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