tessel@tum.bot
Jun9-04, 04:26 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n> "Flip Tomato" <flipt@stanford.edu> asked:\n>\n> > I am undergraduate student who will be taking Quantum Field Theory\n> > next year and I wanted to get as much applied group theory under my\n> > belt as possible over the summer. Does anyone have any recommendations\n> > for books that I can use? (With limited abstract algebra prerequisite)\n\nAaron replied:\n\n> Fulton and Harris is the best textbook I\'ve come across. It doesn\'t do\n> applications, but it has lots of worked examples.\n\nPresumably Aaron means\n\ntitle = {Representation Theory: a First Course},\nauthor = {Fulton, William and Harris, Joe},\nseries = {Graduate Texts in Mathematics},\nvolume = 129,\npublisher = {Springer-Verlag},\nyear = 1991}\n\nThis is an excellent graduate level textbook which does introduce parts of\nthe theory of -Lie- groups, but I suspect the OP might want to first look\nat a sketchy but very readable advanced undergraduate textbook\n\nauthor = {Roger Carter and Graeme Segal and Ian MacDonald},\ntitle = {Lectures on {L}ie Groups and {L}ie Algebras},\nseries = {London Mathematical Society student texts},\nvolume = 32,\npublisher = {Cambridge University Press},\nyear = 1995}\n\nwhich should help motivate studying a more detailed/advanced Lie\ntheory/representation book. Further Lie theory books include:\n\nauthor = {Roe Goodman and Nolan R. Wallach},\ntitle = {Representations and Invariants of the Classical Groups},\nseries = {Encyclopedia of Mathematics and its Applications},\nvolume = 68,\npublisher = {Cambridge University Press},\nyear = 1998}\n\nauthor = {V. S. Varadarajan},\ntitle = {{L}ie Groups, {L}ie Algebras, and their Representations},\nseries = {Graduate texts in mathematics},\nvolume = 102,\npublisher = {Springer-Verlag},\nyear = 1984}\n\nFor more on Lie algebras and Dynkin diagrams, try:\n\nauthor = {James E. Humphreys},\ntitle = {Reflection Groups and {C}oxeter Groups},\npublisher = {Cambridge University Press},\nseries = {Cambridge studies in advanced mathematics},\nvolume = 29,\nyear = 1990}\n\nauthor = {Kane, Richard},\ntitle = {Reflection Groups and Invariant Theory},\npublisher = {Springer-Verlag},\nseries = {CMS books in mathematics},\nvolume = 5,\nyear = 2001}\n\nNote that "invariant theory" is a classical subject which has in the past\nfew decades become increasingly important in physics.\n\nAnd then there is Kleinian geometry (aka "theory of transformation\ngroups"), another classical application of Lie theory which is becoming\nmore important in modern physics:\n\nauthor = {Richard S. Millman},\ntitle = {{K}leinian Transformation Geometry},\njournal = {American Mathematical Monthly},\nvolume = 84,\nyear = 1977,\npages = {338--349}}\n\nAlas, there is apparently no good book yet on this topic, or on the common\ngeneralization of Riemannian geometry and Kleinian geometry, sometimes\ncalled Cartanian geometry. Unfortunately, I don\'t think Sharpe\'s book is\nsatisfactory (among other things, it makes the subject look much too hard,\nand it fails to explain essential connections with differential equations\nand with field theories), but for what its worth the citation is:\n\nauthor = {Richard W. Sharpe},\ntitle = {Differential geometry:\n{C}artan\'s generalization of {K}lein\'s {E}rlangen program},\npublisher = {Springer-Verlag},\nyear = 1997}\n\nBut from the classic\n\nauthor = {Harley Flanders},\ntitle = {Differential Forms with Applications to the Physical Sciences},\npublisher = {Dover},\nyear = 1989}\n\nyou can at least learn the essential topic of invariant volume forms. And\nfrom\n\nauthor = {Frankel, Theodore},\ntitle = {The Geometry of Physics: an Introduction},\npublisher = {Cambridge University Press},\nyear = 1997}\n\nyou can glean some further clues.\n\nBTW, historically, Lie theory and Kleinian geometry were all born in a\nclose collaboration between Sophus Lie and Felix Klein in the early 1870s,\nso the connections between these topics were part of the original vision.\n\nBut "Flip", you should be aware (if are not already aware) that there is\n-much- more to group theory than Lie groups and their\nrepresentations/invariants!\n\nIn particular, the theory of -finite- groups (and group actions) and group\nrepresentations (linear actions) and their invariants is -highly-\nrecommended background for Lie groups. If you have never studied this,\nI\'d recommend starting with this--- it will give you a big advantage when\nyou come to Lie groups and their representations/invariants. For that\nmatter, if you\'ve never studied abstract algebra (groups, rings, modules),\nyou should start with one of these superb textbooks:\n\nauthor = {John B. Fraleigh},\ntitle = {A First Course in Abstract Algebra},\nedition = {Third},\npublisher = {Addison-Wesley},\nyear = 1982}\n\nauthor = {Garrett Birkhoff and Saunders Mac Lane},\ntitle = {A Survey of Modern Algebra},\nedition = {Fourth},\npublisher = {Macmillan},\nyear = 1977}\n\nauthor = {James L. Fisher},\ntitle = {Application Oriented Algebra},\npublisher = {IEP},\naddress = {New York},\nyear = 1977}\n\nFurther applied algebra texts include:\n\nauthor = {Larry L. Dornhoff and Franz E. Hahn},\ntitle = {Applied Modern Algebra},\npublisher = {Macmillan},\nyear = 1978}\n\nFor an efficient introduction to finite groups I highly recommend working\nthrough two problem books:\n\nauthor = {John D. Dixon},\ntitle = {Problems in group theory},\npublisher = {Dover},\nyear = 1973}\n\nand one other which I can\'t find right now, darn. For more on finite\ngroups, I highly recommend these excellent books:\n\nauthor = {Peter M. Neumann and Gabrielle A. Stoy\nand the late Edward C. Thompson},\ntitle = {Groups and Geometry}, publisher = {Oxford University Press},\nyear = 1994}\n\nauthor = {Lyndon, Roger C.},\ntitle = { Groups and geometry},\npublisher = {Cambridge University Press},\nyear = 1895}\n\nNote in particular the beautiful topic of Polya enumeration.\n\nAlso try\n\nauthor = {Larry C. Grove and C. T. Benson},\ntitle = {Finite Reflection Groups},\nedition = {Second},\nseries = {Graduate texts in mathematics},\nvolume = 99,\npublisher = {Springer-Verlag},\nyear = 1985}\n\nand chapter two of the little gem\n\nauthor = {Bernd Sturmfels},\ntitle = {Algorithms in invariant theory},\npublisher = {Springer-Verlag},\nyear = 1993}\n\n(the way cool buzzword here is "Cohen-Macaulay").\n\nI\'d also -highly- recommend reading over John Baez\'s webpages for the\ncollected Weeks, many of which have discussed ADE, invariants of\nreflection groups, and related topics.\n\nGetting a bit off the QFT track: Lie theory is needed for other essential\nareas of mathematical physics. For example, the general approach to\nsolving/studying (systems of) (partial or ordinary) differential equations\n(or integro-differential equations) is also due to Lie--- what we now call\n"Lie theory" was invented as the neccessary background for the main event,\nLie\'s theory of differential equations. For example, if you\'ve heard of\n"Lagrangians" and "Noether\'s theorem", the most useful formulations of\nthis theorem are formulated using Lie\'s theory of differential equations!\nFor example, the equation governing flexural waves in thin rods,\n\nu_(tt) + u_(xxxx)\n\narises from the Lagrangian density\n\nL = [-u_t)^2 + u_(xx)^2]/2\n\n(notice that this is not quadradic in the first derivatives, so the most\nelementary formulations of Noether\'s theorem won\'t cover this Lagrangian).\nThis is invariant under the translations @/@t, @/@x, and the corresponding\nconservation laws have form\n\n@/@t density + @/@x flux = 0\n\n(in higher dimensions, the "flux" is of course a vector; you should\nrecognize here the form of Gauss\'s law, a conservation law for harmonic\nfunctions), where for @/@t we have\n\nenergy density = [(u_t)^2 + u_(xx)^2]/2\n\nenergy flux = u_t u_(xxx) - u_(xx) u_(xt)\n\nOr again, if you\'ve heard of "harmonic analysis" of PDEs, Lie\'s theory\nsupplies very useful perspective. For example, seperation of variables\ndoesn\'t always work--- Lies\'s theory explains how to figure out when it\nwill succeed, and how to use linear equations to [sometimes] solve\nnonlinear ones. More buzzwords: "spherical harmonics", "harmonic\npolynomials", "solitons", "completely integrable Hamiltonian systems",\n"Toda lattice", "instantons". For example, the famous KdV equation\n\nu_t + u u_x + u_(xxx) = 0\n\n(or rather, the PDE satisfied by its potential v, where u = v_x) also\narises from a Lagrangian, and this Lagrangian turns out to have an\n-infinite dimensional- symmetry group and therefore solutions of the KdV\nequation obey an -infinite hierarchy- of convservation laws, e.g. beyond\nconservation of energy and momentum.\n\nSo I -highly- recommend taking some time to look into this, if you\'ve\nmastered some of the basic ideas of undergraduate level Lie theory. In\napproximate order of difficulty, I -highly- recommend looking at all of\nthese:\n\nauthor = {Lawrence Dresner},\ntitle = {Applications of {L}ie\'s theory of ordinary and partial\ndifferential\nequations},\npublisher = {IOP Publishing},\nyear = 1999}\n\nauthor = {Brian J. Cantwell},\ntitle = {Introduction to symmetry analysis},\npublisher = {Cambridge University Press},\nyear = 2002}\n\nauthor = {Bluman, George W., and Kumei, Sukeyuki},\ntitle = {Symmetries and Differential Equations},\nseries = {Applied mathematical sciences},\nvolume =81,\npublisher = {Springer-Verlag},\nyear = {1989}}\n\nauthor = {Peter J. Olver},\ntitle = {Applications of {L}ie Groups to Differential Equations},\nseries = {Graduate Texts in Mathematics},\nvolume = 107,\npublisher = {Springer-Verlag},\nyear = 1993}\n\nauthor = {Peter J. Olver},\ntitle = {Equivalence, Invariants, and Symmetry},\npublisher = {Cambridge University Press},\nyear = 1995}\n\n"Homotopy theory" is needed for studying "covering spaces", "fiber\nbundles", "K-theory", and "noncommutative geometry", among other very\nuseful and sexy topics. If that sounds interesting, you can look at this\namazingly\nbeautiful book:\n\nauthor = {Hatcher, Allen},\ntitle = {Algebraic Topology},\npublisher = {Cambridge University Press},\nyear = 1992}\n\n(note this also offers much information on the topology of famous compact\nLie groups!). If you find you like homotopy, I\'d also recommend these\ntwo:\n\nauthor = {William S. Massey},\ntitle = {Algebraic Topology: An Introduction},\nseries = {Graduate texts in mathematics},\nvolume = 56,\npublisher = {Springer-Verlag},\nyear = 1967}\n\nauthor = {D. L. Johnson},\ntitle = {Presentations of Groups},\npublisher = {Cambridge University Press},\nseries = {London Mathematical Society student series},\nvolume = 15,\nyear = 1990}\n\nAnd if you might want to study superstring theory someday, you\'ll need to\nknow about algebraic geometry and its connections with Lie groups! I\n-highly- recommend:\n\nauthor = {David A. Cox and John Little and Donal O\'Shea},\ntitle = {Ideals, Varieties, and Algorithms},\nseries = {Undergraduate texts in mathematics},\nedition = {Second},\npublisher = {Springer-Verlag},\nyear = 1992}\n\nauthor = {C. G. Gibson },\ntitle = {Elementary geometry of algebraic curves:\nan undergraduate introduction},\ntitle = {Cambridge University Press},\nyear = 1998}\n\nauthor = {Harris, Joe},\ntitle = {Algebraic geometry : a first course},\npublisher = {Springer-Verlag},\nyear = 1992}\n\n(yes, Harris is the coauthor of the book Aaron recommended!--- in fact,\nboth Fulton and Harris are well known for important work on algebraic\ngeometry).\n\nauthor = {Kirwan, Francis},\ntitle = {Complex Algebraic Curves},\npublisher = {Cambridge University Press},\nseries = {London Mathematical Society student series},\nvolume = 23,\nyear = 1992}\n\nThe most important stuff to learn here are the projective groups and their\nactions on algebraic curves, quadratic forms, etc. See also John Baez\'s\n"Weeks", many of which last year had to do with beautiful connections\nbetween this stuff and important topics like the topology of Lie groups,\nelliptic curves, moduli spaces, etc.\n\nThis list is by no means exhaustive---- I haven\'t gotten into symplectic\nand Poisson structures. But perhaps it is already overwhelming, so\ninstead of continuing, maybe I should mention three reassuring points:\n\n1. you have your whole life to read great math/physics books!--- noone\nlearns everything as a student, in fact many experienced researchers will\nprobably agree the stuff they learned -after- graduate school (even apart\nfrom their own research) is the most helpful/important stuff they know,\n\n2. reading math/physics books becomes -exponentially- easier one you\'ve\nreally mastered a handful of really demanding books,\n\n3. there\'s no formula for outstanding success, but I\'ve often noticed a\ncommon characteristic of researchers whose work I admire: people who have\neven -one- area of expertise enjoyed by noone else working in a given\narea, plus some mysterious combination of talent and good luck, often seem\nto do very well.\n\nOr a leading mathematician once put it (words to this effect): many\nleading mathematicians have only a small "bag of tricks", but it is a very\nwell chosen bag.\n\nEnjoy!\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Flip Tomato" <flipt@stanford.edu> asked:
>
> > I am undergraduate student who will be taking Quantum Field Theory
> > next year and I wanted to get as much applied group theory under my
> > belt as possible over the summer. Does anyone have any recommendations
> > for books that I can use? (With limited abstract algebra prerequisite)
Aaron replied:
> Fulton and Harris is the best textbook I've come across. It doesn't do
> applications, but it has lots of worked examples.
Presumably Aaron means
title = {Representation Theory: a First Course},
author = {Fulton, William and Harris, Joe},
series = {Graduate Texts in Mathematics},
volume = 129,
publisher = {Springer-Verlag},
year = 1991}
This is an excellent graduate level textbook which does introduce parts of
the theory of -Lie- groups, but I suspect the OP might want to first look
at a sketchy but very readable advanced undergraduate textbook
author = {Roger Carter and Graeme Segal and Ian MacDonald},
title = {Lectures on {L}ie Groups and {L}ie Algebras},
series = {London Mathematical Society student texts},
volume = 32,
publisher = {Cambridge University Press},
year = 1995}
which should help motivate studying a more detailed/advanced Lie
theory/representation book. Further Lie theory books include:
author = {Roe Goodman and Nolan R. Wallach},
title = {Representations and Invariants of the Classical Groups},
series = {Encyclopedia of Mathematics and its Applications},
volume = 68,
publisher = {Cambridge University Press},
year = 1998}
author = {V. S. Varadarajan},
title = {{L}ie Groups, {L}ie Algebras, and their Representations},
series = {Graduate texts in mathematics},
volume = 102,
publisher = {Springer-Verlag},
year = 1984}
For more on Lie algebras and Dynkin diagrams, try:
author = {James E. Humphreys},
title = {Reflection Groups and {C}oxeter Groups},
publisher = {Cambridge University Press},
series = {Cambridge studies in advanced mathematics},
volume = 29,
year = 1990}
author = {Kane, Richard},
title = {Reflection Groups and Invariant Theory},
publisher = {Springer-Verlag},
series = {CMS books in mathematics},
volume = 5,
year = 2001}
Note that "invariant theory" is a classical subject which has in the past
few decades become increasingly important in physics.
And then there is Kleinian geometry (aka "theory of transformation
groups"), another classical application of Lie theory which is becoming
more important in modern physics:
author = {Richard S. Millman},
title = {{K}leinian Transformation Geometry},
journal = {American Mathematical Monthly},
volume = 84,
year = 1977,
pages = {338--349}}
Alas, there is apparently no good book yet on this topic, or on the common
generalization of Riemannian geometry and Kleinian geometry, sometimes
called Cartanian geometry. Unfortunately, I don't think Sharpe's book is
satisfactory (among other things, it makes the subject look much too hard,
and it fails to explain essential connections with differential equations
and with field theories), but for what its worth the citation is:
author = {Richard W. Sharpe},
title = {Differential geometry:
{C}artan's generalization of {K}lein's {E}rlangen program},
publisher = {Springer-Verlag},
year = 1997}
But from the classic
author = {Harley Flanders},
title = {Differential Forms with Applications to the Physical Sciences},
publisher = {Dover},
year = 1989}
you can at least learn the essential topic of invariant volume forms. And
from
author = {Frankel, Theodore},
title = {The Geometry of Physics: an Introduction},
publisher = {Cambridge University Press},
year = 1997}
you can glean some further clues.
BTW, historically, Lie theory and Kleinian geometry were all born in a
close collaboration between Sophus Lie and Felix Klein in the early 1870s,
so the connections between these topics were part of the original vision.
But "Flip", you should be aware (if are not already aware) that there is
-much- more to group theory than Lie groups and their
representations/invariants!
In particular, the theory of -finite- groups (and group actions) and group
representations (linear actions) and their invariants is -highly-
recommended background for Lie groups. If you have never studied this,
I'd recommend starting with this--- it will give you a big advantage when
you come to Lie groups and their representations/invariants. For that
matter, if you've never studied abstract algebra (groups, rings, modules),
you should start with one of these superb textbooks:
author = {John B. Fraleigh},
title = {A First Course in Abstract Algebra},
edition = {Third},
publisher = {Addison-Wesley},
year = 1982}
author = {Garrett Birkhoff and Saunders Mac Lane},
title = {A Survey of Modern Algebra},
edition = {Fourth},
publisher = {Macmillan},
year = 1977}
author = {James L. Fisher},
title = {Application Oriented Algebra},
publisher = {IEP},
address = {New York},
year = 1977}
Further applied algebra texts include:
author = {Larry L. Dornhoff and Franz E. Hahn},
title = {Applied Modern Algebra},
publisher = {Macmillan},
year = 1978}
For an efficient introduction to finite groups I highly recommend working
through two problem books:
author = {John D. Dixon},
title = {Problems in group theory},
publisher = {Dover},
year = 1973}
and one other which I can't find right now, darn. For more on finite
groups, I highly recommend these excellent books:
author = {Peter M. Neumann and Gabrielle A. Stoy
and the late Edward C. Thompson},
title = {Groups and Geometry}, publisher = {Oxford University Press},
year = 1994}
author = {Lyndon, Roger C.},
title = { Groups and geometry},
publisher = {Cambridge University Press},
year = 1895}
Note in particular the beautiful topic of Polya enumeration.
Also try
author = {Larry C. Grove and C. T. Benson},
title = {Finite Reflection Groups},
edition = {Second},
series = {Graduate texts in mathematics},
volume = 99,
publisher = {Springer-Verlag},
year = 1985}
and chapter two of the little gem
author = {Bernd Sturmfels},
title = {Algorithms in invariant theory},
publisher = {Springer-Verlag},
year = 1993}
(the way cool buzzword here is "Cohen-Macaulay").
I'd also -highly- recommend reading over John Baez's webpages for the
collected Weeks, many of which have discussed ADE, invariants of
reflection groups, and related topics.
Getting a bit off the QFT track: Lie theory is needed for other essential
areas of mathematical physics. For example, the general approach to
solving/studying (systems of) (partial or ordinary) differential equations
(or integro-differential equations) is also due to Lie--- what we now call
"Lie theory" was invented as the neccessary background for the main event,
Lie's theory of differential equations. For example, if you've heard of
"Lagrangians" and "Noether's theorem", the most useful formulations of
this theorem are formulated using Lie's theory of differential equations!
For example, the equation governing flexural waves in thin rods,
u_(tt) + u_(xxxx)
arises from the Lagrangian density
L = [-u_t)^2 + u_(xx)^2]/2
(notice that this is not quadradic in the first derivatives, so the most
elementary formulations of Noether's theorem won't cover this Lagrangian).
This is invariant under the translations @/@t, @/@x, and the corresponding
conservation laws have form
@/@t[/itex] density + @/@x flux =
(in higher dimensions, the "flux" is of course a vector; you should
recognize here the form of Gauss's law, a conservation law for harmonic
functions), where for @/@t we have
energy density [itex]= [(u_t)^2 + u_(xx)^2]/2
energy flux = u_t u_(xxx) - u_(xx) u_(xt)
Or again, if you've heard of "harmonic analysis" of PDEs, Lie's theory
supplies very useful perspective. For example, seperation of variables
doesn't always work--- Lies's theory explains how to figure out when it
will succeed, and how to use linear equations to [sometimes] solve
nonlinear ones. More buzzwords: "spherical harmonics", "harmonic
polynomials", "solitons", "completely integrable Hamiltonian systems",
"Toda lattice", "instantons". For example, the famous KdV equation
u_t + u u_x + u_(xxx) =
(or rather, the PDE satisfied by its potential v, where u = v_x) also
arises from a Lagrangian, and this Lagrangian turns out to have an
-infinite dimensional- symmetry group and therefore solutions of the KdV
equation obey an -infinite hierarchy- of convservation laws, e.g. beyond
conservation of energy and momentum.
So I -highly- recommend taking some time to look into this, if you've
mastered some of the basic ideas of undergraduate level Lie theory. In
approximate order of difficulty, I -highly- recommend looking at all of
these:
author = {Lawrence Dresner},
title = {Applications of {L}ie's theory of ordinary and partial
differential
equations},
publisher = {IOP Publishing},
year = 1999}
author = {Brian J. Cantwell},
title = {Introduction to symmetry analysis},
publisher = {Cambridge University Press},
year = 2002}
author = {Bluman, George W., and Kumei, Sukeyuki},
title = {Symmetries and Differential Equations},
series = {Applied mathematical sciences},
volume =81,
publisher = {Springer-Verlag},
year = {1989}}
author = {Peter J. Olver},
title = {Applications of {L}ie Groups to Differential Equations},
series = {Graduate Texts in Mathematics},
volume = 107,
publisher = {Springer-Verlag},
year = 1993}
author = {Peter J. Olver},
title = {Equivalence, Invariants, and Symmetry},
publisher = {Cambridge University Press},
year = 1995}
"Homotopy theory" is needed for studying "covering spaces", "fiber
bundles", "K-theory", and "noncommutative geometry", among other very
useful and sexy topics. If that sounds interesting, you can look at this
amazingly
beautiful book:
author = {Hatcher, Allen},
title = {Algebraic Topology},
publisher = {Cambridge University Press},
year = 1992}
(note this also offers much information on the topology of famous compact
Lie groups!). If you find you like homotopy, I'd also recommend these
two:
author = {William S. Massey},
title = {Algebraic Topology: An Introduction},
series = {Graduate texts in mathematics},
volume = 56,
publisher = {Springer-Verlag},
year = 1967}
author = {D. L. Johnson},
title = {Presentations of Groups},
publisher = {Cambridge University Press},
series = {London Mathematical Society student series},
volume = 15,
year = 1990}
And if you might want to study superstring theory someday, you'll need to
know about algebraic geometry and its connections with Lie groups! I
-highly- recommend:
author = {David A. Cox and John Little and Donal O'Shea},
title = {Ideals, Varieties, and Algorithms},
series = {Undergraduate texts in mathematics},
edition = {Second},
publisher = {Springer-Verlag},
year = 1992}
author = {C. G. Gibson },
title = {Elementary geometry of algebraic curves:
an undergraduate introduction},
title = {Cambridge University Press},
year = 1998}
author = {Harris, Joe},
title = {Algebraic geometry : a first course},
publisher = {Springer-Verlag},
year = 1992}
(yes, Harris is the coauthor of the book Aaron recommended!--- in fact,
both Fulton and Harris are well known for important work on algebraic
geometry).
author = {Kirwan, Francis},
title = {Complex Algebraic Curves},
publisher = {Cambridge University Press},
series = {London Mathematical Society student series},
volume = 23,
year = 1992}
The most important stuff to learn here are the projective groups and their
actions on algebraic curves, quadratic forms, etc. See also John Baez's
"Weeks", many of which last year had to do with beautiful connections
between this stuff and important topics like the topology of Lie groups,
elliptic curves, moduli spaces, etc.
This list is by no means exhaustive---- I haven't gotten into symplectic
and Poisson structures. But perhaps it is already overwhelming, so
instead of continuing, maybe I should mention three reassuring points:
1. you have your whole life to read great math/physics books!--- noone
learns everything as a student, in fact many experienced researchers will
probably agree the stuff they learned -after- graduate school (even apart
from their own research) is the most helpful/important stuff they know,
2. reading math/physics books becomes -exponentially- easier one you've
really mastered a handful of really demanding books,
3. there's no formula for outstanding success, but I've often noticed a
common characteristic of researchers whose work I admire: people who have
even -one- area of expertise enjoyed by noone else working in a given
area, plus some mysterious combination of talent and good luck, often seem
to do very well.
Or a leading mathematician once put it (words to this effect): many
leading mathematicians have only a small "bag of tricks", but it is a very
well chosen bag.
Enjoy!
"T. Essel" (hiding somewhere in cyberspace)
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