Geometry and Topology, orhowdifferent amug andadoughnut ReallyAre. Victoria LEBED, Research Fellow in Maths
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1 Geometry and Topology, orhowdifferent amug andadoughnut ReallyAre Victoria LEBED, Research Fellow in Maths
2 Is itthis? 1 What is mathematics?
3 And this? 1 What is mathematics?
4 Sometimes yes: 1 What is mathematics? NASA scientists with their board of calculations, 1961, Life magazine.
5 Sometimes yes: 1 What is mathematics? A screenshot from Hidden Figures.
6 1 What is mathematics? But mathsis alsoallof this:
7 Maths is about understanding mechanisms and seeing patterns, not only about computing! It involves creativity, and is sometimes closer to arts and philosophy than to sciences.
8 Maths is about understanding mechanisms and seeing patterns, not only about computing! It involves creativity, and is sometimes closer to arts and philosophy than to sciences. Fortune 500 s Most Valued Characteristics in an Employee: Characteristics Teamwork 1 10 Problem Solving 2 12 Interpersonal Skills 3 13 Oral Communication 4 4 Listening Skills 5 5 Personal Career Development 6 6 Creative Thinking 7 7 Leadership 8 8 Goal Setting/Motivation 9 9 Writing 10 1 Organizational Effectiveness Computational Skills 12 2 Reading Skills 13 3
9 2 Mugs and doughnuts Geometry Distances, angles, curvature. Metallic mathematics. Topology Overall shape, deformations. Clay mathematics.
10 2 Mugs and doughnuts Geometry Distances, angles, curvature. Metallic mathematics. mug doughnut Topology Overall shape, deformations. Clay mathematics. mug =doughnut
11 2 Mugs and doughnuts Geometry Distances, angles, curvature. Metallic mathematics. mug doughnut Examples of applications: 1) surveying; 2) construction; 3) astronomy; 4)the shapeof theuniverse. Topology Overall shape, deformations. Clay mathematics. mug =doughnut
12 2 Mugs and doughnuts Geometry Distances, angles, curvature. Metallic mathematics. mug doughnut Examples of applications: 1) surveying; 2) construction; 3) astronomy; 4)the shapeof theuniverse. Topology Overall shape, deformations. Clay mathematics. mug =doughnut Examples of applications: 1) Seven Bridges of Königsberg Problem;
13 2 Mugs and doughnuts Geometry Distances, angles, curvature. Metallic mathematics. mug doughnut Examples of applications: 1) surveying; 2) construction; 3) astronomy; 4)the shapeof theuniverse. Topology Overall shape, deformations. Clay mathematics. mug =doughnut Examples of applications: 1) Seven Bridges of Königsberg Problem; 2) knot theory.
14 3 Knots and braids Knots and braids surround us: rope knots in sailing, mountaineering
15 3 Knots and braids ties hairstyles
16 3 Knots and braids decoration, religion quipu: usedby Incas for recording information
17 4 Knots and braids in mathematics
18 4 Knots and braids in mathematics nature mathematics many objects: different material, size, usage one abstraction: shape
19 4 Knots and braids in mathematics nature mathematics many objects: different material, size, usage one abstraction: shape Mathematics begins when many objects are replaced with one abstraction.
20 4 Knots and braids in mathematics In mathematics, knots and braids are considered up to deformation: =
21 4 Knots and braids in mathematics In mathematics, knots and braids are considered up to deformation: = =
22 5 How to distinguish knots?
23 5 How to distinguish knots? =
24 5 How to distinguish knots? = = =
25 5 How to distinguish knots?
26 5 How to distinguish knots? It is a difficult problem: =
27 6 Knot invariants??
28 6 Knot invariants?? Knot colourings: 3 colours Rule: Ifaknot K isadeformation of aknot K, andkcanbe coloured by 3 colours, then K canbe coloured by3colours aswell.
29 6 Knot invariants Canbe coloured? no yes no
30 7 Knot theory: history 1867: Peter Tait experimented with smoke rings;
31 7 Knot theory: history 1867: Peter Tait experimented with smoke rings; Lord Kelvin (Thomson): atoms = knotted tubes of ether
32 7 Knot theory: history? =
33 7 Knot theory: history? = This is false!!!
34 8 Knot theory: applications Biology: DNA molecules.
35 8 Knot theory: applications Biology: DNA molecules. Some enzymes can cut, twist, and reconnect the DNA. This changes the properties of the DNA. Knot theory isused todetect the actionof enzymes.
36 8 Knot theory: applications Textile: weaving patterns.
37 8 Knot theory: applications Textile: weaving patterns. The shape of the universe question: all 3-dimensional spaces can be encoded by knots.
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