PPT-Lecture 6: Creating a simplicial complex from data

in a series of preparatory lectures for the Fall 2013 online course MATH7450 22M305 Topics in Topology Scientific and Engineering Applications of Algebraic Topology Target Audience Anyone interested in . Step 0.) Start by adding 0-dimensional vertices . (0-simplices). Creating a simplicial complex. 1. .) . A. dding . 1. -dimensional edges (1-simplices). Add an edge between data points that are “close”. (and cell complexes). . in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology. Target Audience: Anyone interested in . Applications I. ©2008 I.K. Darcy. All rights reserved. This work was partially supported by  the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285).. Isabel K. Darcy. Shalev Ben-David. Definition. Given a function. . and oracle access to , determine f(x) with minimum number of queries. E.g. f is OR on the bits of x – Grover search. D(f) is the deterministic query complexity. Step 0.) Start by adding 0-dimensional vertices . (0-simplices). Creating a simplicial complex. 1. .) . A. dding . 1. -dimensional edges (1-simplices). Add an edge between data points that are “close”. simplicial. . complexes. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology. Target Audience: Anyone interested in . complex = CW . complex. Building block: n-cells = { x in . R. n. : || x || ≤ 1 } . 2-cell = open disk = { x in R. 2. : ||x || < 1 }. Examples: . 0-cell = { x in R. 0. : ||x || < 1 }. Applications II. This work was partially supported by  the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285).. Isabel K. Darcy. Mathematics Department . simplicial. . complexes. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology. Target Audience: Anyone interested in . complex = CW . complex. Building block: n-cells = { x in . R. n. : || x || ≤ 1 } . 2-cell = open disk = { x in R. 2. : ||x || < 1 }. Examples: . 0-cell = { x in R. 0. : ||x || < 1 }. /article/10.1007/s00454-004-1146-y. A . filtered complex . is an increasing sequence of simplicial complexes: C. 0. C. 1. C. 2. …. U. U. U. a, b is in . C. 0. C. 1. Sept 4, 2013. Clustering Via Persistent Homology. Creating a simplicial complex. Step 0.) Start by adding 0-dimensional vertices . (0-simplices). Creating a simplicial complex. 1. .) . A. dding . 1. https://writingcenter.uiowa.edu/#. services. . Math 3900 students will be assigned writing fellows.. For Feb 22: Focus on explaining the TDA mapper algorithm including benefits and negatives.. For March 9. arxiv.org/abs/1601.01580. . Microcircuit . of neurons in the neocortex of . a two-week-old rat.. 8 million connections between 31,000 . neurons.. Data from . Cell..  2015 Oct 8;163(2):456-92. . doi.