topological data analysis and stochastic topology yuliy baryshnikov waikiki, march 2013
|
|
|
- Earl Douglas
- 5 years ago
- Views:
Transcription
1 topological data analysis and stochastic topology yuliy baryshnikov waikiki, march 2013
2 Promise of topological data analysis: extract the structure from the data. In: point clouds Out: hidden structure
3 Underlying structure: manifolds? stratified sets? Two ways to look at them: Poincare's and Riemann: Correspondingly, there are two approaches in dimensionality reduction: Poincare view - use ambient coordinates - PCA, Kernel PCA,... Riemann view - only intrinsic proximity makes sense - IsoMap, EigenMap,...
4 Topological Data Analysis aims at recovery of the most stable invariants of the data. In particular, complete scale invariance is sought This is both a curse and a blessing: very susceptible to noise only vague notion of proximity matters
5 Invariants of topological spaces: two spaces are the same if they are homeomorphic? diffeomorphic? homotopic?
6 Invariants of topological spaces: Dimension? Homotopy groups? Homology and cohomology
7 Homology and cohomology come in different flavors and colors singular, de Rham, simplicial Simplicial - easiest to deal with. Works well with meshes...
8 Algebraic formalism includes chains, cycles and boundaries, and the idiosyncratic relation In essence, linear algebra encoding combinatorics of adjacencies.
9 Homology groups depend on the coefficient ring: what are the entries of the boundary matrix. Sometimes there is torsion: Working over a field ranks of the homology groups are called Betti numbers: numeric invariants of a topological space
10 Morse theory an important tool to generate the cellular partition of the underlying manifold, and estimate the Betti numbers
11 What happens when one adds a simplex of dimension d: either a cycle in dimension d-1 disappears, or a cycle of dimension d appears. This implies existence of an aggregate invariant, the Euler characteristic.
12 Back to data: given a point cloud, how to arrive at a topological space? Taking into account desirable scale? Old idea: alpha-shapes (Robbins, Edelsbrunner...) Place a ball around each point in the dataset and construct the Čech complex: glue in a simplex for any non-empty intersection of the balls. Nerve Theorem: If all intersections are contractible, the union of balls is homotopy equivalent to the Čech complex.
13 Noise is a problem, however: how to get rid of the small features? Answer: increase the scale. and again and again Until the features disappear altogether Persistence is a way to address the onslaught of irrelevant features
14 Tidal traces to be ignored: only the islands survive.
15 Persistence homology Start with a filtration, and record when a cycle appears, and when it dies.
16 Encoding: either barcodes, or persistence diagram
17 Important result: stability theorem small perturbations of the filtration change the persistence diagram little
18 Important tool: Morse theory
19 Some examples Mumford dataset and the Klein bottle
20 Krioukov et al. Hyperbolic mapping of complex networks
21 Simple test: what is the dimension of the internet? How we find the dimension knowing a sample from a manifold?
22 Testing on a special (stratified) set
23 Testing on a special (stratified) set
24 Autonomous Systems ( Internet )
25 Stochastic Topology: dealing with the baseline case Niyogi, Smale, Weinberger: Cech complex for small enough r and dense enough sample with has the homotopy type of the underlying manifold
26 Big Picture: from dust to foam to monolith
27 Big Picture: timescale
28 Big Picture: foam phase
29 Foam phase: oscillation of Euler characteristic Euler characteristic in dim=7
30 Foam phase: concentrations of Betti numbers
31 Life in the ambient space: crackle phenomenon If the noise is Gaussian, the balls form a solid ball. In exponential and subexponential cases, the homologies form in layers.
32 Plenty of further models and questions in stochastic topology Persistence diagrams for Brownian bridge
33 Plenty of further models and questions in stochastic topology Topology of the level sets of random functions Random simplicial complexes Random configuration spaces...
34
A Multicover Nerve for Geometric Inference. Don Sheehy INRIA Saclay, France
A Multicover Nerve for Geometric Inference Don Sheehy INRIA Saclay, France Computational geometers use topology to certify geometric constructions. Computational geometers use topology to certify geometric
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
4.2 Simplicial Homology Groups
4.2. SIMPLICIAL HOMOLOGY GROUPS 93 4.2 Simplicial Homology Groups 4.2.1 Simplicial Complexes Let p 0, p 1,... p k be k + 1 points in R n, with k n. We identify points in R n with the vectors that point
Homology and Persistent Homology Bootcamp Notes 2017
Homology and Persistent Homology Bootcamp Notes Summer@ICERM 2017 Melissa McGuirl September 19, 2017 1 Preface This bootcamp is intended to be a review session for the material covered in Week 1 of Summer@ICERM
66 III Complexes. R p (r) }.
66 III Complexes III.4 Alpha Complexes In this section, we use a radius constraint to introduce a family of subcomplexes of the Delaunay complex. These complexes are similar to the Čech complexes but differ
4. Simplicial Complexes and Simplicial Homology
MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 4. Simplicial Complexes and Simplicial Homology Geometric simplicial complexes 4.1 Definition. A finite subset { v 0, v 1,..., v r } R n
Topological estimation using witness complexes. Vin de Silva, Stanford University
Topological estimation using witness complexes, Acknowledgements Gunnar Carlsson (Mathematics, Stanford) principal collaborator Afra Zomorodian (CS/Robotics, Stanford) persistent homology software Josh
Brian Hamrick. October 26, 2009
Efficient Computation of Homology Groups of Simplicial Complexes Embedded in Euclidean Space TJHSST Senior Research Project Computer Systems Lab 2009-2010 Brian Hamrick October 26, 2009 1 Abstract Homology
Homology of Simplicial Complexes
Homology of Simplicial Complexes Math, David Perkinson Introduction. This is an introduction to the homology of simplicial complexes suitable for a first course in linear algebra. It uses little more than
Stratified Structure of Laplacian Eigenmaps Embedding
Stratified Structure of Laplacian Eigenmaps Embedding Abstract We construct a locality preserving weight matrix for Laplacian eigenmaps algorithm used in dimension reduction. Our point cloud data is sampled
Diffusion Maps and Topological Data Analysis
Diffusion Maps and Topological Data Analysis Melissa R. McGuirl McGuirl (Brown University) Diffusion Maps and Topological Data Analysis 1 / 19 Introduction OVERVIEW Topological Data Analysis The use of
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
Observing Information: Applied Computational Topology.
Observing Information: Applied Computational Topology. Bangor University, and NUI Galway April 21, 2008 What is the geometric information that can be gleaned from a data cloud? Some ideas either already
Data Analysis, Persistent homology and Computational Morse-Novikov theory
Data Analysis, Persistent homology and Computational Morse-Novikov theory Dan Burghelea Department of mathematics Ohio State University, Columbuus, OH Bowling Green, November 2013 AMN theory (computational
Topological Data Analysis
Proceedings of Symposia in Applied Mathematics Topological Data Analysis Afra Zomorodian Abstract. Scientific data is often in the form of a finite set of noisy points, sampled from an unknown space, and
TOPOLOGICAL DATA ANALYSIS
TOPOLOGICAL DATA ANALYSIS BARCODES Ghrist, Barcodes: The persistent topology of data Topaz, Ziegelmeier, and Halverson 2015: Topological Data Analysis of Biological Aggregation Models 1 Questions in data
THE RISE, FALL AND RISE OF SIMPLICIAL COMPLEXES. Andrew Ranicki (Edinburgh) aar Warwick 2nd February, 2018
1 THE RISE, FALL AND RISE OF SIMPLICIAL COMPLEXES Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ aar Warwick 2nd February, 2018 2 Simplicial complexes A simplicial complex is a combinatorial scheme
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,
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
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,
A fast and robust algorithm to count topologically persistent holes in noisy clouds
A fast and robust algorithm to count topologically persistent holes in noisy clouds Vitaliy Kurlin Durham University Department of Mathematical Sciences, Durham, DH1 3LE, United Kingdom vitaliy.kurlin@gmail.com,
Lecture 5: Simplicial Complex
Lecture 5: Simplicial Complex 2-Manifolds, Simplex and Simplicial Complex Scribed by: Lei Wang First part of this lecture finishes 2-Manifolds. Rest part of this lecture talks about simplicial complex.
PERSISTENT HOMOLOGY OF FINITE TOPOLOGICAL SPACES
PERSISTENT HOMOLOGY OF FINITE TOPOLOGICAL SPACES HANEY MAXWELL Abstract. We introduce homology and finite topological spaces. From the basis of that introduction, persistent homology is applied to finite
Persistent Homology and Nested Dissection
Persistent Homology and Nested Dissection Don Sheehy University of Connecticut joint work with Michael Kerber and Primoz Skraba A Topological Data Analysis Pipeline A Topological Data Analysis Pipeline
THE TOPOLOGICAL APPROACH TO SOCIAL CHOICE
THE TOPOLOGICAL APPROACH TO SOCIAL CHOICE WEI HAN CHIA Abstract. Social Choice Theory is a field in economics which studies the possibility of fair aggregation maps of choices. One particularly famous
Euler s Theorem. Brett Chenoweth. February 26, 2013
Euler s Theorem Brett Chenoweth February 26, 2013 1 Introduction This summer I have spent six weeks of my holidays working on a research project funded by the AMSI. The title of my project was Euler s
APPLICATIONS OF TOPOLOGICAL GRAPH THEORY TO 2-MANIFOLD LEARNING
APPLICATIONS OF TOPOLOGICAL GRAPH THEORY TO 2-MANIFOLD LEARNING T. BERRY AND S. SCHLUCHTER Abstract. We show how, given a sufficiently large point cloud sampled from an embedded 2-manifold in R n, we may
Random Simplicial Complexes
Random Simplicial Complexes Duke University CAT-School 2015 Oxford 8/9/2015 Part I Random Combinatorial Complexes Contents Introduction The Erdős Rényi Random Graph The Random d-complex The Random Clique
Kernel Methods for Topological Data Analysis
Kernel Methods for Topological Data Analysis Toward Topological Statistics Kenji Fukumizu The Institute of Statistical Mathematics (Tokyo Japan) Joint work with Genki Kusano and Yasuaki Hiraoka (Tohoku
Cell-Like Maps (Lecture 5)
Cell-Like Maps (Lecture 5) September 15, 2014 In the last two lectures, we discussed the notion of a simple homotopy equivalences between finite CW complexes. A priori, the question of whether or not a
Computational Topology in Reconstruction, Mesh Generation, and Data Analysis
Computational Topology in Reconstruction, Mesh Generation, and Data Analysis Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2014) Computational Topology CCCG
CSE 5559 Computational Topology: Theory, algorithms, and applications to data analysis. Lecture 0: Introduction. Instructor: Yusu Wang
CSE 5559 Computational Topology: Theory, algorithms, and applications to data analysis Lecture 0: Introduction Instructor: Yusu Wang Lecture 0: Introduction What is topology Why should we be interested
Analysis of high dimensional data via Topology. Louis Xiang. Oak Ridge National Laboratory. Oak Ridge, Tennessee
Analysis of high dimensional data via Topology Louis Xiang Oak Ridge National Laboratory Oak Ridge, Tennessee Contents Abstract iii 1 Overview 1 2 Data Set 1 3 Simplicial Complex 5 4 Computation of homology
What will be new in GUDHI library version 2.0.0
What will be new in GUDHI library version 2.0.0 Jean-Daniel Boissonnat, Paweł Dłotko, Marc Glisse, François Godi, Clément Jamin, Siargey Kachanovich, Clément Maria, Vincent Rouvreau and David Salinas DataShape,
HOMOLOGY. Equivalently,
HOMOLOGY JOHN ROGNES 1. Homology 1.1. The Euler characteristic. The Euler characteristic of a compact triangulated surface X is defined to be the alternating sum χ(x) = V E + F where V, E and F are the
Topology and the Analysis of High-Dimensional Data
Topology and the Analysis of High-Dimensional Data Workshop on Algorithms for Modern Massive Data Sets June 23, 2006 Stanford Gunnar Carlsson Department of Mathematics Stanford University Stanford, California
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
Effective Computational Geometry for Curves and Surfaces. Chapter 7 Computational Topology: An Introduction
Effective Computational Geometry for Curves and Surfaces Chapter 7 Computational Topology: An Introduction Günter Rote and Gert Vegter We give an introduction to combinatorial topology, with an emphasis
Random Simplicial Complexes
Random Simplicial Complexes Duke University CAT-School 2015 Oxford 10/9/2015 Part III Extensions & Applications Contents Morse Theory for the Distance Function Persistent Homology and Maximal Cycles Contents
Applied Topology. Instructor: Sara Kališnik Verovšek. Office Hours: Room 304, Tu/Th 4-5 pm or by appointment.
Applied Topology Instructor: Sara Kališnik Verovšek Office Hours: Room 304, Tu/Th 4-5 pm or by appointment. Textbooks We will not follow a single textbook for the entire course. For point-set topology,
Random Simplicial Complexes
Random Simplicial Complexes Duke University CAT-School 2015 Oxford 9/9/2015 Part II Random Geometric Complexes Contents Probabilistic Ingredients Random Geometric Graphs Definitions Random Geometric Complexes
SOCIAL CHOICE AMONG COMPLEX OBJECTS:MATHEMATICAL TOOLS
SOCIAL CHOICE AMONG COMPLEX OBJECTS:MATHEMATICAL TOOLS SIMONA SETTEPANELLA Abstract. Here the reader can find some basic definitions and notations in order to better understand the model for social choise
On the Shape of Data
On the Shape of Data Caren Marzban!, Ulvi Yurtsever 2 1 Applied Physics Lab, and Dept of Statistics, University of Washington, Seattle, WA 98195 2 MathSense Analytics, 1273 Sunny Oaks Circle, Altadena,
Exact discrete Morse functions on surfaces. To the memory of Professor Mircea-Eugen Craioveanu ( )
Stud. Univ. Babeş-Bolyai Math. 58(2013), No. 4, 469 476 Exact discrete Morse functions on surfaces Vasile Revnic To the memory of Professor Mircea-Eugen Craioveanu (1942-2012) Abstract. In this paper,
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
Algebraic Topology: A brief introduction
Algebraic Topology: A brief introduction Harish Chintakunta This chapter is intended to serve as a brief, and far from comprehensive, introduction to Algebraic Topology to help the reading flow of this
Computational Homology of cubical and permutahedral complexes
Computational Homology of cubical and permutahedral complexes by Fintan Hegarty School of Mathematics, Statistics, and Applied Mathematics Supervised by Prof. Graham Ellis SUBMITTED IN PARTIAL FULFILLMENT
Topological Persistence
Topological Persistence Topological Persistence (in a nutshell) X topological space f : X f persistence X Dg f signature: persistence diagram encodes the topological structure of the pair (X, f) 1 Topological
arxiv: v1 [math.st] 11 Oct 2017
![arxiv: v1 [math.st] 11 Oct 2017](/thumbs/83/87139311.jpg "arxiv: v1 [math.st] 11 Oct 2017")
An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists Frédéric Chazal and Bertrand Michel October 12, 2017 arxiv:1710.04019v1 [math.st] 11 Oct 2017 Abstract
New Results on the Stability of Persistence Diagrams
New Results on the Stability of Persistence Diagrams Primoz Skraba Jozef Stefan Institute TAGS - Linking Topology to Algebraic Geometry and Statistics joint work with Katharine Turner and D. Yogeshwaran
Topology and Data. Gunnar Carlsson Department of Mathematics, Stanford University Stanford, California October 2, 2008
Topology and Data Gunnar Carlsson Department of Mathematics, Stanford University Stanford, California 94305 October 2, 2008 1 Introduction An important feature of modern science and engineering is that
The Cyclic Cycle Complex of a Surface
The Cyclic Cycle Complex of a Surface Allen Hatcher A recent paper [BBM] by Bestvina, Bux, and Margalit contains a construction of a cell complex that gives a combinatorial model for the collection of
arxiv: v7 [math.at] 12 Sep 2017
![arxiv: v7 [math.at] 12 Sep 2017](/thumbs/83/87225858.jpg "arxiv: v7 [math.at] 12 Sep 2017")
A Roadmap for the Computation of Persistent Homology arxiv:1506.08903v7 [math.at] 12 Sep 2017 Nina Otter (otter@maths.ox.ac.uk) Mason A. Porter (mason@math.ucla.edu) Ulrike Tillmann (tillmann@maths.ox.ac.uk)
JPlex. CSRI Workshop on Combinatorial Algebraic Topology August 29, 2009 Henry Adams
JPlex CSRI Workshop on Combinatorial Algebraic Topology August 29, 2009 Henry Adams Plex: Vin de Silva and Patrick Perry (2000-2006) JPlex: Harlan Sexton and Mikael Vejdemo- Johansson (2008-present) JPlex
Topological Classification of Data Sets without an Explicit Metric
Topological Classification of Data Sets without an Explicit Metric Tim Harrington, Andrew Tausz and Guillaume Troianowski December 10, 2008 A contemporary problem in data analysis is understanding the
Geometric Data. Goal: describe the structure of the geometry underlying the data, for interpretation or summary
Geometric Data - ma recherche s inscrit dans le contexte de l analyse exploratoire des donnees, dont l obj Input: point cloud equipped with a metric or (dis-)similarity measure data point image/patch,
arxiv: v1 [cs.it] 10 May 2016
![arxiv: v1 [cs.it] 10 May 2016](/thumbs/93/112959514.jpg "arxiv: v1 [cs.it] 10 May 2016")
Separating Topological Noise from Features using Persistent Entropy Nieves Atienza 1, Rocio Gonzalez-Diaz 1, and Matteo Rucco 2 arxiv:1605.02885v1 [cs.it] 10 May 2016 1 Applied Math Department, School
1 Euler characteristics
Tutorials: MA342: Tutorial Problems 2014-15 Tuesday, 1-2pm, Venue = AC214 Wednesday, 2-3pm, Venue = AC201 Tutor: Adib Makroon 1 Euler characteristics 1. Draw a graph on a sphere S 2 PROBLEMS in such a
Computational Statistics and Mathematics for Cyber Security
and Mathematics for Cyber Security David J. Marchette Sept, 0 Acknowledgment: This work funded in part by the NSWC In-House Laboratory Independent Research (ILIR) program. NSWCDD-PN--00 Topics NSWCDD-PN--00
Clustering algorithms and introduction to persistent homology
Foundations of Geometric Methods in Data Analysis 2017-18 Clustering algorithms and introduction to persistent homology Frédéric Chazal INRIA Saclay - Ile-de-France frederic.chazal@inria.fr Introduction
Computational Topology in Neuroscience
Computational Topology in Neuroscience Bernadette Stolz Lincoln College University of Oxford A dissertation submitted for the degree of Master of Science in Mathematical Modelling & Scientific Computing
6.2 Classification of Closed Surfaces
Table 6.1: A polygon diagram 6.1.2 Second Proof: Compactifying Teichmuller Space 6.2 Classification of Closed Surfaces We saw that each surface has a triangulation. Compact surfaces have finite triangulations.
From Topology to Formal Languages: a multiscale analysis of large data set
From Topology to Formal Languages: a multiscale analysis of large data set Emanuela Merelli University of Camerino, Italy Multiscale Modelling and Computing Workshop at Lorentz Center Leiden 11th April
Research in Computational Differential Geomet
Research in Computational Differential Geometry November 5, 2014 Approximations Often we have a series of approximations which we think are getting close to looking like some shape. Approximations Often
COMBINATORIAL METHODS IN ALGEBRAIC TOPOLOGY
COMBINATORIAL METHODS IN ALGEBRAIC TOPOLOGY 1. Geometric and abstract simplicial complexes Let v 0, v 1,..., v k be points in R n. These points determine a hyperplane in R n, consisting of linear combinations
Topological Perspectives On Stratification Learning
Topological Perspectives On Stratification Learning Bei Wang School of Computing Scientific Computing and Imaging Institute (SCI) University of Utah www.sci.utah.edu/~beiwang August 9, 2018 Talk Overview
Basics of Combinatorial Topology
Chapter 7 Basics of Combinatorial Topology 7.1 Simplicial and Polyhedral Complexes In order to study and manipulate complex shapes it is convenient to discretize these shapes and to view them as the union
Topic: Orientation, Surfaces, and Euler characteristic
Topic: Orientation, Surfaces, and Euler characteristic The material in these notes is motivated by Chapter 2 of Cromwell. A source I used for smooth manifolds is do Carmo s Riemannian Geometry. Ideas of
Lectures on topology. S. K. Lando
Lectures on topology S. K. Lando Contents 1 Reminder 2 1.1 Topological spaces and continuous mappings.......... 3 1.2 Examples............................. 4 1.3 Properties of topological spaces.................
Manifolds. Chapter X. 44. Locally Euclidean Spaces
Chapter X Manifolds 44. Locally Euclidean Spaces 44 1. Definition of Locally Euclidean Space Let n be a non-negative integer. A topological space X is called a locally Euclidean space of dimension n if
A THINNING ALGORITHM FOR TOPOLOGICALLY CORRECT 3D SURFACE RECONSTRUCTION
A THINNING ALGORITHM FOR TOPOLOGICALLY CORRECT 3D SURFACE RECONSTRUCTION Leonid Tcherniavski Department of Informatics University of Hamburg tcherniavski@informatik.uni-hamburg.de ABSTRACT The existing
The Classification Problem for 3-Manifolds
The Classification Problem for 3-Manifolds Program from ca. 1980: 1. Canonical decomposition into simpler pieces. 2. Explicit classification of special types of pieces. 3. Generic pieces are hyperbolic
The Topology of Skeletons and Offsets
The Topology of Skeletons and Offsets Stefan Huber 1 1 B&R Industrial Automation stefan.huber@br-automation.com Abstract Given a polygonal shape with holes, we investigate the topology of two types of
A Multicover Nerve for Geometric Inference
A Multicover Nerve for Geometric Inference Donald R. Sheehy Abstract We show that filtering the barycentric decomposition of a Čech complex by the cardinality of the vertices captures precisely the topology
Western TDA Learning Seminar. June 7, 2018
The The Western TDA Learning Seminar Department of Mathematics Western University June 7, 2018 The The is an important tool used in TDA for data visualization. Input point cloud; filter function; covering
Classifying Spaces and Spectral Sequences
Classifying Spaces and Spectral Sequences Introduction Christian Carrick December 2, 2016 These are a set of expository notes I wrote in preparation for a talk given in the MIT Kan Seminar on December
The virtual cohomology dimension of Teichmüller modular groups: the first results and a road not taken
The virtual cohomology dimension of Teichmüller modular groups: the first results and a road not taken To Leningrad Branch of Steklov Mathematical Institute, with gratitude for giving me freedom to pursue
Recent Advances and Trends in Applied Algebraic Topology
Recent Advances and Trends in Applied Algebraic Topology Mikael Vejdemo-Johansson School of Computer Science University of St Andrews Scotland July 8, 2012 M Vejdemo-Johansson (St Andrews) Trends in applied
Geom/Top: Homework 1 (due Monday, 10/08/12)
Geom/Top: Homework (due Monday, 0/08/2). Read Farb notes. 2. Read Hatcher, Intro to Chapter 2 and Section 2.. 3. (Do not hand this in) Let φ : A B and ψ : B A be group homomorphisms. Suppose that φ ψ =
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU /858B Fall 2017
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B Fall 2017 LECTURE 2: THE SIMPLICIAL COMPLEX DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU
3. Persistence. CBMS Lecture Series, Macalester College, June Vin de Silva Pomona College
Vin de Silva Pomona College CBMS Lecture Series, Macalester College, June 2017 o o The homology of a planar region Chain complex We have constructed a diagram of vector spaces and linear maps @ 0 o 1 C
Estimating Geometry and Topology from Voronoi Diagrams
Estimating Geometry and Topology from Voronoi Diagrams Tamal K. Dey The Ohio State University Tamal Dey OSU Chicago 07 p.1/44 Voronoi diagrams Tamal Dey OSU Chicago 07 p.2/44 Voronoi diagrams Tamal Dey
Stable and Multiscale Topological Signatures
Stable and Multiscale Topological Signatures Mathieu Carrière, Steve Oudot, Maks Ovsjanikov Inria Saclay Geometrica April 21, 2015 1 / 31 Shape = point cloud in R d (d = 3) 2 / 31 Signature = mathematical
GEOMETRY OF SURFACES. b3 course Nigel Hitchin
GEOMETRY OF SURFACES b3 course 2004 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on surfaces. Your mental image of a surface should be something like this: or this However we
Namita Lokare, Daniel Benavides, Sahil Juneja and Edgar Lobaton
#analyticsx HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY ABSTRACT We propose a hierarchical activity clustering methodology, which incorporates the use of topological
Discrete Differential Geometry. Differential Geometry
Discrete Differential Geometry Yiying Tong CSE 891 Sect 004 Differential Geometry Why do we care? theory: special surfaces minimal, CMC, integrable, etc. computation: simulation/processing Grape (u. of
Clique homological simplification problem is NP-hard
Clique homological simplification problem is NP-hard NingNing Peng Department of Mathematics, Wuhan University of Technology pnn@whut.edu.cn Foundations of Mathematics September 9, 2017 Outline 1 Homology
Small CW -models for Eilenberg-Mac Lane spaces
Small CW -models for Eilenberg-Mac Lane spaces in honour of Prof. Dr. Hans-Joachim Baues Bonn, March 2008 Clemens Berger (Nice) 1 Part 1. Simplicial sets. The simplex category is the category of finite
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.
Surfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
Rational Homotopy Theory
Yves Felix Stephen Halperin Jean-Claude Thomas Rational Homotopy Theory Springer Contents Introduction Table of Examples vii xxvii I Homotopy Theory, Resolutions for Fibrations, and P- local Spaces 0 Topological
Topological Data Analysis
Topological Data Analysis Deepak Choudhary(11234) and Samarth Bansal(11630) April 25, 2014 Contents 1 Introduction 2 2 Barcodes 2 2.1 Simplical Complexes.................................... 2 2.1.1 Representation
Algorithmic aspects of embedding simplicial complexes in R d
Algorithmic aspects of embedding simplicial complexes in R d Jiří Matoušek Charles University, Prague and ETH Zurich joint work with Martin Čadek, Marek Krčál, Eric Sedgwick, Francis Sergeraert, Martin
Topological Inference via Meshing
Topological Inference via Meshing Benoit Hudson, Gary Miller, Steve Oudot and Don Sheehy SoCG 2010 Mesh Generation and Persistent Homology The Problem Input: Points in Euclidean space sampled from some
Mapper, Manifolds, and More! Topological Data Analysis and Mapper
Mapper, Manifolds, and More! Topological Data Analysis and Mapper Matt Piekenbrock Data Science and Security Cluster (DSSC) Talk: March 28th, 2018 Topology Topology is a branch of mathematics concerned
Topological Invariance under Line Graph Transformations
Symmetry 2012, 4, 329-335; doi:103390/sym4020329 Article OPEN ACCESS symmetry ISSN 2073-8994 wwwmdpicom/journal/symmetry Topological Invariance under Line Graph Transformations Allen D Parks Electromagnetic
Computational Geometry Learning
Computational Geometry Learning Jean-Daniel Boissonnat Frédéric Chazal Geometrica, INRIA http://www-sop.inria.fr/geometrica Lectures at MPRI MPRI Computational Geometry Learning Lectures at MPRI 1 / 14
Introduction to geometry
1 2 Manifolds A topological space in which every point has a neighborhood homeomorphic to (topological disc) is called an n-dimensional (or n-) manifold Introduction to geometry The German way 2-manifold
Topological Data Analysis
Mikael Vejdemo-Johansson School of Computer Science University of St Andrews Scotland 22 November 2011 M Vejdemo-Johansson (St Andrews) TDA 22 November 2011 1 / 32 e Shape of Data Outline 1 e Shape of
arxiv: v1 [math.ca] 29 Apr 2017
![arxiv: v1 [math.ca] 29 Apr 2017](/thumbs/93/111256484.jpg "arxiv: v1 [math.ca] 29 Apr 2017")
1 arxiv:1705.00970v1 [math.ca] 29 Apr 2017 1 Inference of a Dyadic Measure and its Simplicial Geometry from Binary Feature Data and Application to Data Quality Linda Ness Rutgers University linda.ness@rutgers.edu