Recent Advances and Trends in Applied Algebraic Topology
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1 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 algebraic topology July 8, / 15
2 History and trends in applied algebraic topology Outline 1 History and trends in applied algebraic topology 2 Showcasing the next generation M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
3 History and trends in applied algebraic topology e History of the eld 1845 Kirchhoff formulates homological circuit laws 1895 Analysis Situs by Henri Poincaré de nes homology Noether, Mayer & Vietoris categorify homology 1993 Size theory 1996 Incremental Betti numbers 2002 Persistent homology 2005 Algebraic persistent homology 2007 Stability theorems 2007 Extended persistence 2007 Multi-dimensional persistence 2008 Zig-zag persistence 2009 Algebraic stability 2009 Persistent cohomology 2010 Well groups 2011 Dualities (absolute/relative (co)homology share barcodes) 2012 Categori cation of persistence M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
4 History and trends in applied algebraic topology Current trends Formalization: k[t]-modules Order representations Categorical diagrams Enlargement: more topological techniques are introduced into the eld Speed up: faster computation means larger problems; parallelization, simplifying pre-processing More application elds: information ow networks, ethology, phylogenetics, sports, M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
5 Outline 1 History and trends in applied algebraic topology 2 Showcasing the next generation M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
6 Paweł Dłotko Cohomology in electromagnetic modeling Based on reformulation of Maxwell s laws in a discrete setting Local version of laws (Ampere, Faraday,) formulated on primal/dual cell complex by using (co)boundary operator We impose global version of the laws by adding cohomological information We show that rst cohomology basis of the insulating region is the missing dof that make the computations consistent We provide efficient software to compute cohomology groups & generators M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
7 Paweł Dłotko Homology for regular CW-complexes Usually (co)homology of simplicial or cubical complexes is considered in applied science We have introduced algorithm providing (co)homology of any regular CW complex Gain when the data do not t the simplicial or cubical structure Input: list of faces of every cell Output: (co)homology Incidences of cells computed by the algorithm Example: homology of non regular cubical grid which efficiently approximates homology of nodal domains of functions Technique used in statistical simulation of spinodal decomposition in alloys, providing orders of magnitude better performance M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
8 Showcasing the next generation Bei Wang, University of Utah Local homology measures behavior very close to a point Everything outside a small circle collapses to a single point n-dimensional manifolds turn into n-spheres; we can use local homology to measure dimensionality More importantly, singular or branching behavior corresponds to wedges of spheres x (a) (b) (f) (e) x x x w w (c) (g) (d) x (h) (i) x M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
9 Bei Wang, University of Utah Results Can classify strata in strati ed spaces; turn up inter alia when analyzing conformation spaces in robotics or in chemistry Can classify branching points in self-intersecting time series data M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
10 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
11 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
12 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
13 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
14 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
15 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
16 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
17 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
18 Vidit Nanda, Rutgers M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
19 Vidit Nanda, Rutgers Main result Discrete Morse theory is compatible with persistent homology A ltered complex contracts to a ltered complex This speeds up computations in applied algebraic topology Implementation available Part of the Perseus project Available at M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
20 Showcasing the next generation Ulrich Bauer, IST Austria Uses discrete morse theory for topological simpli cation M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
21 Elizabeth Munch, Duke M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
22 Elizabeth Munch, Duke M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
23 Elizabeth Munch, Duke M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
24 Elizabeth Munch, Duke M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
25 Elizabeth Munch, Duke M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
26 Elizabeth Munch, Duke M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
27 Elizabeth Munch, Duke Results Suppose sensors fail with some probability Then: Computing the probability of failure of the H 2 criterion is #P-hard If imminent failure can be detected by a monitoring system, an algorithm is provided M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
28 Jennifer Gamble, NCSU A sensor network that varies through time, where sensors are mobile or transient, poses additional challenges Zig-zag homology provides the tools needed to track coverage holes across several snapshots M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
29 Hungry for more? Survey papers Topology and Data by Gunnar Carlsson Barcodes: the persistent topology of data by Robert Ghrist Topic conferences ATMCS biennial conference on applied and computational algebraic topology ACM SoCG computational geometry conference with strong computational topology sessions Aggregating websites Homepage of Gunnar Carlsson s workgroup Contains conference and preprint aggregation yet to be launched; coordinating webpage for the community, centered around ATMCS M Vejdemo-Johansson (St Andrews) Trends in applied algebraic topology July 8, / 15
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