Spot the Difference:How to tell when two things are the same

(and when they’re not!)

Raymond Vozzo

School of Mathematical SciencesUndergraduate 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 (A,+,×, R-mult)(non)-example: (R3,×) Lie algebra

R (cts properties) Topological space

Sameness Problems

ProblemHow 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 IdeaChange object but preserve structureWant maps V →W which preserve the vector space (or group,or algebra, or . . .) structure

I e.g. NOT R3 → R2; (x , y , z) 7→ (x2y , sin(z)/√

x)

Examples:Object Maps

Vector spaces linear mapsGroups homomorphismsAlgebra (algebra) homomorphismsTopological spaces continuous functions

Sameness Problems

Equivalence: Find maps both ways (i.e. maps with inverses)

ProblemHow can you tell if 2 objects are equivalent or not?

I A ' B: Find oneI A 6' B: Extract info which doesn’t change under

equivalence and show it’s different for A and B

invariantsExample: (finite-dimensional) vector spaces:

V 'W ⇔ dim V = dim W

Outline

Introduction

Topology

Category Theory

TopologyIdea: Throw away notions of measurement, distances, etc. . .Keep ideas of “closeness” and “connectedness”Example:

2 2

22

2

4

8 8 6

2

4

4

65

41

32

43

3

33 1

1

3

3

59 7 7Special fares apply

5

56

4

4River Thames

A

B

C

D

E

F

1 2 3 4 5 6 7 8 9

1 2 3 4 5 76 8 9

A

B

C

D

E

F

Transport for LondonMAYOR OF LONDON

Transport for LondonThis 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

AmershamChorleywood

Mill Hill East

Rickmansworth

Perivale

KentishTown West

CamdenRoad

Dalston Kingsland

Hackney CentralWanstead Park

Vauxhall

Hanger Lane

Edgware

Burnt Oak

Colindale

Hendon Central

Brent Cross

Golders Green

West Silvertown

Pontoon Dock

London City Airport

Woolwich Arsenal

King George V

Hampstead

Belsize Park

Chalk Farm

Chalfont &Latimer

Chesham

New Cross Gate

Moor Park

NorthwoodNorthwoodHills

Pinner

North Harrow

Custom House for ExCeL

Prince Regent

Royal Albert

Beckton Park

Cyprus

Gallions Reach

Beckton

Watford

Croxley

Fulham Broadway

LambethNorth

HeathrowTerminal 4

Harrow-on-the-Hill

KensalRise

Canonbury

BethnalGreen

Westferry

SevenSisters

Blackwall

BrondesburyPark

HampsteadHeath

HarringayGreen Lanes

LeytonstoneHigh Road

LeytonMidland Road

NorthwickPark

PrestonRoad

RoyalVictoria

WembleyPark

Rayners Lane

Watford High Street

RuislipGardens

South Ruislip

Greenford

Northolt

South Harrow

Sudbury Hill

Sudbury Town

Alperton

Pimlico

Park Royal

North Ealing

Acton Central

South Acton

Ealing Broadway

Watford Junction

West Ruislip

Bushey

Carpenders Park

Hatch End

North Wembley

West Brompton

Ealing Common

South Kenton

Kenton

Wembley Central

Kensal Green

Queen’s Park

Gunnersbury

Kew Gardens

Richmond

Stockwell

Bow Church

Stonebridge Park

Harlesden

Camden Town

Willesden Junction

Headstone Lane

Parsons Green

Putney Bridge

East Putney

Southfields

Wimbledon Park

Wimbledon

Island Gardens

Greenwich

Deptford Bridge

South Quay

Crossharbour

Mudchute

Heron Quays

West India Quay

Elverson Road

Oakwood

Cockfosters

Southgate

Arnos Grove

Bounds Green

Theydon Bois

Epping

Debden

Loughton

Buckhurst Hill

WalthamstowQueen’s Road

Woodgrange Park

Leytonstone

Leyton

Wood Green

Turnpike Lane

Manor House

Stanmore

Canons Park

Queensbury

Kingsbury

High Barnet

Totteridge & Whetstone

Woodside Park

West Finchley

Finchley Central

Woodford

South Woodford

Snaresbrook

Hainault

Fairlop

Barkingside

Newbury Park

East Finchley

Highgate

Archway

Devons Road

Langdon Park

All Saints

Tufnell Park

Kentish Town

Neasden

Dollis Hill

Willesden Green

South Tottenham

Swiss Cottage

ImperialWharf

Brixton

Kilburn

West Hampstead

Blackhorse Road

Acton Town

CanningTown

Finchley Road

Highbury &Islington

Canary Wharf

Stratford

FinsburyPark

Elephant & Castle

Stepney Green

Barking

East HamUpton Park

Plaistow

Poplar

West Ham

Upper Holloway

PuddingMill Lane

Kennington

Borough

Elm ParkDagenhamEast

DagenhamHeathway

Becontree

Upney

Heathrow Terminal 5

Finchley Road& Frognal

Crouch Hill

Northfields

Boston Manor

South Ealing

Osterley

Hounslow Central

Hounslow East

Clapham North

Oval

Clapham Common

Clapham South

Balham

Tooting Bec

Tooting Broadway

Colliers Wood

South Wimbledon

Arsenal

Holloway Road

Caledonian Road

Morden

HounslowWest

Hatton Cross

HeathrowTerminals 1, 2, 3

ClaphamJunction

WestHarrow

BrondesburyCaledonian

Road &Barnsbury

TottenhamHale

WalthamstowCentral

HackneyWick

Homerton

WestActon

Limehouse EastIndia

ChiswickPark

RodingValley

GrangeHill

Chigwell

Redbridge

GantsHill

Wanstead

NorthGreenwich

Ickenham

TurnhamGreen

Uxbridge

Hillingdon Ruislip

GospelOak

Mile End

Bow Road

Bromley-by-Bow

Upminster

Upminster Bridge

Hornchurch

Harrow &Wealdstone

Cutty Sark for Maritime Greenwich

RuislipManor

Eastcote

Wapping

Shadwell

New Cross

CanadaWater

Surrey Quays

Whitechapel

Lewisham

Kilburn Park

Regent’s Park

KilburnHigh Road

EdgwareRoad

SouthHampstead

GoodgeStreet

Shepherd’s BushMarket

Goldhawk Road

Hammersmith

Bayswater

Warren Street

Aldgate

Euston

Farringdon

BarbicanRussellSquare

Kensington(Olympia)

MorningtonCrescent

High StreetKensington

Old Street

St. John’s Wood

Green Park

BakerStreet

NottingHill Gate

Victoria

AldgateEast

Blackfriars

Mansion House

Cannon Street

OxfordCircus

BondStreet

TowerHill

Westminster

TottenhamCourt Road

PiccadillyCircus

CharingCross

Holborn

Tower Gateway

Monument

Moorgate

Leicester Square

London Bridge

St. Paul’s

Hyde Park Corner

Knightsbridge

StamfordBrook

RavenscourtPark

WestKensington

NorthActon

HollandPark

Marylebone

Angel

Queensway MarbleArch

SouthKensington

Earl’sCourt

SloaneSquare

Covent Garden

LiverpoolStreet

GreatPortland

Street

Bank

EastActon

ChanceryLane

LancasterGate

Warwick AvenueMaida Vale

Fenchurch Street

Paddington

BaronsCourt

GloucesterRoad St. James’s

Park Temple

Latimer RoadLadbroke Grove

Royal Oak

Westbourne Park

Bermondsey

Rotherhithe

ShoreditchHigh Street

Dalston Junction

Haggerston

Hoxton

Wood Lane

Shepherd’sBush

WhiteCity

King’s CrossSt. Pancras

EustonSquareEdgware

Road

Waterloo

Southwark

Embankment

Tube map

Key to lines Check before you travelBakerloo

Central

No special arrangements

Circle

ChigwellGrange HillRoding Valley

Served until about 2400

Hammersmith& City

No special arrangements

Canary WharfJubilee Step-free interchange between Underground, Canary Wharf DLR andHeron Quays DLR stations at street level

Metropolitan Chesham Change at Chalfont & Latimer on most trains

Northern

Blackfriars

Cannon Street

Underground station closed until late 2011Open until 2100 Mondays to Fridays.Closed Saturdaysand Sundays

District Blackfriars

Cannon Street

Kensington (Olympia)

Underground station closed until late 2011Open until 2100 Mondays to Fridays.Closed Saturdaysand Sundays

Camden Town

Charing Cross branch

Mill Hill East

Open for interchange and exit only from 1300 until 1730 Saturdays and SundaysChange at Kennington at off-peak times if travelling towards or from MordenChange at Finchley Central at off-peaktimes

Piccadilly Covent Garden

Eastcote to Uxbridge

Heathrow Terminal 4

Hounslow West

Turnham Green

A short walk from either Leicester Square (6 minutes) or Holborn (9 minutes)Not served by Piccadilly line trains early morningsOpen until 2400 Mondays to Saturdays and until 2330 Sundays. Trains may wait for eight minutes before continuing to Terminals 1,2,3Step-free access for wheelchair users onlyServed by Piccadilly line trains early mornings and late evenings only

No special arrangements

Heron Quays

West India Quay

Overground

DLR

Victoria

Waterloo & City Bank to Waterloo

Step-free interchange between Heron Quays and Canary Wharf Underground station at street level.Not served by DLR trainsfrom Bank towards Lewisham at peak times

Open 0615 until 2148 Mondays to Fridays and 0800 until 1830 Saturdays. Closed Sundays and public holidays

Dalston Junction to New Cross / New Cross Gate

Open 0700 until 2000Mondays to Fridays only.Closed Saturdays andSundays

Served 0700 until 2345 Mondays to Saturdays and 0800 until 2345 Sundays

Topology

Equivalency condition: Continuous deformations(homeomorphisms)

I i.e. X f−→ Y and Y f−1−−→ X continuous

Example: R→ R; x 7→ x3 vs R→ R; x 7→

{1/x , x 6= 00, x = 0

ManifoldsManifolds are important examples of topological spacesRoughly: A manifold is a topological space which locally lookslike (i.e. is homeomorphic to) Rn

Examples:1D S1

2D S2

T2

Other surfaces3D S3 = {(x , y , z, w) ∈ R4 | x2 +y2 +z2 +w2 = 1}4D The Universe!10D The Universe?

Algebraic Topology

QuestionHow can you tell if 2 manifolds are homeomorphic?Answer: Look for invariants!Example Question: S2 ?' T2

Note: It’s not necessarily obvious. Consider?'

The Fundamental Group

Consider loops:{γ : [0, 1]→ X | γ(0) = γ(1) = x0}

X

x0

!

γ1 ∼ γ2 ⇐⇒ γ1cts

deformn///o/o/o/o/o/o/o γ2

X

x0

γ1

!2e.g. γ 6∼ γ′

!!

!

The Fundamental Group

Define:I γ1 ∗ γ2 = γ1 then γ2

!1

!2

I γ−1 = γ backwards!

!!1

=⇒ It’s a group!

The Fundamental Group, π1(X )

Important Property: X ' Y =⇒ π1(X ) ' π1(Y )

The Fundamental GroupBack to question: S2 ?' T2 is π1(S2)

?' π1(T2)π1(S2) :

!" !" !"

x0

=⇒ π1(S2) ' {0} (i.e. S2 is simply connected)π1(T2) :

&

=⇒ π1(T2) ' Z⊕ Z

π1(S2) 6' π1(T2) =⇒ S2 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 ' S3

More about π1

Important Property: f : X → Y induces f∗ : π1(X )→ π1(Y )

[0, 1]

!"

!!!!!!!!!!!!!!!"

!"

f

f !!

Y

y0

f !!

!

X

x0

!

I π1 is an example of a functor

Outline

Introduction

Topology

Category Theory

Category Theory

I Formalism for talking about structures and maps whichpreserve them

I A category consists ofI ObjectsI Arrows between them

Examples:Top Topological spaces and continuous mapsGrp Groups and group homomorphismsVect Vector spaces and linear mapsCat Category of Categories!

I Invariants for topological spaces←→ Functors Top→ Grp

More functors: πn, Hn