LunchMaths x MUMS seminar: Hyperbolic geometry and knots (Professor Jessica Purcell)

Professor Jessica Purcell of Monash University introduces knot theory, which is the study of circles embedded in 3D space. Along the way, we also encounter hyperbolic geometry, which is a geometry in which there are *multiple* "lines", passing through a *single* point, which are all "parallel" to a given line! We also learn about manifolds, which are "locally" look like n-dimensional space, but "globally" could be vastly different. Lastly, the tools of knot theory, topology, and hyperbolic geometry converge (at infinity, we suppose?) to give an overview of how hyperbolic geometry can be used to study knots.

Professor Purcell has been a professor at Monash University since 2015, and has previously held roles in the UK and the US. She is a recipient of an ARC Future Fellowship to support outstanding Australian mid-career researchers, and her research interests include geometry and topology, especially 3-manifolds, hyperbolic geometry, knot theory, and how these relate to other areas of mathematics. She is currently writing a textbook on hyperbolic knot theory, aimed at advanced students, which will be published in November 2020.

Chapters:
00:00 Speaker introduction
01:07 Part I - Knot theory
05:22 History of (classification of knots in) knot theory
10:43 Part II - Hyperbolic geometry
14:22 Upper half plane model for hyperbolic geometry
19:39 Part III - Geometric structures
22:29 Tori and universal covers
26:00 More interesting manifolds
30:21 Part IV - Hyperbolic knot theory
32:11 Cutting the figure-8 knot complement into tetrahedra
39:25 Hyperbolic tetrahedra
42:18 SnapPy program demo
45:58 Recent advances in classification of knots
48:11 Conclusion

Speaker: Professor Jessica Purcell
Title: "Hyperbolic geometry and knots"
Abstract: 𝘒𝘯𝘰𝘵 𝘵𝘩𝘦𝘰𝘳𝘺 𝘪𝘴 𝘵𝘩𝘦 𝘮𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘤𝘢𝘭 𝘴𝘵𝘶𝘥𝘺 𝘰𝘧 𝘬𝘯𝘰𝘵𝘵𝘦𝘥 𝘤𝘪𝘳𝘤𝘭𝘦𝘴. 𝘐𝘵
𝘢𝘥𝘥𝘳𝘦𝘴𝘴𝘦𝘴 𝘲𝘶𝘦𝘴𝘵𝘪𝘰𝘯𝘴 𝘴𝘶𝘤𝘩 𝘢𝘴, 𝘸𝘩𝘦𝘯 𝘢𝘳𝘦 𝘵𝘸𝘰 𝘬𝘯𝘰𝘵𝘴 𝘵𝘩𝘦 𝘴𝘢𝘮𝘦? 𝘞𝘩𝘦𝘯 𝘤𝘢𝘯 𝘢 𝘬𝘯𝘰𝘵 𝘣𝘦 𝘶𝘯𝘵𝘪𝘦𝘥 𝘸𝘪𝘵𝘩𝘰𝘶𝘵 𝘤𝘶𝘵𝘵𝘪𝘯𝘨? 𝘔𝘰𝘥𝘦𝘳𝘯 𝘬𝘯𝘰𝘵 𝘵𝘩𝘦𝘰𝘳𝘺 𝘪𝘯𝘷𝘦𝘴𝘵𝘪𝘨𝘢𝘵𝘦𝘴 𝘵𝘩𝘦𝘴𝘦 𝘲𝘶𝘦𝘴𝘵𝘪𝘰𝘯𝘴 𝘶𝘴𝘪𝘯𝘨 𝘵𝘰𝘰𝘭𝘴 𝘢𝘳𝘪𝘴𝘪𝘯𝘨 𝘧𝘳𝘰𝘮 𝘸𝘪𝘥𝘦 𝘢𝘳𝘦𝘢𝘴 𝘰𝘧 𝘮𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘤𝘴. 𝘞𝘦 𝘸𝘪𝘭𝘭 𝘪𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘦 𝘰𝘯𝘦 𝘴𝘶𝘤𝘩 𝘵𝘰𝘰𝘭: 𝘵𝘩𝘦 𝘩𝘺𝘱𝘦𝘳𝘣𝘰𝘭𝘪𝘤 𝘨𝘦𝘰𝘮𝘦𝘵𝘳𝘺 𝘰𝘧 𝘓𝘰𝘣𝘢𝘤𝘩𝘦𝘷𝘴𝘬𝘺 𝘢𝘯𝘥 𝘉𝘰𝘭𝘺𝘢𝘪.

This seminar, recorded on 18 Sept. 2020, was the third in a series of collaborations with MUMS - the Melbourne Uni Maths and Stats Society. The target audience is undergraduate university students who study maths, or have an interest in maths.

LunchMaths is student-run and is not affiliated with any Monash University club.

#LunchMaths #KnotTheory #HyperbolicGeometry #MUMS

[physira]

I understand upto part 3 but part 4 is currently beyond me!
Still a very cool talk
Thanks very much