Homological Algebra and Its Application: A Descriptive Study

Kaushal Rana

doi:10.55544/ijrah.2.1.47

Authors

Dr. Kaushal Rana Assistant Professor, Department of Mathematics, Dau Dayal Institute of Vocational Education, Dr. Bhimrao Ambedkar University, Agra, Uttar Pradesh, INDIA.

DOI:

https://doi.org/10.55544/ijrah.2.1.47

Keywords:

Homological algebra, torsion functors, extension functors, E'tale sheaf theory

Abstract

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTⁿ(-,-) , the torsion functors TOR_n (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

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References

J. Hocking and G. Young. Topology. Dover Press, 1988. 30

X. Jiang, L.-H. Lim, Y. Yao, and Y. Ye. Statistical ranking and combinatorial Hodge theory. Math. Program., 127(1, Ser. B):203–244, 2011. 39

H. Adams and G. Carlsson. Evasion paths in mobile sensor networks. International Journal of Robotics Research, 34:90–104, 2014. 46, 47

R. J. Adler. The Geometry of Random Fields. Society for Industrial and Applied Mathematics, 1981. 9

R. J. Adler, O. Bobrowski, M. S. Borman, E. Subag, and S. Weinberger. Persistent homology for random fields and complexes. In Borrowing Strength: Theory Powering Applications, pages 124–143. IMS Collections, 2010. 13

R. J. Adler and J. E. Taylor. Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York, 2007. 9, 49

R. H. Atkin. Combinatorial Connectivities in Social Systems. Springer Basel AG, 1977. 6

A. Banyaga and D. Hurtubise. Morse Homology. Springer, 2004. 32

Y. Baryshnikov and R. Ghrist. Target enumeration via Euler characteristic integrals. SIAM J. Appl. Math., 70(3):825–844, 2009. 9, 44

Y. Baryshnikov and R. Ghrist. Euler integration over definable functions. Proc. Natl. Acad. Sci. USA, 107(21):9525–9530, 2010. 9, 44

Y. Baryshnikov, R. Ghrist, and D. Lipsky. Inversion of Euler integral transforms with applications to sensor data. Inverse Problems, 27(12), 2011. 9

U. Bauer and M. Lesnick. Induced matchingsand the algebraic stability of persistence barcodes. Discrete & Computational Geometry, 6(2):162–191, 2015. 21

P. Bendich, H. Edelsbrunner, and M. Kerber. Computing roburobustand persistence for images. IEEE Trans. Visual and Comput. Graphics, pages 1251–1260, 2010. 22

P. Bendich, J. Marron, E. Miller, A. Pielcoh, and S. Skwerer. Persistent homology analysis of brain artery trees. To appear in Ann. Appl. Stat., 2016. 22

S. Bhattacharya, R. Ghrist, and V. Kumar. Persistent homology for path planning in uncertain environments. IEEE Trans. on Robotics, 31(3):578–590, 2015. 23

L. J. Billera. Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. Amer. Math. Soc., 310(1):325–340, 1988. 44

L. J. Billera, S. P. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees. Adv. in Appl. Math., 27(4):733–767, 2001. 7

M. Botnan and M. Lesnick. Algebraic stability of zigzag persistence modules. ArXiv:160400655v2, Apr 2016. 21, 44

R. Bott and L. Tu. Differential Forms in Algebraic Topology. Springer, 1982. 39

G. Bredon. Sheaf Theory. Springer, 1997. 40

P. Bubenik, V. de Silva, and J. Scott. Metrics for generalized persistence modules. Found. Comput. Math., 15(6):1501–1531, 2015. 21, 44

P. Bubenik and J. A. Scott. Categorification of persistent homology. Discrete Comput. Geom., 51(3):600–627, 2014. 21

G. Carlsson. The shape of data. In Foundations of computational mathematics, Budapest 2011, volume 403 of London Math. Soc. Lecture Note Ser., pages 16–44. Cambridge Univ. Press, Cambridge, 2013. 49

G. Carlsson and V. de Silva. Zigzag persistence. Found. Comput. Math., 10(4):367–405, 2010. 44, 45

G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian. On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1):1–12, Jan. 2008. 22

G. Carlsson and F. Mémoli. Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res., 11:1425–1470, Aug. 2010. 22

G. Carlsson and F. Mémoli. Classifying clustering schemes. Found. Comput. Math., 13(2):221–252, 2013. 22

G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete Comput. Geom., 42(1):71–93, 2009. 44

S. Carson, V. Ruta, L. Abbott, and R. Axel. Random convergence of olfactory inputs in the drosophila mushroom body. Nature, 497(7447):113–117, 2013. 14

F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules. Arxiv preprint arXiv:1207.3674, 2012. 21

D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete Com-put. Geom., 37(1):103–120, 2007. 21

A. Collins, A. Zomorodian, G. Carlsson, and L. Guibas. A barcode shape descriptor for curve point cloud data. In M. Alexa and S. Rusinkiewicz, editors, Eurographics Symposium on Point-Based Graphics, ETH, Zürich, Switzerland, 2004. 20

J. Curry. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, 2014. 44, 45

J. Curry and A. Patel. Classification of constructible cosheaves. ArXiv 1603.01587, Mar 2016. 45

C. Curto and V. Itskov. Cell groups reveal structure of stimulus space. PLoS Comput. Biol., 4(10):e1000205, 13, 2008. 13

M. d’Amico, P. Frosini, and C. Landi. Optimal matching between reduced size functions. Techni-cal Report 35, DISMI, Univ. degli Studi di Modena e Reggio Emilia, Italy, 2003. 20

V. de Silva and G. Carlsson. Topological estimation using witness complexes. In M. Alexa and S. Rusinkiewicz, editors, Eurographics Symposium on Point-based Graphics, 2004. 6

J. Derenick, A. Speranzon, and R. Ghrist. Homological sensing for mobile robot localization. In Proc. Intl. Conf. Robotics & Aut., 2012. 24

C. Dowker. Homology groups of relations. Annals of Mathematics, pages 84–95, 1952. 6

H. Edelsbrunner and J. Harer. Computational Topology: an Introduction. American Mathematical Society, Providence, RI, 2010. 3, 49

H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete and Computational Geometry, 28:511–533, 2002. 20

M. Farber. Invitation to Topological Robotics. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. 9

D. Farley and L. Sabalka. On the cohomology rings of tree braid groups. J. Pure Appl. Algebra, 212(1):53–71, 2008. 34

D. Farley and L. Sabalka. Presentations of graph braid groups. Forum Math., 24(4):827–859, 2012. 34

R. Forman. Morse theory for cell complexes. Adv. Math., 134(1):90–145, 1998. 33

R. Forman. A user’s guide to discrete Morse theory. Sém. Lothar. Combin., 48, 2002. 33

S. R. Gal. Euler characteristic of the configuration space of a complex. Colloq. Math., 89(1):61–67, 2001. 9

M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow, and V. Nanda. Topological mea-surement of protein compressibility via persistent diagrams. Japan J. Industrial & Applied Mathe-matics, 32(1):1–17, Oct 2014. 24

T. Gao, J. Brodzki, and S. Mukherjee. The geometry of synchronization problems and learning group actions. ArXiv:1610.09051, 2016. 39

S. I. Gelfand and Y. I. Manin. Methods of Homological Algebra. Springer Monographs in Mathemat-ics. Springer-Verlag, Berlin, second edition, 2003. 27, 48

R. Ghrist. Elementary Applied Topology. Createspace, 1.0 edition, 2014. 3, 7, 10, 31, 39, 48

R. Ghrist and S. Krishnan. Positive Alexander duality for pursuit and evasion. ArXiv:1507.04741, 2015. 46, 47, 48

R. Ghrist and S. M. Lavalle. Nonpositive curvature and Pareto optimal coordination of robots. SIAM J. Control Optim., 45(5):1697–1713, 2006. 7

R. Ghrist, D. Lipsky, J. Derenick, and A. Speranzon. Topological landmark-based navigation and mapping. Preprint, 2012. 6, 24

R. Ghrist and V. Peterson. The geometry and topology of reconfiguration. Adv. in Appl. Math., 38(3):302–323, 2007. 7

C. Giusti, E. Pastalkova, C. Curto, and V. Itskov. Clique topology reveal intrinsic structure in neural connections. Proc. Nat. Acad. Sci., 112(44):13455–13460, 2015. 12, 13

L. J. Guibas and S. Y. Oudot. Reconstruction using witness complexes. In Proc. 18th ACM-SIAM Sympos. on Discrete Algorithms, pages 1076–1085, 2007. 6

A. Hatcher. Algebraic Topology. Cambridge University Press, 2002. 9, 10, 27, 28, 31

G. Henselman and R. Ghrist. Matroid filtrations and computational persistent homology. ArXiv:1606.00199 [math.AT], Jun 2016. 36