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