Spot the Difference:
How to tell when two things are the same
(and when they’re not!)
Raymond Vozzo
School of Mathematical Sciences
Undergraduate Seminar
May 2010
Outline
Introduction
Topology
Category Theory
Outline
Introduction
Topology
Category Theory
Examples of Mathematical Structures
Example
General concept
Rn
Vector space
(V , +, R-mult)
Symmetries
Group
(G, ·)
Functions on R
Algebra
(non)-example:
(A, +, ×, R-mult)
(R3 , ×)
Lie algebra
R (cts properties)
Topological space
Sameness Problems
Problem
How to come up with a natural notion of “sameness” for these
(& other) structures?
I
“=” is usually too strict
e.g. {x-y plane} ⊆ R3
R2
{x-y } =
6 R2
but {x-y } ' R2
Similarly: any 2D subspace of any Rn ' R2
Sameness Problems
Key Idea
Change object but preserve structure
Want maps V → W which preserve the vector space (or group,
or algebra, or . . .) structure
√
I e.g. NOT R3 → R2 ;
(x, y , z) 7→ (x 2 y , sin(z)/ x)
Examples:
Object
Vector spaces
Groups
Algebra
Topological spaces
Maps
linear maps
homomorphisms
(algebra) homomorphisms
continuous functions
Sameness Problems
Equivalence: Find maps both ways (i.e. maps with inverses)
Problem
How can you tell if 2 objects are equivalent or not?
I
A ' B: Find one
I
A 6' B: Extract info which doesn’t change under
equivalence and show it’s different for A and B
invariants
Example: (finite-dimensional) vector spaces:
V ' W ⇔ dim V = dim W
Outline
Introduction
Topology
Category Theory
Topology
Idea: Throw away notions of measurement, distances, etc. . .
Keep ideas of “closeness” and “connectedness”
Tube map
Example:
1
9
Chesham
Chalfont &
Latimer
Amersham
2
8
Watford
A
Rickmansworth
Uxbridge
South Ruislip
B
Harrowon-the-Hill
West
Harrow
South Harrow
Edgware
Stanmore
Harrow &
Wealdstone
Northwick
Park
Kenton
Queensbury
Preston
Road
Kingsbury
Wembley
Park
Wembley Central
Sudbury Hill
Stonebridge Park
Willesden Junction
Kensal
Rise
Queen’s Park
Kilburn Park
Maida Vale
Warwick Avenue
East
Acton
West
Acton
3
North
Acton
Acton Central
Ealing Common
South Acton
D
Acton Town
South Ealing
Chiswick
Park
Shepherd’s Bush
Market
2
Turnham Stamford Ravenscourt
Brook
Park
Green
West
Kensington
Bond
Street
Gloucester
Road
Earl’s
Court
Victoria
Hounslow East
Kew Gardens
Hounslow Central
Hatton Cross
Richmond
4
Heathrow
5
3
2
Fulham Broadway
Parsons Green
Cannon Street
Aldgate
East
Tower
Hill
Blackfriars
Stepney Green
Whitechapel
Vauxhall
Southfields
Rotherhithe
London
Bridge
Bermondsey
Mudchute
Clapham North
Bromleyby-Bow
New Cross
Blackwall
Poplar
Kennington
3
4
D
Royal Albert
West Silvertown
Pontoon Dock
Cyprus
F
Tooting Broadway
Colliers Wood
South Wimbledon
Morden
Transport for London
London City Airport
2
2
3
4
3
Woolwich Arsenal
E
Victoria
Greenwich
Waterloo & City
4
Elverson Road
Lewisham
Overground
3
F
DLR
This diagram is an evolution of the original design conceived in 1931 by Harry Beck · 04.10
Correct at time of going to print, April 2010
5
6
7
A short walk from
either Leicester Squ
(6 minutes) or Holbo
(9 minutes)
Eastcote to
Uxbridge
Not served by
Piccadilly line trains
early mornings
Heathrow
Terminal 4
Open until 2400
Mondays to Saturda
and until 2330 Sund
Trains may wait for
eight minutes befor
continuing to
Terminals 1,2,3
Hounslow West
Step-free access for
wheelchair users on
Turnham Green
Served by Piccadilly
trains early morning
and late evenings o
Beckton
King George V
Brixton
4
Covent Garden
Gallions Reach
Balham
Tooting Bec
Change at Chalfont
Latimer on most tra
Beckton Park
North
Greenwich
8
9
Transport for London
Step-free interchan
between Undergrou
Canary Wharf DLR a
Heron Quays DLR
stations at street le
Open for interchang
and exit only from
1300 until 1730
Saturdays and Sund
Change at Kenningt
at off-peak times
if travelling towards
or from Morden
Change at Finchley
Central at off-peak
times
Mill Hill East
Piccadilly
Served 0700 until 23
Mondays to Saturda
and 0800 until 2345
Sundays
Camden Town
Custom House for ExCeL
Prince Regent
Clapham Common
Clapham South
Northern
Underground statio
closed until late 201
Open until 2100
Mondays to Fridays.
Closed Saturdays
and Sundays
Chesham
Charing Cross
branch
Royal
Victoria
East
India
Deptford Bridge
Stockwell
C
West Ham
Canning
Town
Cutty Sark for
Maritime Greenwich
New Cross Gate
East Ham
Langdon Park
All Saints
Canary Wharf
Becontree
Metropolitan
Served until
about 2400
Underground statio
closed until late 201
Open until 2100
Mondays to Fridays.
Closed Saturdays
and Sundays
No special arrangements
Plaistow
Island Gardens
Borough
Elephant & Castle
Jubilee
Dagenham
Heathway
Wanstead Park
Devons Road
Heron Quays
Crossharbour
Hammersmith
& City
Elm Park
Barking
West
India Quay
South Quay
Lambeth
North
Hornchurch
Dagenham
East
Upton Park
Kensington
(Olympia)
B
Upminster Bridge
Gants
Hill
Upney
Canary Wharf
Canada
Water
Southwark
Oval
Clapham
Junction
Wimbledon Park
2
Westferry
Limehouse
Wapping
River Thames
Temple
Embankment
2
Blackfriars
Cannon Street
Upminster
Woodgrange Park
Bow Road
Shadwell
Tower
Gateway
District
Fairlop
Mile End
Shoreditch
High Street
Aldgate
Fenchurch Street
Blackfriars
Cannon Street
Leytonstone
High Road
Stratford
Bow Church
Monument
A
Hainault
Hackney
Wick
Pudding
Mill Lane
Bethnal
Green
1
Bank
Mansion House
Charing
Cross
2
Homerton
Moorgate
St. Paul’s
Leicester Square
1
Pimlico
Imperial
Wharf
East Putney
Wimbledon
1
Barbican
Chigwell
Grange Hill
Roding Valley
Barkingside
Wanstead
Leyton
Hackney Central
Surrey Quays
6
MAYOR OF LONDON
Hoxton
Liverpool
Street
Chancery
Lane
Holborn
No special arrangements
Central
Circle
Leytonstone
Leyton
Midland Road
Dalston Kingsland
Haggerston
Covent Garden
Check before you travel
Bakerloo
Newbury Park
3
Dalston Junction
Farringdon
Russell
Square
5
Redbridge
Walthamstow
Queen’s Road
Waterloo
Putney Bridge
Snaresbrook
Walthamstow
Central
Canonbury
Old Street
Euston
Square
Westminster
West Brompton
E Terminals 1, 2, 3
Caledonian
Road &
Barnsbury
Piccadilly
Circus
Sloane
Square
South
Kensington
Finsbury
Park
Highbury &
Islington
King’s Cross
St. Pancras
Goodge
Street
St. James’s
Park
South Woodford
Blackhorse
Road
Holloway Road
Euston
Tottenham
Court Road
1
Woodford
South Tottenham
Tottenham
Hale
Caledonian Road
Mornington
Crescent
Green Park
Hyde Park Corner
Seven
Sisters
Angel
Oxford
Circus
Knightsbridge
Harringay
Green Lanes
Manor House
Arsenal
Grange
Hill
4
Wood Green
Turnpike Lane
Camden
Road
Camden Town
Warren Street
Marble
Arch
Key to lines
Chigwell
Upper Holloway
Kentish Town
Chalk Farm
2
Roding
Valley
Crouch Hill
Tufnell Park
Kentish
Town West
Belsize Park
Great
Portland
Street
Baker
Street
Gospel
Oak
Buckhurst Hill
Arnos Grove
Bounds Green
Archway
Regent’s Park
Lancaster
Gate
Queensway
High Street
Kensington
Barons
Court
Hammersmith
Gunnersbury
Heathrow Terminal 5
Notting
Hill Gate
Holland
Park
Kensington
(Olympia)
Goldhawk Road
Highgate
Hampstead
Heath
Bayswater
Shepherd’s
Bush
White
City
Wood Lane
Osterley
Heathrow
Terminal 4
St. John’s Wood
Edgware
Road
Latimer Road
North Ealing
Hounslow
West
Swiss Cottage
South
Hampstead
Edgware
Marylebone
Road
Ladbroke Grove
Ealing Broadway
Northfields
Kilburn
High Road
Paddington
Westbourne Park
Park Royal
Boston Manor
Finchley Road
Royal Oak
Hanger Lane
5 4
West Hampstead
Brondesbury
Kensal Green
Alperton
Greenford
Perivale
Finchley Road
& Frognal
Kilburn
Brondesbury
Park
Southgate
West Finchley
Finchley Central
Hampstead
Dollis Hill
Willesden Green
Harlesden
9
6
Debden
Loughton
Oakwood
East Finchley
Golders Green
Sudbury Town
Cockfosters
Woodside Park
Mill Hill East
3
Brent Cross
Neasden
8
Epping
Colindale
Hendon Central
South Kenton
North Wembley
Northolt
C
Burnt Oak
Canons Park
North Harrow
Rayners Lane
7
Theydon Bois
4
Hatch End
Headstone Lane
Pinner
Eastcote
Ruislip
Gardens
6
High Barnet
Totteridge & Whetstone
Carpenders Park
Ruislip
Manor
Ickenham
5
5
Bushey
Northwood
Northwood
Hills
Ruislip
4
8 7 6
Watford Junction
Watford High Street
Moor Park
West Ruislip
Hillingdon
3
Special fares apply
7
Croxley
Chorleywood
No special arrangements
Bank to Waterloo Open 0615 until 2148
Mondays to Fridays
and 0800 until
1830 Saturdays.
Closed Sundays and
public holidays
Dalston Junction Open 0700 until 200
to New Cross / Mondays to Fridays
New Cross Gate Closed Saturdays an
Sundays
Heron Quays
Step-free interchan
between Heron Qua
and Canary Wharf
Underground statio
street level.
West India Quay Not served by DLR t
from Bank towards
Lewisham at peak ti
Topology
Equivalency condition: Continuous deformations
(homeomorphisms)
I
f
f −1
i.e. X −
→ Y and Y −−→ X continuous
Example: R → R; x 7→
x3
vs
(
1/x, x 6= 0
R → R; x →
7
0,
x =0
Manifolds
Manifolds are important examples of topological spaces
Roughly: A manifold is a topological space which locally looks
like (i.e. is homeomorphic to) Rn
Examples:
1D
S1
2D
S2
T2
3D
4D
10D
Other surfaces
S 3 = {(x, y , z, w) ∈ R4 | x 2 + y 2 + z 2 + w 2 = 1}
The Universe!
The Universe?
Algebraic Topology
Question
How can you tell if 2 manifolds are homeomorphic?
Answer: Look for invariants!
?
Example Question: S 2 ' T2
Note: It’s not necessarily obvious. Consider
?
'
The Fundamental Group
Consider loops:
X
{γ : [0, 1] → X | γ(0) = γ(1) = x0 }
γ
x0
cts
γ1 ∼ γ2 ⇐⇒ γ1 /o /o /o /o /o n /o o/ / γ2
deform
e.g. γ 6∼ γ 0
X
γ1
x0
γ
γ!
γ2
The Fundamental Group
γ1
Define:
I
γ1 ∗ γ2 = γ1 then γ2
I
γ −1
γ2
γ
= γ backwards
γ
−1
=⇒ It’s a group!
The Fundamental Group, π1 (X )
Important Property: X ' Y =⇒ π1 (X ) ' π1 (Y )
The Fundamental Group
?
Back to question: S 2 ' T2
π1 (S 2 ) :
−→
?
is π1 (S 2 ) ' π1 (T2 )
−→
−→
x0
=⇒ π1 (S 2 ) ' {0}
(i.e. S 2 is simply connected)
π1 (T2 ) :
&
=⇒ π1 (T2 ) ' Z ⊕ Z
π1 (S 2 ) 6' π1 (T2 ) =⇒ S 2 6' T2
Classification of Surfaces
In general π1 doesn’t completely characterise X
I
2D: it does
−→
−→
Interesting Question: What about higher dimensions?
?
e.g. π1 (X 3 ) = 0 =⇒ X 3 ' S 3
−→
...
More about π1
Important Property: f : X → Y induces f∗ : π1 (X ) → π1 (Y )
[0, 1]
−→
γ
−−−
−−− f ◦γ
−−−
−−−
−−−
→
X
γ
x0
Y
f
−→
f ◦γ
y0
I
π1 is an example of a functor
Outline
Introduction
Topology
Category Theory
Category Theory
I
I
Formalism for talking about structures and maps which
preserve them
A category consists of
I
I
Objects
Arrows between them
Examples:
Top
Grp
Vect
Cat
I
Topological spaces and continuous maps
Groups and group homomorphisms
Vector spaces and linear maps
Category of Categories!
Invariants for topological spaces ←→ Functors Top → Grp
More functors: πn , Hn