sets A and B such that A B = X, we