Unlocking Braid Group Cryptography: Topology Meets Cybersecurity 🔐

Goal. Explaining basic concepts of geometric topology in an intuitive way. This time. What is...braid group cryptography? Or: Applications 2 (topology in cybersecurity). Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. Geometric topology is usually the study of manifolds. This can mean manifold things ;-) So to be precise, this video series is mostly about knots, surfaces, three and four manifolds. Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-geotop-2022.html Thumbnail. Pictures created with tikz Geometric topology. https://en.wikipedia.org/wiki/Geometric_topology https://en.wikipedia.org/wiki/List_of_geometric_topology_topics https://en.wikipedia.org/wiki/Manifold https://en.wikipedia.org/wiki/Knot_(mathematics) https://en.wikipedia.org/wiki/Surface_(topology) https://en.wikipedia.org/wiki/3-manifold Concepts in geometric topology. https://en.wikipedia.org/wiki/Knot_invariant https://en.wikipedia.org/wiki/Braid_group https://en.wikipedia.org/wiki/Euler_characteristic https://en.wikipedia.org/wiki/Mapping_class_group https://en.wikipedia.org/wiki/Exotic_sphere https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture https://en.wikipedia.org/wiki/Thurston%27s_geometrization_conjecture https://en.wikipedia.org/wiki/Cobordism https://en.wikipedia.org/wiki/Unknotting_problem Applications of geometric topology. https://physics.stackexchange.com/questions/27051/applications-of-geometric-topology-to-theoretical-physics https://mathoverflow.net/questions/48222/applications-of-knot-theory https://math.mit.edu/research/highschool/primes/circle/documents/2019/Lim_Martin_2019.pdf The wild world of four manifolds. https://en.wikipedia.org/wiki/4-manifold https://en.wikipedia.org/wiki/Category:4-manifolds https://arxiv.org/abs/math/0610700 https://web.stanford.edu/~cm5/4D.pdf https://bookstore.ams.org/gsm-20 https://global.oup.com/academic/product/4-manifolds-9780198784869?cc=au&lang=en& https://en.wikipedia.org/wiki/Four-dimensional_space History of topology. https://www.elsevier.com/books/history-of-topology/james/978-0-444-82375-5 https://en.wikipedia.org/wiki/Analysis_Situs_(paper) https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7.image https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf Poincare conjecture and h-cobordism. https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture https://www.jstor.org/stable/1970239?origin=crossref https://en.wikipedia.org/wiki/H-cobordism https://mathworld.wolfram.com/h-CobordismTheorem.html Braid groups and cryptography. https://arxiv.org/pdf/0711.3941.pdf https://people.math.wisc.edu/~nboston/mahlburg.pdf https://courses.cs.washington.edu/courses/csep590/06wi/finalprojects/anandam.pdf https://en.wikipedia.org/wiki/Braid_group https://www.youtube.com/watch?v=_ScO_ugBd6c (Shameful selfpromotion) Pictures used. Pictures from https://www.youtube.com/watch?v=ONZvm06wsJc Picture from https://arxiv.org/pdf/0711.3941.pdf Pictures created with tikz https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange#/media/File:Diffie-Hellman_Key_Exchange.svg Picture from https://www.youtube.com/watch?v=_ScO_ugBd6c Some books I am using (I sometimes steal some pictures from there). https://www.math.cuhk.edu.hk/course_builder/1920/math4900e/Adams--The%20Knot%20Book.pdf https://arxiv.org/abs/1610.02592?context=math https://www.degruyter.com/document/doi/10.1515/9781400865321/html?lang=en https://bookstore.ams.org/fourman https://www.degruyter.com/document/doi/10.1515/9783110250367/html?lang=en https://bookstore.ams.org/surv-55 https://bookstore.ams.org/gsm-20 KnotAtlas. http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table SnapPy. https://snappy.math.uic.edu/ Mathematica. http://katlas.org/wiki/The_Mathematica_Package_KnotTheory%60 https://demonstrations.wolfram.com/ChartForATorus/ SageMath. https://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html #geometrictopology #topology #mathematics