Lec 17 | Abstract Algebra

00:46:01
https://www.youtube.com/watch?v=tx2yCz8MEvU

Resumo

TLDREl vídeo explora la classificació dels grups de simetria en matemàtiques, enfocant-se en les accions de grups i les seves implicacions en les estructures geomètriques. S'analitza com un operador gamma bar ha de preservar les característiques d'un subgrup i com les rotacions i reflexions influeixen en aquestes estructures. A més, es discuteix l'ús de les matrius relacionades amb angles de rotació i detallades equacions que determinen les accions de rotació. Es tracta la importància de l'acció transitiva i la classificació de xarxes de punts, així com la seva connexió amb la teoria de nombres, destacant el paper fonamental d'aquest tema en la matemàtica moderna.

Conclusões

  • 🔍 Gamma bar és crucial per preservar subgrups.
  • 📏 Les rotacions i reflexions tenen efectes significatius.
  • 🔗 Les accions transitives connecten diversos punts en un conjunt.
  • 🧮 L'equació característica determina les propietats de rotació.
  • 📊 La classificació de xarxes és fonamental en geometria.

Linha do tempo

  • 00:00:00 - 00:05:00

    La discussió tracta sobre les possibilitats dels subgrups generats per gamma bar, discutint les implicacions de preservar la distància i l'angle de rotació. Es posa èmfasi en les condicions necessàries per a l'existència d'un subgrup evident, ja que ha de ser simètric respecte a l'origen i la seva distància. Es conclou que la única rotació possible és un comptador o rotacions d'angle 0 o pi.

  • 00:05:00 - 00:10:00

    S'analitzen les rotacions dins d'un grup i el seu impacte en les possibilitats de gamma bar, ressaltant la rellevància de la polinòmia característica d'ordres imposats per la rotació i subgrups. Això condueix a la classificació dels grups possibles segons la seva estructura i informació, revelant similituds amb estructures conegudes.

  • 00:10:00 - 00:15:00

    Es destaca que qualsevol relació d'angles relacionats amb rotacions ha de tenir en compte els valors de 2 cosinus theta, limitant així les possibilitats a valors enteros compresos entre -2 i 2. Aquesta restricció ajuda a establir el comportament del grup i la seva relació amb la geometria.

  • 00:15:00 - 00:20:00

    Es fa referència a la conservació de les xarxes en el context de gamma, discutint el seu impacte en la classificació dels grups i subgrups, i la proporció de vectors amb matrius respectives. Això configura una visió general de les possibles estructures que es poden observar en els diferents espais.

  • 00:20:00 - 00:25:00

    El document continua explorant les accions de grup sobre espais, parlant sobre la necessitat d'escalar i rotar els vectors per obtenir les representacions apropiades. Estableix que les mides dels vectors i les seves direccions són crucials per les interaccions dins de les xarxes; així s'assegura la distinció entre els diferents grups.

  • 00:25:00 - 00:30:00

    A mesura que la discussió avança, s'introdueix la noció de traducció i reflexió com a operacions essencials per mantenir els propòsits de grup i la seva congruència, amb l'objectiu d'una correcta classificació del subgrup de vectors. Aquesta reflexió és vital per establir les similituds que avancen el concepte de xarxes espontànies.

  • 00:30:00 - 00:35:00

    S'observa la classificació dels grups d'acord amb les propietats de latituds complexes, amb un èmfasi en les relacions de simetria que es poden derivar d'aquestes accions. L'objectiu aquí és proporcionar una base per investigar les estructures i les seves possibles formes d'interacció en dimensions més altes.

  • 00:35:00 - 00:40:00

    El professor es referència a les similituds amb patrons de paper pintat en art, conectant els conceptes matemàtics amb aplicacions pràctiques en la vida quotidiana i l'art que sorgeixen de la comprensió de patrons grupals. Aquesta analogia il·lustra la rellevància del tema dins del context més ampli de les matemàtiques i la seva interacció amb altres disciplines.

  • 00:40:00 - 00:46:01

    Finalment, es conclou que l'estudi de l'acció de grup i la seva classificació en espais de latitud és fonamental per a la comprensió dels moviments en R2 i altres dimensions, indicando que la matemàtica moderna busca confrontar amb objectes complexos a partir de les seves bases essencials.

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Mapa mental

Vídeo de perguntas e respostas

  • Què és gamma bar i com es relaciona amb els subgrups?

    Gamma bar és un operador que ha de preservar les característiques d'un subgrup específic en el context de la simetria.

  • Com s'analitzen les rotacions i reflexions en el vídeo?

    S'analitzen mitjançant l'equació característica associada i la seva influència en les configuracions geomètriques.

  • Què implica l'acció transitiva en un grup?

    Una acció transitiva significa que cada punt del conjunt pot ser mapejat a qualsevol altre punt mitjançant una acció del grup.

  • Quina és la relació entre el vídeo i la teoria de nombres?

    El vídeo explora com les accions de grups estan íntimament relacionades amb la teoria de nombres, mostrant la seva importància en les matemàtiques modernes.

  • Quina importància té la classificació de xarxes en matemàtiques?

    La classificació de xarxes permet comprendre les estructures geomètriques i les propietats de simetria de manera més profunda.

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Legendas
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Rolagem automática:
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    would be into one reflection what could
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    be running water
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    the client or that's it possibilities
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    for gamma bar anyway yesterday before
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    the prove is easy think the line
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    generated by a understanding here's to a
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    as minus a 0
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    now this subgroup has to be preserved by
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    gamma bar so if I a family memoir when I
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    applied hours a day what is the
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    possibility can okay well it not to be
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    an evident than a subgroup that's the
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    first thing
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    so for multiple vane and the second
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    thing is and I thought the same distance
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    from the origin as any because gamma is
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    an element in Y orthogonal group right
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    Reserve's distance now there are only
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    two things are the same distance from
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    the origin remain here and those are a
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    minus a so this has to be plus or minus
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    okay well that's it about us because if
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    it were a rotation
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    so to it then the identity so the only
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    rotations in the group have water one or
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    two because once you in relation can
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    already take this element that tells you
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    the angular rotation t 0 or pi and once
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    I know what rotations are the group that
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    restricts what run die so that says the
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    only cyclic groups I could have a guess
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    the only possible value for groups I
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    could have is this is why I appeared I
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    need to provide out rotation a different
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    angle and in fact you can get the
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    psychical group because we can also pick
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    a minus ed by taking the reflection
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    around this particular one right and
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    that commutes with the rotation through
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    for 180 degrees so the group leader just
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    done it could be the rotation that we
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    see - it could be the reflection around
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    this line would you even 52 or it could
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    be the Kline word remote from Haven
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    reflect those possibilities thinking
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    might say ok so once I know the lattices
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    of this one the sub 1 guess I'm joking L
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    plus B then the LT possibilities for
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    gamma bar
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    FC n to n where hands
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    one two three four four six in the
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    largest group people have got water 12
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    water forward and I gave you up what
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    proof of that based on what had a
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    polynomial satisfied by the way my group
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    last time I was saying now log is in
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    down the bar in addition so again the
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    proved essential you have to show 22
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    stations in town I have order one two
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    three four six because I have a
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    different value throw group you will
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    have locations of order n right D 2 n
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    has a location water and so if I have a
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    location I'm just have to classify that
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    and I look at the characteristic
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    polynomial or gamma it looks like x
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    squared well I'll write it like this
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    that doesn't make me even go on is 2
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    cosine theta X plus 1 so there's the
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    characteristic polynomial where theta is
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    the angle of rotation so this number
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    whatever it is is an absolute value less
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    than or equal to 2 because cosine theta
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    is
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    - cosine theta less than two on the
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    other hand since gala preserves this
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    last that we can write down a matrix of
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    gamma with respect to the bases a and B
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    of this lattice and the 2x2 matrix
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    analysis issues what the matrix a of
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    gamma with respect to the basis maybe it
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    is integral doesn't just have real-life
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    things that has integer entries because
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    it takes a will end your own mold a plus
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    in your movie and easy to write
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    therefore it's trace is a ganger so
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    integral Chris so therefore whatever
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    this angle is two cosine theta is some
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    numbers between -2 and 2 which is an
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    integer there are only five integers
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    between minus 2 2 minus 2 minus 1 0 1 2
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    and once I know what 2 cosine theta is
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    cosine theta 2 minus 3 minus 1 0 1 or 2
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    and that's definitely what cosine 8 is
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    minus 1 minus 1/2 0 1 half and 1 I know
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    that I know it's angle is and you find
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    exactly these financials so that's the
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    proof that you know yet those now you
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    might want to know why do you get one
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    root when you get the other so that's
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    how I'm going to finish today
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    if you want to you can finish reading
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    all about the wallpaper patterns in art
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    not a whole lot on that because when you
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    think course in chemistry of the walls
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    wallpaper patterns will also three
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    dimensions which is your mortgage so if
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    your general places what is the
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    classification lattices well the
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    classification have seen you might just
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    ask for all subgroups of r2 but if
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    you're trying to figure out what groups
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    can act on them to determine what
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    Calabar and see what what what work is
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    that you can have if I change L and I
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    act on our people I know that doesn't
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    change the possibilities for gamma bar
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    just conjugate down bar on going so then
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    you l - L won't you just years
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    down the bar I'm not I'm working Vala
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    right now so you got a similar subgroup
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    of Oh two similar finance on who
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    provoked to Kate by no your goggles and
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    also changing L to a multiple level
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    let's say a see time detail is in our
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    star doesn't change that and you just
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    take anything if you go a lot of AD and
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    you're scaling a to be out by the same
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    fish scaler CA CB that anytime you have
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    a rotation to preserve talent abruptly
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    we preserve this because rotations
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    commute with multiplication by scaling
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    matrices so this just means take
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    everything in elements here now so I
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    really want to just classify lattices up
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    the action on the space of lattices by
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    Helena Senora gotta mental and scalars
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    I'll tell you all asses up to the
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    audience of the orthogonal group and
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    scalars as the famous mathematical
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    picture and then for each labs I'll tell
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    you what the possibilities are
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    now be the end of this so l up to action
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    okay and tartar well we know that the
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    lightest as a short is better right
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    because get their vectors is the last
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    part journey home to the origin this
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    isn't a lot as the vectors we found away
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    from each other okay
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    used by action of scaling on black to
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    make the shortest vector have length 1
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    right that I can do yeah so the shorter
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    Spector
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    these arms are also sort inspector has
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    length one and then then I'm really only
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    need positive things with that but
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    because you only have to scale by
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    positive you get me one and then using
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    hello you rotate Hey
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    so that because does that wait one is
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    actually equal to the vector one so by
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    scaling my lattice and by rotating my
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    shortest vector I can assume that the
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    shortest vector arises Patridge the
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    number one okay everybody agree the
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    first thing I do is a scale that in
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    short inspector on the circle and then I
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    have unique put that left or even circle
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    and I have unique rotation so that the
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    shortest vector has length one
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    I still have a reflection left and I
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    still have a negative left for work okay
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    now here is the second vector on which
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    lacks well first of all the second
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    vector has to be outside of this circle
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    bless the vector B
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    the first thing we know the absolute
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    value will be has to be at least one
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    okay and so it lies outside the circle
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    and we can also replace V by minus BD
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    which gives the same subgroup za plus ZB
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    is the same thing and make sure the
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    imaginary part is positive we know it's
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    imaginary part is nonzero
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    that's our it's been second coordinate
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    and y coordinate is nonzero because in
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    the y coordinate was hero would be
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    linearly dependent with a but we know a
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    B are linearly independent
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    so whatever B is he going to love here
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    or below here but it's not on this line
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    so I'm part of placing it by minus B we
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    can assume that the y coordinate is
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    positive so it's in this it's outside of
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    the circle up here somewhere and finally
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    we got the same subgroup so this doesn't
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    in fact are really interested in what
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    this subgroup is we can also replace B
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    by H multiple leg and we'll still get
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    the same subgroup so replace B by a
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    multiple day to shift it back so that
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    gets x coordinate lies between 1/2 and
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    minus am
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    x-coordinate is between 1/2 and minus
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    1/2 is just a convention we could make
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    it between 0 and 1 we could make it
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    between 3 & 4 we can get an interval
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    length one right by shifting by
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    multiples overhead so I did it between a
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    half of my setup which puts the in the
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    following region it's the life time you
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    got out - ahem then outside the circle
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    and then the line going up and a half
  • 00:12:51
    like that okay so B is somewhere in this
  • 00:12:56
    region and gives us the same lies that
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    we're interested in up through these
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    locations and now it's no I really
  • 00:13:03
    flexion that we could use so we can
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    still reflect the lives around this line
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    and that would take am I to say which we
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    could replace it to and it would take me
  • 00:13:12
    to the other change this on the other
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    side of this region so really the
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    classification of lattices is in this
  • 00:13:18
    half of this region so the B's that live
  • 00:13:23
    in the region classify the lattices that
  • 00:13:26
    are possible up to the action in your
  • 00:13:28
    boggle group and scale so this would be
  • 00:13:33
    like a movie and reflected in a four
  • 00:13:36
    percent reflection a we get minus a
  • 00:13:39
    replaced by a cool one but this would
  • 00:13:41
    look like the same class so the
  • 00:13:44
    classification of lattices is really
  • 00:13:46
    just the points in this bizarre region
  • 00:13:48
    of space so he'll decison work it or ll
  • 00:13:55
    be or be in the one where L of B is Z
  • 00:14:04
    plus Z B because we're using the vector
  • 00:14:06
    agent for fabric yes how do you know
  • 00:14:12
    that when you add subtract multiples of
  • 00:14:14
    a to get these exporters which we gave a
  • 00:14:16
    half and a half then B still has trailer
  • 00:14:20
    ah very good they have to translate it
  • 00:14:24
    back up
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    very good point maybe I could have
  • 00:14:26
    started out here and then I could have
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    translated it back in here and then I
  • 00:14:31
    have to replace it by the vector on the
  • 00:14:33
    other side of it you're absolutely right
  • 00:14:35
    there's a so important in the
  • 00:14:38
    translation of such elements you might
  • 00:14:40
    end up inside at the circle in which
  • 00:14:43
    case you're not going to replace the
  • 00:14:44
    lattice by the last where you still
  • 00:14:46
    again took that vector 2 1 and solver in
  • 00:14:49
    vector a 1 but I'm asking you to believe
  • 00:14:52
    that this process eventually lands you
  • 00:14:55
    in this region is not completely obvious
  • 00:14:56
    thank you that was the subtlety I was
  • 00:14:59
    coming
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    that's why is there element
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    because in the translation you might
  • 00:15:07
    have come to a short effect ah that's a
  • 00:15:10
    good point
  • 00:15:10
    very good point maybe if we started with
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    a Becker in this region that if that's
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    the answer to question budget lately we
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    cannot buy any translation none of these
  • 00:15:24
    factors are legally inside the circle
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    because our assumption was that the in
  • 00:15:29
    order to settle a stop so the game is
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    the short effect therefore matter what
  • 00:15:34
    vector we replace T by it's the last I
  • 00:15:36
    have somebody figures in one so that if
  • 00:15:39
    we in starting off with the last out
  • 00:15:40
    here it went in a shorter vector than 18
  • 00:15:44
    English translation especially the
  • 00:15:47
    penguin we would have been with the
  • 00:15:48
    incorrect recipe thank you but then by
  • 00:15:51
    the same reasoning I'm just saying that
  • 00:15:53
    other types more to little region
  • 00:15:56
    honey's then prefer you can certainly
  • 00:16:01
    choose to be in your region such that
  • 00:16:05
    you get a problem you be in this region
  • 00:16:09
    problem ah the all these beer what
  • 00:16:13
    lengths water the end and any translated
  • 00:16:17
    them has meant by arm are you
  • 00:16:21
    considering be like a rotation no no be
  • 00:16:25
    is a second okay so if the lattice would
  • 00:16:28
    be generated by one and B and I claim
  • 00:16:32
    the bow sizes are in equivalent of your
  • 00:16:34
    pocket group and count the zl1 to none
  • 00:16:37
    okay now there are two very special
  • 00:16:40
    points of this region
  • 00:16:43
    maybe this point here and this point
  • 00:16:46
    here sort of boundary points in the
  • 00:16:49
    region and if you can't be to be this
  • 00:16:52
    point here then you've got a very famous
  • 00:16:55
    letter
  • 00:16:55
    it looks like this that one and drove an
  • 00:16:59
    angle of 60 degrees but you'll figure
  • 00:17:01
    out the value list 60 degrees for that
  • 00:17:03
    big meets and then you have these
  • 00:17:05
    vectors and there are six factors around
  • 00:17:07
    the origin that have the same way the
  • 00:17:09
    same 6.00 this is the famous exact
  • 00:17:17
    analyze if you use this guy's the past
  • 00:17:18
    years like putting a sphere in each set
  • 00:17:22
    that were putting a circle in each
  • 00:17:24
    lattice point
  • 00:17:25
    a radius 1/2 sort of the spheres just
  • 00:17:28
    punch each other one is can get the best
  • 00:17:33
    packing of two-dimensional space if you
  • 00:17:34
    can arrange with its peers covers the
  • 00:17:37
    most of two-dimensional space but there
  • 00:17:42
    are six peers six then I put seven lines
  • 00:17:47
    here problem thank you
  • 00:17:52
    there are six years around the origins
  • 00:17:55
    here and it covers about 75% of states
  • 00:17:59
    with circles and this is the most
  • 00:18:02
    efficient way that would Sears a regular
  • 00:18:04
    way into to space and it's the way that
  • 00:18:06
    was discovered by the bees long before
  • 00:18:09
    we would work so you have a nickel
  • 00:18:11
    honeycomb and you like to make circular
  • 00:18:13
    objects and you want to let the waste
  • 00:18:15
    any space in your heart this is what you
  • 00:18:17
    need okay
  • 00:18:19
    the last the nice basis vector here is
  • 00:18:23
    the same as rectangular lattice so they
  • 00:18:25
    are four vectors of length one and
  • 00:18:28
    orthogonal to each other and that's the
  • 00:18:31
    maximum spacing you know we have this
  • 00:18:34
    danger vise
  • 00:18:36
    that's a nice vacuum but it's not
  • 00:18:38
    sufficient is that exactly and the
  • 00:18:41
    mountains and the L boxes that
  • 00:18:43
    correspond to vectors along this little
  • 00:18:45
    word line are the lysis where you have
  • 00:18:47
    more than two factors of the same length
  • 00:18:49
    but not in Nice angles like this here's
  • 00:18:53
    my memories now the general axe is
  • 00:18:58
    inside of David then the only
  • 00:19:01
    possibility for Delabar is like we did
  • 00:19:03
    in the gum inside a ring actually in
  • 00:19:16
    sake but not on this boundary point not
  • 00:19:19
    on this boundary Circle not on this
  • 00:19:21
    boundary point then only possibility or
  • 00:19:29
    gamma bar is one or
  • 00:19:37
    yes correct in situ just means minus the
  • 00:19:41
    identity minus the identity that
  • 00:19:43
    preserves any - because it takes a - - a
  • 00:19:46
    video - but there are no rotations that
  • 00:19:49
    preserve the map because these any
  • 00:19:51
    location without if a - a vector of the
  • 00:19:53
    same way as a and so this group is
  • 00:19:57
    really the group I sure hope sorry yeah
  • 00:20:02
    the only way the only vector of the same
  • 00:20:05
    thing today is my name and the
  • 00:20:08
    reflection doesn't work because they see
  • 00:20:10
    out of this region son see if these were
  • 00:20:16
    bottom of it so your lattice looks like
  • 00:20:19
    this so that would be a vector for this
  • 00:20:25
    one then to get out of R equal to 1 C -
  • 00:20:35
    but you can also have the reflection
  • 00:20:38
    around this line with it e to minus B
  • 00:20:40
    and likewise around
  • 00:20:42
    within a 2 minus a but you can also have
  • 00:20:45
    the group which contains those two
  • 00:20:50
    reflections the two orthogonal
  • 00:20:51
    directions okay or it could be a
  • 00:20:57
    separate would be work would be one of
  • 00:20:58
    the reflections and none have that
  • 00:21:00
    rotation and the all those glasses
  • 00:21:07
    Nestor rotations in them are these two
  • 00:21:11
    lexes from this last we have never bar
  • 00:21:14
    equal to 1 C 2 and there's also rotating
  • 00:21:17
    work or or and go to eat or 88 and this
  • 00:21:24
    led us the hexagonal eyes and have down
  • 00:21:26
    a bar the wanting rotation water 3 C 3 C
  • 00:21:32
    6 or a pigeon were six thanks inspector
  • 00:21:35
    back after 26 and those these are all
  • 00:21:40
    the two lattices which you can get a
  • 00:21:42
    larger we've gotta mark work
  • 00:21:43
    okay season oh yeah of course yeah
  • 00:21:48
    Portugal's
  • 00:21:51
    likewise but not before okay so these
  • 00:21:58
    are the possibilities for down apartment
  • 00:22:00
    I analyzed these the possibilities for
  • 00:22:02
    gala bar in the square lattice these are
  • 00:22:04
    the possibilities for Gaby barkay have
  • 00:22:06
    with eigenvectors and here's the
  • 00:22:08
    possibilities for generalize then you
  • 00:22:11
    have to say okay if I have Li have down
  • 00:22:13
    bar what can I say Val gamma because it
  • 00:22:16
    that's an argument exactly like exactly
  • 00:22:28
    right exactly mainly we prove that if we
  • 00:22:32
    have a location in the last half water
  • 00:22:34
    two three four or six okay here the
  • 00:22:38
    obligations of one or two that's easy to
  • 00:22:42
    spread lies its effect of tonight's
  • 00:22:44
    event but if we are a rotation of water
  • 00:22:47
    three we have to have this vector in the
  • 00:22:49
    lines right given the back of beyond
  • 00:22:51
    vector one we have this second allows if
  • 00:22:54
    you add one to it you get this vector in
  • 00:22:57
    the last so we're already in the
  • 00:22:58
    hexagonal lattice case if you have a
  • 00:23:00
    rotation Waterford likewise if you
  • 00:23:02
    engine or six and we have a location to
  • 00:23:04
    work for and we have one of the last we
  • 00:23:06
    have that this vector device okay
  • 00:23:10
    when I write one I should be writing I'm
  • 00:23:12
    sorry I should be writing one zero
  • 00:23:14
    keep writing why because I'm thinking of
  • 00:23:16
    this playing as the complex point that's
  • 00:23:20
    the thick of all this really and yet do
  • 00:23:22
    this we do complex now successive
  • 00:23:26
    addition so we can complex like I'm
  • 00:23:28
    sorry I keep like this I I think this is
  • 00:23:31
    a perfect unity
  • 00:23:35
    it is and this is the element of trace
  • 00:23:38
    to cosine theta is minus 159 0 7 a
  • 00:23:47
    displaced okay so this is a combination
  • 00:23:50
    of our classification alliances here and
  • 00:23:53
    our description of wonderful
  • 00:23:56
    possible rotation okay that one is but I
  • 00:24:01
    promise you as you guys go long and you
  • 00:24:03
    study it turns out that this region is
  • 00:24:05
    central to the study of moment here and
  • 00:24:07
    what we'd really like to know is what
  • 00:24:09
    that region is arises in our a and we
  • 00:24:13
    know it for all the way up to about
  • 00:24:15
    eight dimensions and then just a
  • 00:24:19
    statements so we very much like that a
  • 00:24:21
    classification of all glasses is t1 plus
  • 00:24:27
    I am inside our end up to ceiling and
  • 00:24:33
    boy that's one of the most central
  • 00:24:37
    objects in modern mathematics and we
  • 00:24:40
    know this for Emma
  • 00:24:42
    you know what this region was like
  • 00:24:44
    what's the ecology looks like everything
  • 00:24:46
    and this that it's I would say may no
  • 00:24:50
    reflection in number theory so you
  • 00:24:53
    should at least be exposed to within our
  • 00:24:55
    people this is related to work on the
  • 00:25:00
    sphere of mapping and they know that
  • 00:25:03
    Professor L bees in the department and
  • 00:25:05
    big breakthrough about five years ago
  • 00:25:09
    constructing during dense appear packets
  • 00:25:11
    of high the internal space as I said the
  • 00:25:14
    hexagonal packing is the best type of
  • 00:25:15
    knowing two-dimensional space it's going
  • 00:25:18
    to be proved pretty soon that we know
  • 00:25:19
    the best patents about the eight
  • 00:25:21
    dimensions and in twenty four
  • 00:25:23
    dimensional space with the best Packer
  • 00:25:24
    stores and the other space two packs
  • 00:25:28
    ears and since we don't know the
  • 00:25:29
    dimension the universe this will be
  • 00:25:31
    within
  • 00:25:33
    even in three-dimension it was only
  • 00:25:35
    recently proved that the packing of
  • 00:25:37
    spheres that you see on engine to
  • 00:25:39
    England come where they stack
  • 00:25:41
    cannonballs by first thing to the
  • 00:25:43
    hexagonal packing in one layer and then
  • 00:25:45
    stack the walls on the holes and not
  • 00:25:47
    later when stacked balls on the holes of
  • 00:25:49
    math ledger so that's also the way to
  • 00:25:51
    find oranges stack that cannot hit the
  • 00:25:53
    best the best packing of spheres in
  • 00:25:56
    3-space so that's been known since the
  • 00:25:58
    15th century but no improve did not
  • 00:26:01
    matter there's a horrible prove out
  • 00:26:03
    there which I'm variants unhappy but
  • 00:26:06
    it's approved its uses the computer to
  • 00:26:07
    analyze thousands and thousands of
  • 00:26:09
    possible cases turns it into the linear
  • 00:26:12
    programming problem will be extremely
  • 00:26:14
    nice a nice anyhow professor Elvis has
  • 00:26:23
    got this great method where you know
  • 00:26:25
    determine the best packing of spheres in
  • 00:26:27
    a dimensional space okay now let me go
  • 00:26:33
    on with a little bit of language
  • 00:26:35
    you can use it all the time study the
  • 00:26:38
    action of the group G which is hard to I
  • 00:26:41
    hope to emotions on r2 now aren't you is
  • 00:26:49
    just a set the set of points now you
  • 00:26:53
    might want to study group actions on a
  • 00:26:55
    set as the general group t happy actions
  • 00:27:02
    and general set s and develop some
  • 00:27:08
    language for because it's country comes
  • 00:27:10
    up all the nine eleven so a group action
  • 00:27:12
    of the set is a mapping from he cross s
  • 00:27:15
    context without a helmet and a pair and
  • 00:27:18
    here s the constant to what the element
  • 00:27:22
    moved on to yelled into the set and it
  • 00:27:25
    has two basic properties the identity
  • 00:27:27
    proof is the identity map of the set so
  • 00:27:30
    here s s and its associated if you take
  • 00:27:37
    th and you find s FG up
  • 00:27:42
    if so when you look at this group of
  • 00:27:49
    motions on our to this takes effect of s
  • 00:27:51
    going to the vector a s plus B where you
  • 00:27:56
    have the LM a mess S Plus T conditions
  • 00:28:02
    but with a difference X here we can also
  • 00:28:05
    take the action the concoction of cheese
  • 00:28:10
    on the set
  • 00:28:13
    align L in RT because if you have a
  • 00:28:19
    points of r2 by the group emotion to
  • 00:28:21
    preserve lives is actually preserves
  • 00:28:22
    points on the lot
  • 00:28:24
    let's do another one or we could then
  • 00:28:27
    yeah I'll cheese on the set of triangles
  • 00:28:36
    they are key because if they are a
  • 00:28:39
    triangle as you get the image of a and
  • 00:28:44
    the image of the inhibitive see those
  • 00:28:46
    are three points this one is taken to
  • 00:28:48
    the line through it
  • 00:28:49
    so goes to another time in this case the
  • 00:28:56
    different images of this are all
  • 00:28:57
    triangles congruent to that in the plane
  • 00:28:59
    because remember deep reserves including
  • 00:29:02
    Iran the distance aside to the final the
  • 00:29:07
    converse classic triangle you going to a
  • 00:29:10
    comma
  • 00:29:11
    okay now some terminology we call all
  • 00:29:17
    the messes it s defined to set what is a
  • 00:29:22
    subset of s or the order and the other
  • 00:29:30
    is a subgroup of G of s so OS are all
  • 00:29:42
    points that he takes s to so this is the
  • 00:29:45
    7 F prime equal G of s some GD and the
  • 00:29:53
    stabilizer is the something that bitch's
  • 00:29:55
    ass
  • 00:30:01
    so those things are used all the time
  • 00:30:03
    for true vacuum the two most important
  • 00:30:05
    things to do for each point you're going
  • 00:30:07
    to warm it and you have to stabilize now
  • 00:30:09
    let's take a look at some orbits and
  • 00:30:11
    stabilizers if we take the normal action
  • 00:30:15
    of this group on the plane
  • 00:30:19
    what is the ordinate of zero what points
  • 00:30:28
    can you take to zero everything warm is
  • 00:30:34
    zero here is Audrey because I think
  • 00:30:37
    white United translates if I want to get
  • 00:30:39
    zero too because you translation might
  • 00:30:41
    be and what's the stabilizer of zero
  • 00:30:49
    that's the group of two
  • 00:30:52
    those are the linear transformations and
  • 00:30:54
    experiments no translation fixes it but
  • 00:30:57
    everything in here this okay now a very
  • 00:31:02
    interesting question is if I think so if
  • 00:31:04
    I look at this action and I fix a line
  • 00:31:06
    through the origin and you just line
  • 00:31:10
    what's the orbit of that line under the
  • 00:31:14
    group of motions of the plane and woodsy
  • 00:31:17
    this is a hard question anyone see what
  • 00:31:20
    the orbit is this line what would be the
  • 00:31:21
    orbit under the rotation any time
  • 00:31:26
    through the origin right because if I do
  • 00:31:29
    the origin of some fixed angle from the
  • 00:31:30
    origin rotating by that
  • 00:31:32
    you go there so we have over over to
  • 00:31:34
    your everyone
  • 00:31:38
    Oh L is equal to s because if I have any
  • 00:31:47
    line there I can translate it to a line
  • 00:31:50
    through the origin as a base at home and
  • 00:31:52
    I wrote a delta that and so this is a
  • 00:31:56
    case where on the page line at any other
  • 00:31:58
    line in this case I can't make every
  • 00:32:01
    triangle to every other triangle because
  • 00:32:04
    I might a triangle like this I'm going
  • 00:32:06
    to take it into a triangle like that
  • 00:32:07
    they're not tied with each other so the
  • 00:32:10
    ordinate here happens to be the prime is
  • 00:32:12
    confidence I'm not going to prove that
  • 00:32:14
    so sometimes your dick is a subset of s
  • 00:32:17
    and sometimes the orbit is all elements
  • 00:32:19
    if the orbit is all of s we call the
  • 00:32:22
    action transitive
  • 00:32:37
    so these are three cute
  • 00:32:39
    terminology points in trust a point that
  • 00:32:42
    can be taken at any point and the action
  • 00:32:44
    is changing so this action is transit
  • 00:32:46
    the action of o2 on points is not
  • 00:32:50
    transitive
  • 00:32:51
    because it preservers distance so if you
  • 00:32:53
    have just the action of o2 on the plane
  • 00:32:56
    you could about at this point this
  • 00:32:58
    vector into this egg because resistance
  • 00:33:02
    okay so some actions are transitive so
  • 00:33:04
    heart let me give you a very general
  • 00:33:07
    transitive action that you don't need a
  • 00:33:09
    space for you honking anything if you
  • 00:33:10
    start with any group so this is in some
  • 00:33:16
    sense a model case a transitive action
  • 00:33:29
    and GB your group any group - they H any
  • 00:33:33
    subgroup G any something and let s be
  • 00:33:43
    the set up left Cossacks that's a
  • 00:33:50
    perfectly nice set the G option is by
  • 00:33:54
    left translation
  • 00:34:05
    it has some by Death Eater code sent to
  • 00:34:09
    the translating coaster that is the
  • 00:34:12
    identity preserves all the protest and
  • 00:34:14
    its associated I claim this is
  • 00:34:17
    transitive
  • 00:34:25
    No
  • 00:34:27
    well I just have to find one goes there
  • 00:34:29
    that I committed anything and if I think
  • 00:34:33
    it goes that age and I want to take you
  • 00:34:35
    to the Cosette aah I just translate my
  • 00:34:37
    Jane
  • 00:34:44
    on you go from edge any other code sent
  • 00:34:47
    by the school battery so before middle
  • 00:34:50
    age okay
  • 00:34:55
    what is the stabilizer in this case
  • 00:35:06
    the set of elements in G then fixes the
  • 00:35:10
    Cosette so this is the 7 cg sub CGH once
  • 00:35:18
    then so then if I want to make a
  • 00:35:26
    transitive action internally from the
  • 00:35:28
    group I think any suburb i1l exactly you
  • 00:35:31
    are far from I take the set if you the
  • 00:35:35
    set of all concepts I did the action if
  • 00:35:38
    you love transitive let love translation
  • 00:35:40
    this is transitive and the stabilizer
  • 00:35:42
    retirements myself ok what is the
  • 00:35:48
    stabilizer of another poster well it's
  • 00:35:57
    the set of G such the GA
  • 00:36:10
    so I claim that this is a and any
  • 00:36:14
    interest right if you workout this again
  • 00:36:21
    you just say you know your body but also
  • 00:36:23
    it's easy to see
  • 00:36:26
    x age right age and those that hate even
  • 00:36:33
    to anything and students cancel a
  • 00:36:41
    converse anything of that point to
  • 00:36:43
    Johnson's so it's a conjugate of the
  • 00:36:49
    original subgroup and that happens in
  • 00:36:52
    any transaction or generally more
  • 00:37:01
    general gene acts transitively on X and
  • 00:37:12
    G sub s and you want to compare and she
  • 00:37:18
    said as bride
  • 00:37:19
    so saying G of s is like s prime so I'm
  • 00:37:24
    fine 2.2 s and we find a g2 takes us
  • 00:37:29
    from s to s prime and we want to know
  • 00:37:32
    what the stabilizer of s prime is in
  • 00:37:34
    terms of the stabilizer s is gonna hit
  • 00:37:37
    vs inside so the stabilizers are
  • 00:37:43
    conjugate
  • 00:37:51
    this is a turkey Allegan you work it out
  • 00:37:54
    I just see it so suppose I have
  • 00:37:56
    something that stabilizes X so here's an
  • 00:37:58
    elevation GS and I see that an element
  • 00:38:02
    of this form stabilizes s Prime
  • 00:38:04
    well G inverse takes s prime to s that I
  • 00:38:07
    know that could stabilize desk then I go
  • 00:38:10
    back at s prime Y T so it's clear that
  • 00:38:13
    this is contained in the stabilizers s
  • 00:38:15
    prime then you have to show them
  • 00:38:17
    something of s Prime when you conjugate
  • 00:38:18
    it can't analyze your best which is the
  • 00:38:20
    same argument backers in fact this is
  • 00:38:29
    really no more general this is a
  • 00:38:39
    surprising thing I want you to take out
  • 00:38:40
    of this lecture so we have this very
  • 00:38:42
    general notion in the G action on set
  • 00:38:45
    and when we have a G action on a set it
  • 00:38:49
    breaks the set up into orbit
  • 00:38:50
    o as
  • 00:38:55
    and each one of these orbits we have a
  • 00:38:57
    transitive action so if you want to
  • 00:39:03
    break up any actions broken into a bunch
  • 00:39:05
    of trans interactions so the key is to
  • 00:39:07
    understand what a transitive action
  • 00:39:09
    looks like and here's an example of a
  • 00:39:11
    transient action on this process of a
  • 00:39:13
    sudden and the reason that this result
  • 00:39:17
    is no more general than this result is
  • 00:39:21
    that any transitive action is isomorphic
  • 00:39:24
    to the action on the coast that's an
  • 00:39:26
    assumption what if the acts and Ridley
  • 00:39:35
    on s and s and s and stabilizer piece of
  • 00:39:46
    s then there is it ejection of G sets
  • 00:39:58
    concepts with the G action the things
  • 00:40:00
    that we're looking at now between s and
  • 00:40:03
    the cosets of GS g GS is by technically
  • 00:40:09
    identified to this it takes an element
  • 00:40:12
    here and it knocks it to the l a-- g of
  • 00:40:16
    s now you see now that only the members
  • 00:40:21
    Cosette
  • 00:40:22
    Monti s because if I define modified
  • 00:40:25
    Jeep I was
  • 00:40:27
    fix's s this element is still well
  • 00:40:30
    determined right first like why L didn't
  • 00:40:33
    fix and s like stay in s that I find G
  • 00:40:35
    so this is well the fun has only begins
  • 00:40:48
    on rosette ggs if you get every element
  • 00:40:56
    in s this way it's a surjective Mac
  • 00:40:58
    because the action is transient so if I
  • 00:41:00
    want to get to some s prime I find some
  • 00:41:02
    gg-good things has to s prime it's a
  • 00:41:04
    transitive action so it's surjective a
  • 00:41:07
    be always the fist and yeah and this and
  • 00:41:13
    it's also injected the things that hate
  • 00:41:15
    the take a co set the inverse image of
  • 00:41:20
    each point here is exactly constant and
  • 00:41:24
    then if you use the left action of G of
  • 00:41:26
    this set by left multiplication of the
  • 00:41:28
    normal action G is there converted to
  • 00:41:30
    each other so that we can identify all
  • 00:41:33
    transitive actions with this kind of
  • 00:41:37
    action a transitive action determines a
  • 00:41:40
    subgroup the stabilizer before and once
  • 00:41:44
    you know that subgroup the action on the
  • 00:41:48
    it really doesn't determine uh something
  • 00:41:49
    pontifical on it because they get this
  • 00:41:51
    piece of s we have to pick a point in
  • 00:41:54
    the second right if we chose a different
  • 00:41:57
    point in the set the defining stabilizer
  • 00:42:00
    we get it conjugates eyes so that's the
  • 00:42:04
    real moral of transactions in Trinity
  • 00:42:13
    rationalism
  • 00:42:23
    and so in every time we have an action
  • 00:42:27
    of G on a set we break into orbits and
  • 00:42:32
    each word rate you get a conjugacy class
  • 00:42:34
    of submarines up to the orbit is
  • 00:42:35
    identified in the action on code sense
  • 00:42:40
    now so far we haven't had too many
  • 00:42:44
    interesting actions but we're going to
  • 00:42:45
    get a ton of interesting actions coming
  • 00:42:47
    up if by the way G is a finite group and
  • 00:42:52
    s is a finite set then we know G and s
  • 00:42:59
    papaya that we conclude from this of the
  • 00:43:04
    order of G is you our best time to be
  • 00:43:08
    ordered a watch because the number of
  • 00:43:11
    elements in this set is the same key
  • 00:43:13
    element of s we already proved that the
  • 00:43:15
    90 subgroup before 10 times its index is
  • 00:43:17
    your energy so we get a little for
  • 00:43:20
    counting formula now we're going to use
  • 00:43:26
    counted for nodes like this for all
  • 00:43:28
    kinds of interesting G actions so let's
  • 00:43:31
    say that this left translation is a
  • 00:43:34
    stupid action but we could have G icon
  • 00:43:37
    itself as an example of interesting that
  • 00:43:40
    you forgive all kinds account
  • 00:43:41
    one of those yes SCP by conjugation and
  • 00:43:51
    we have go to GS universe your board is
  • 00:43:57
    our economist Alexis and we're going to
  • 00:44:04
    want to count and G sub s is to
  • 00:44:07
    centralize your less elements in the
  • 00:44:13
    group so we're going to count communist
  • 00:44:19
    classic elements in the conjugacy class
  • 00:44:21
    by calculating what their centralizes
  • 00:44:23
    are etc and we'll be using this kind of
  • 00:44:26
    coordinate formula throughout the rest
  • 00:44:28
    of the talk the residence of our group
  • 00:44:30
    theory you're going to leave your
  • 00:44:31
    committee emotion we're going to go into
  • 00:44:33
    the abstract theory of finite groups
  • 00:44:35
    counting conjugation classes what are
  • 00:44:38
    called the seal off the herbs predicting
  • 00:44:39
    the existence of subgroups in certain
  • 00:44:41
    order and actually take us another two
  • 00:44:44
    weeks of group here before we go into
  • 00:44:46
    race okay I didn't go into the complete
  • 00:44:50
    wall paper designs I say this is a
  • 00:44:52
    fascinating subject it's a bun or
  • 00:44:54
    disinterested chemists not just into
  • 00:44:57
    this but in three steps because if you
  • 00:44:59
    can imagine the same problem in 3-space
  • 00:45:01
    when you look for lattices under the
  • 00:45:03
    orthogonal group of mention three of the
  • 00:45:06
    translation questions that might
  • 00:45:08
    uniforms of molecules kind of segmented
  • 00:45:14
    complete classification the discrete
  • 00:45:17
    circles in motion groups in free space
  • 00:45:19
    that allows them to forget how various
  • 00:45:23
    things going on I will do some things
  • 00:45:25
    about the fact that we may want to
  • 00:45:28
    classify this treat something to this
  • 00:45:30
    oak tree like we did press on - they
  • 00:45:33
    were all cyclic Brussels three you've
  • 00:45:35
    got some very interesting discrete
  • 00:45:36
    groups in particular a by can all fit
  • 00:45:39
    any move on five letters here's on the
  • 00:45:41
    comedic street some so3 and that group
  • 00:45:44
    which recently been promoted as the
  • 00:45:45
    model for the universe which I think we
  • 00:45:47
    are around so I think I will tell you
  • 00:45:50
    about 85 the
Etiquetas
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