Spin: a first look at modern Geometry - University of...

Spin: a first look at modern Geometry

Thomas Prince

Magdalen College, Oxford

30 September 2017

Thomas Prince (Magdalen College, Oxford) Spin 30 September 2017 1 / 23

RI Masterclasses

The RI masterclass program was set in motion by Professor Zeeman afterhis Christmas lectures in 1978 (those lectures have also been continueduntil today).

Today we’ll see a little of Prof. Zeeman’s own subject: geometry andtopology.

You can find out more here: http://www.rigb.org/education/masterclasses


RI Masterclasses

The RI masterclass program was set in motion by Professor Zeeman afterhis Christmas lectures in 1978 (those lectures have also been continueduntil today).Today we’ll see a little of Prof. Zeeman’s own subject: geometry andtopology.



RI Masterclasses

The RI masterclass program was set in motion by Professor Zeeman afterhis Christmas lectures in 1978 (those lectures have also been continueduntil today).Today we’ll see a little of Prof. Zeeman’s own subject: geometry andtopology.



Introducing Geometry

What does the word geometry mean to you?

Can you think of any results in Geometry you’ve covered in class?

Pythagorus’ theorem,

Circle theorems,

Trigonometry.






Circle theorems,

Trigonometry.






Circle theorems,

Trigonometry.


A little history

The geometry you will have seen in classes is some of the most ancientmaterial taught in schools (including in history classes).

There is a clay tablet from around 1800BC listing a collection oftriples a, b, c such that a2 = b2 + c2.

Euclid’s Elements written around 300BC contains many results oncircles, cones and cylinders that are well known to you.

Here are some faces associated to results you will know...


A little history






A little history






Modern Geometry

Geometry remains a central subject within mathematics - with connectionsto number theory, mathematical physics, fluid mechanics,...

Exercises

The exercises and activities in this session are all basic or fundamentalexamples in various areas of modern research, I’ll try and say a little bitabout each as we go. In particular I want to reach a connection withmathematical physics which shows how the kind of geometry we look attoday can be used to understand aspects of nature.

Research

Another goal of the masterclass is to give a bit of an idea of what researchmathematics involves, why it is a creative pursuit, and what can attractpeople to it .


Modern Geometry


Exercises


Research



Modern Geometry


Exercises


Research



Rotations

You are hopefully at least a little familiar with rotational symmetry in 2D,but we’ll quickly review it.

Rotation in 2D

Given an angle a and a point p construct acircular arc centred at (0, 0) of angle adrawn anti-clockwise from p, the operationtaking every point p to te other end of thearc formed in this way is a rotation of anglea. You will see ways of making this precisein the next two years.

Recall that a shape in the plane has rotational symmetry with angle a ifthe rotation of angle a takes the shape to itself. How do we count thesesymmetries?


Rotations


Rotation in 2D




Rotations


Rotation in 2D




Rotations

We will use the analogous notion in 3D space.

Rotation in 3D

Fix a line l in 3D space, and an angle a (between 0 and 360◦). The set ofpoints at right angles to a point on l forms a plane. A rotation in 3D fixesl and rotates each plane at right angles to l by a.

Just as for two dimensional shapes, three dimensions objects can haverotational symmetry - with the same definition.


Rotations


Rotation in 3D




Rotations


Rotation in 3D




Exercises

How many rotational symmetries of a cube can you find?

How many rotations can I find which fix a given corner?

How many rotations can I find which fix a given face?

Can I find a pair of rotations which commute? Can I find a pair whichdoes not?


Combining rotations

Performing one rotation then another defines an operation on the set ofrotations, called a group.

Composing rotations

We can think of this as a kind of multiplication on the set of rotationalsymmetries of a cube. Notice that in this context multiplication a× b doesnot need to be the same as b × a.

The classification of finite (simple) groups was a fundamental problem inalgebra (more or less completed in the 1980s). For example, the rotationalsymmetries of a cube form a famous example (the symmetric group S4).


Combining rotations


Composing rotations




Combining rotations


Composing rotations




An Introduction to Topology

We now change direction (we’ll come back to 3D rotation at the endthough), and look at another important area of modern mathematics:topology.

Topology

Topology is the abstract study of shape, which is an active field ofresearch today. To a topologist shapes are ‘the same’ if they can bedeformed into each other by any ‘smooth’ process (no cutting or tearing),if you open a book these deformations are called ‘homotopies’.




Topology





Topology



Surfaces

One consequence of making any shape you can deform into another one‘the same’ is that you probably haven’t seen that many examples. Howmany topologically different shapes can you think of?

A list of shapes (up to deformation)

point (line?),

circle (square, triangle),

torus (bagel),

sphere,...

The last two can be made by gluing together a rectangle in certain ways.Can we see how? Can we see why we call these two-dimensional, orsurfaces.


Surfaces



point (line?),


torus (bagel),

sphere,...



Surfaces



point (line?),


torus (bagel),

sphere,...



Constructing surfaces

In the next exercise you will construct a number of different surfaces.These constructions involve using certain diagrams describing how to glueedges of simple shapes together.

Gluing diagrams

Sphere from square, disc from triangle.


Constructing surfaces

In the next exercise you will construct a number of different surfaces.These constructions involve using certain diagrams describing how to glueedges of simple shapes together.

Gluing diagrams

Sphere from square, disc from triangle.


Klein bottles and projective planes

The last diagram shows the construction of a Klein bottle.

Like the Mobius strip this has no ‘inner’ and ‘outer’ face (non-orientable).What is the other possible gluing? What shape do we get if we performthe other gluing?










Invariants

One basic question in geometry and topology is how to tell shapes apart.To do this one uses numbers we can attach to a shape which don’t dependon whatever we are allowed to do to our shape which keeps it the ‘same’.

Congruence

You are already familiar with this idea in a different context. If you wantto show two triangles (up to rigid transformations) are ‘the same’ you haverules (SSS,SAS,...) to see this.

Euler number

An analogous notion in topology is the Euler number. We can calculatethis by chopping the surface up into pieces.

Can we compute this in some examples? Is it a complete invariant.


Invariants


Congruence


Euler number




Invariants


Congruence


Euler number




Invariants


Congruence


Euler number




Euler number examples

Below I have two diagrams showing how to cut up a sphere into pieces.

What (regular) shape does the diagram on the left represent? Can wedraw the one on the right? What are the Euler numbers?


Break


Parameter spaces

You might wonder why all the things we’ve discussed is really ‘maths’ - itis quite different from maths you are used to.

Classification in Geometry

Some problems in geometry involve classification (symmetries of a cube),some involve finding invariants (Euler characterstic), there is anothertypical problem: describing a ‘space’ or shape of objects in some class.

Circles

Discuss the parameter space of circles in the plane? (fixing the centre?)


Parameter spaces




Circles



Parameter spaces




Circles



Exercises

I would like to leave you to think about the parameter space of lines in theplane - which you should be familiar with.

Lines in the plane

There are two sheets, covering two different problems

The first is about lines passing through the origin, to introduce you tothe ideas I have in mind.

The second is about all lines in the plane, probably the most algebrayou’ll do in this masterclass.

Have you seen any spherical or hyperbolic geometry? What is the ‘spaceof lines’ in each of those contexts?


Exercises


Lines in the plane






Exercises


Lines in the plane






Dirac’s belt trick

In this last section I want to move onto a slightly more dramaticapplication of topology, this time to Mathematical Physics.

Belt trick

There are two experiments to do, both involve fixing one end of yourribbon securely:

Rotate the free end 360◦ (two half twists). Check that you cannotmake the ribbon untwisted without rotating the unsecured end again.

Rotate the free end 720◦ (four half twists). What happens now?


Dirac’s belt trick

In this last section I want to move onto a slightly more dramaticapplication of topology, this time to Mathematical Physics.

Belt trick

There are two experiments to do, both involve fixing one end of yourribbon securely:

Rotate the free end 360◦ (two half twists). Check that you cannotmake the ribbon untwisted without rotating the unsecured end again.

Rotate the free end 720◦ (four half twists). What happens now?


Spaces of rotations

In fact this is a consequence of a property of the space of rotations in 3D.Just as for the space of lines we can think about how to algebraicallydescribe all rotations and see what shape we find.

Rotations in 2D

We’ve already seen the space of rotations in 2D (if you like you can thinkof these as the rotational symmetries of a circle). What space do theyform?


Spaces of rotations

In fact this is a consequence of a property of the space of rotations in 3D.Just as for the space of lines we can think about how to algebraicallydescribe all rotations and see what shape we find.

Rotations in 2D

We’ve already seen the space of rotations in 2D (if you like you can thinkof these as the rotational symmetries of a circle). What space do theyform?


Rotation in 3D

The final exercise on the sheet will take you through the construction ofthe space of rotations in 3D. We construct the space in stages as follows.

Space of rotations

Identify each point in a ball of radius 180◦ with a rotation as follows.

The point p lies on a line containing the point (0, 0, 0) - this is theaxis of th rotation.

The length from (0, 0, 0) to p is the angle we twist by.

Points on the boundary on the same line represent the same rotation.

After gluing opposite points on the boundary the path along the linebetween them in the ball is a loop.


Rotation in 3D


Space of rotations







Rotation in 3D


Space of rotations







Rotation in 4D?

The space of rotations in 2D is a circle, the space of rotations in 3D spaceis a 3D shape which is nearly a sphere, but has the non-contractible loopswe saw in the belt trick. What about rotations in 4D?

4 dimensional space

Modern Geometry often involves working in spaces of dimension biggerthan three. That is, just define a space with a number of independentco-ordinates (x1, x2, x3, x4). There is a notion of length and angle, just asin 3D.

There is a notion of rotation in 4D too, how can we define this? There is aname for the space of rotations in any dimension, the special orthogonalgroup SO(n), and it is well understood in dimension 4.


Rotation in 4D?


4 dimensional space




Rotation in 4D?


4 dimensional space


There is a notion of rotation in 4D too, how can we define this?

There is aname for the space of rotations in any dimension, the special orthogonalgroup SO(n), and it is well understood in dimension 4.


Rotation in 4D?


4 dimensional space




The End


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