Tuesday, Thursday, 2pm - 3:20 pm
Recommended reading for weeks 1-3:
Buhovsky L., Payette J., Polterovich I., Polterovich L., Shelukhin E., Stojisavljević V. Coarse nodal count and topological persistence. Journal of the European Mathematical Society. 2024
Carlsson G. Topological pattern recognition for point cloud data. Acta Numerica. 2014, 289-368.
Ghrist R. Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society. 2008, 45, 61-75.
Polterovich L., Rosen D., Samvelyan K., Zhang J. Topological persistence in geometry and analysis. American Mathematical Soc.; 2020
Polterovich L, Rosen D. Function theory on symplectic manifolds. American Mathematical Soc.; 2014 (Chapter 10.2)
Weinberger S. Interpolation, the rudimentary geometry of spaces of Lipschitz functions, and geometric complexity. Foundations of Computational Mathematics. 2019, 5, 991-1011.
Weinberger S. What is...persistent homology. Notices of the AMS, 2011, 58, 36-39.
Syllabus for weeks 1-3:
Persistence modules and their morphisms.
Main examples: Morse theory and the Vietoris- Rips complexes for finite metric spaces.
The normal form theorem and barcodes.
Stability theorems.
The isometry theorem.
Filtered chain complexes. Simplex counting.
Recommended reading for weeks 4-5:
1. Adler, R. and Taylor, J., Random Fields and Geometry, Springer Monographs in Mathematics. 2007
2. JD Boissonnat, F Chazal, M Yvinec Geometric and topological inference Cambridge University Press 2018
3. Robert Ghrist, "Barcodes: the persistent topology of data" (PDF), Bull. Amer. Math. Soc., 45: (2008) 61–75
4. O.Bobrowski and S.Weinberger On the vanishing of homology in random Čech complexes, Prob Theory and Related Fields. 51 (2017) no. 1, 14–51.
5. Bobrowski, G Oliveira - Random Čech complexes on Riemannian manifolds Random Structures & Algorithms, 54 (3), 373-412 (2019)
6. Gunnar Carlsson, Topological pattern recognition for point cloud data, Acta Numerica 33 (2024), 289-368
7. A.Edelman and E.Kostlan, How many zeroes of a random polynomial are real? Bull AMS (1995) 1-37
8. Niyogi, P., Smale, S. and Weinberger, S. (2008), ‘Finding the homology of submanifolds with high confidence from random samples’, Discrete Comput. Geom. 39, 419–441
Recommended reading for weeks 6-8:
1. Matt Clay and Dan Margalit, Office hours with a geometric group theorist. Princeton University Press 2017
2. John Roe, Lectures on Coarse Geometry, AMS 2003
3. S.Gersten, Dehn Functions and l1-norms of Finite Presentations, Algorithms and classification in combinatorial group theory 1992
4. Yuri Manin, A course in mathematical logic for mathematicians, Springer 2010
5. J.Rotman, An introduction to the theory of groups, Springer 1995
6. J.Jost, Riemannian geometry and geometric analysis Springer 2008
7. S. Weinberger, What is Persistent Homology? Notices AMS 2011
8. P. Pansu, Théorie de l’homotopie quantitative, Bourbaki Janvier 2025 (which has many useful references and a beautiful discussion)
Further reading for week 8:
A. Berdnikov, L. Guth, and F. Manin, Degrees of maps and multiscale geometry, Forum Math. Pi 12 (2024), Paper No. e2, 48.
A. Berdnikov and F. Manin, Scalable spaces, Invent. Math. 229 (2022), no. 3, 1055–1100.
G. R. Chambers, D. Dotterrer, F. Manin, and Sh. Weinberger, Quantitative null-cobordism, J. Amer. Math. Soc. 31 (2018), no. 4, 1165–1203, With an appendix by Manin and Weinberger.
G. R. Chambers, F. Manin, and Sh. Weinberger, Quantitative nullhomotopy and rational homotopy type, Geom. Funct. Anal. 28 (2018), no. 3, 563–588.
S. Ferry and Sh. Weinberger, Quantitative algebraic topology and Lipschitz homotopy, Proc. Natl.Acad. Sci. USA 110 (2013), no. 48, 19246–19250.
M. Gromov Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhauser Boston, Inc., Boston, MA, 1999, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates.
M. Gromov Quantitative homotopy theory, Prospects in mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 45–49.
F. Manin, Plato’s cave and differential forms, Geom. Topol. 23 (2019), no. 6, 3141–3202.