390 Lecture for §11.1, April 23.
One benefit to this is that I'm not a limited by time. I'm trying
to find positives - truly I wish I could be there and I wish that we
could discuss this in person. Alas.
Please reply with "classtime reactions" to both this and the reading reactions for today.
This is mostly written *before* the quick answers (with a few minor
adjustments), which will be written during the moments before class as
usual.
Here is our cast of characters (it is astoundingly long - it just gets more and more difficult to do this as we go forward):
Kovaleskaya
differential equations and mathematical physics - I don't know much
more. Unfortunately, this is still one of my weak spots.
Lobachevsky
This is the primary name associated with hyperbolic geometry.
There are certainly others who contributed in important ways -
Beltrami, Bolyai &c (including Gauß who says he knew about
it, but didn't tell anyone), but Lobachevsky is the one who is most
famous for it. What is the consequence? Here's the story,
and why it is important. From the beginning of the Elements,
there has been suspicion about the fifth postulate - even Euclid was
hesitant to use it (he proved everything he could without it
first). Many people tried to prove the fifth postulate.
al-Khayyami and al-Haytham did significant work concerning "what if
it's not true". This was mostly repeated by Saccheri and Lambert
(the people who get credit for it in the Wallace/West geometry
book). The attempt mostly in those cases was to prove by
contradiction that the fifth postulate needed to be true.
Sometimes people thought they had proven the fifth postulate, but in
each case they had assumed something equivalent to the fifth postulate.
Just about the time that this was become serious, shortly before the
big breakthrough, the philosopher Immanuel Kant gave Euclidean geometry
as an example of ultimate truth. In fact, when Gauß says he
knew but didn't tell anyone he said it was because he was afraid of the
ramifications. Many people have attached the Truth of Euclidean
geometry to religious meanings.
But, then Lobachevsky (and others) demonstrated that there is a
geometry that is consistent in which all of the first four Euclidean
postulates are true, but so is the hyperbolic postulate - that thru any
point not on a line there is more than one line thru that point not
intersecting the given line. This was demonstrated consistent by
producing a "model", which is more like a map (akin to a map of the
sphere). This is accomplished mostly by redefining what is
considered straight. It wasn't until years later that a model
(more like a ball is a model of a sphere) was produced. In this
model you can *see* the hyperbolic geometry in a natural setting -
straight lines are what you would think of as straight on the
surface. If you want to see a model of this, there is one on the
bottom of my display case in the library. One more important
point about all this - it is frequently said that hyperbolic geometry
is the first non-Euclidean geometry. This is simply not
true. Spherical geometry is non-Euclidean (the opposite of the
hyperbolic postulate is true - namely that thru any point not on a line
there are *no* lines thru the given line that do not intersect).
Hyperbolic geometry was a big step against universal and ultimate
truth. There was an even bigger step to come in the 20th century.
Chebyshev
Made significant advances and proved important results in probability and statistics.
Markov
Devised "Markov chains" or sequential probabilities - used to analyse
long-term processes when knowing the probability of going from one
stage to another.
Mittag-Leffler
In addition to his work in publishing and potentially his part in their
not being a Nobel prize in mathematics, Mittag-Leffler proved important
results in complex analysis. It is worth nothing that
Mittag-Leffler is one person, so the Mittag-Leffler theorem (unlike the
Cauchy-Reimann equations, for example) is proven by one person.
Poincare
Poincare was one of the last people who could see and understand all of
mathematics (Hilbert was one of the others, after them it is basically
taken to be impossible for one person to understand it all - there's
just too much out there). Among many other things in many other
branches of mathematics, Poincare was responsible for devising modern
topology. In this context it seems appropriate to discuss the
Poincare conjecture - undoubtedly the most important and most famous
problem solved in the past 5 years. Poincare devised ways to
attach algebraic objects (groups and things like them) to
geometric-like objects (ok, they are topological objects - what is a
topological object? Well, it is a geometric object where you
forget about measuring anything. Or it's a geometric object where
you may ignore the bumps [a bumpy sphere is the same as a smooth sphere
as topological objects]). The simplest (and most important)
example of Poincare's idea is to consider the group of ways in which
loops can be made on a topological object. He then generalises
this to the group of ways in which spheres, and higher dimensional
sphere can be put on a topological object. Poincare's conjecture
says that any object that has all the same groups (as given by the
above) as the n-sphere must be topologically indistinguishable from the
n-sphere.
This conjecture has a curious history in its development. It is
obvious for the 1-sphere (also known as a circle). It is pretty
easy to prove (and is proven in 338) for the 2-sphere (this is the
surface of your everyday ball, or the earth). It turned out it
was pretty easy to prove for spheres of dimension 5 and higher.
That was surprising, but done in the second half of the 20th
century. In the 1980s it was proven that it is true for the
4-sphere, and not easily. Finally, about five years ago Perelman
proved it for 3-spheres, thus settling the question. One thing
that's different about this and the famous FLT is that topologists had
previously proven numerous conditionals of the form "if the Poincare
conjecture is true then …" So, when the Poincare
conjecture was proven, suddenly we knew much more about the
possibilities of topological objects.
Cantor
Cantor did for sets what hyperbolic did for geometry - he shook the
world. Cantor devised an intuitively simple way to compare the
size of sets by considering whether they can be paired up. He
then followed the consequence of this idea to a sequence of results
which must be surprising somehow. His first results were more
concerned with sets which were the same size: he proved that the
natural numbers were the same size as the natural numbers with
one extra element, the same size as the integers, the same size as all
the rational numbers. By this point it was starting to look
merely that all infinite sets were the same size.
Next Cantor's diagonalisation argument showed that if you thought you
had a list of all the decimals you could construct one not on the list
by making one that is different in the nth decimal places from the nth
number on the list. This shows that the set of real numbers is
strictly larger than the naturals (and hence rationals), that it is
uncountable. From this point Cantor notices several things that
make the situation even worse: 1. since the reals are
uncountable but the rationals are countable; this means that the
irrationals, those leftover, are uncountable (in spite of the fact that
the rationals are more familiar). 2. not only are the
rationals countable, but so are the algebraics (remember, all numbers
which are solutions to polynomial equations); this means that the
transcendentals left over are uncountable (in spite of the fact that we
really only two three good examples of these (pi, e, and Liouville's
artificial transcendental number)). 3. not only are the
reals larger than the naturals, but that gives an idea - the reals are
the same size as the power set (the set of all subsets) of that
naturals (by considering each natural number to tell us whether a
decimal place is occupied or not). Following this idea Cantor
proves that *every* power set is larger than the set itself. As a
consequence of this -- there are infinitely many sizes of
infinity. And *that* is just a little unsettling.
Nobel
not much to say here. There ultimately isn't a Nobel prize in
mathematics. The top prize in mathematics is the Fields medal,
which is in fact, not rewarded to anyone over 40 (except for Andrew
Wiles, who was close to 40 when he completed his work on FLT).
Hilbert
Hilbert worked with logic, geometry, and set theory, among many
others. But, he will always be best known for his problem
list. He presented it at the International Congress of
Mathematicians in Paris in 1900. Here are some links describing
the problems:
http://aleph0.clarku.edu/~djoyce/hilbert/toc.html
http://www.mathacademy.com/pr/prime/articles/hilbert_prob/index.asp?PRE=hilber&TAL=Y&TAN=Y&TBI=Y&TCA=Y&TCS=Y&TEC=Y&TFO=Y&TGE=Y&TNT=Y&TPH=Y&TST=Y&TTO=Y&TTR=Y&TAD=
http://mathworld.wolfram.com/HilbertsProblems.html
http://en.wikipedia.org/wiki/Hilbert's_problems
It would be easy to design a year-long course around considering
Hilbert's problems, and still that would only be touching the
surface. It is still interesting to see what what important and
what progress has been made in over 100 years. I recommend
glancing through the lists and discussions, but not reading them in
depth, because there will surely be much you don't understand (and much
I don't understand for that matter).
In related matters, two similar meetings were held in 2000 - one in LA
(which I attended) and one in Canada at the Fields institute
(check) which I did not. To compare the breadth of mathematics
100 years later, whereas Hilbert gave one talk outlining the 1900
problems, there were (I think 5) days of talks describing what people
that was important in 2000.
Einstein (do you really need a picture?)
Well, he's famous, right? Clearly he devised relativity, but also
the geometry necessary to go along with it. As I indicated
before, the steps to end there are not so far from a few
predecessors: Maxwell, and Riemann in particular. Notice
that a large amount of mathematics comes together to produce relativity
- algebra via Hamilton to Maxwell, geometry from Riemann, and Lie
groups. It is, indeed, interesting to compare Poincare to
Einstein. Poincare was pursuing similar results, but naturally
isn't as well known for them. Ultimately, this brings an
important point that many times mathematical ideas are generated more
as a consequence of related ideas than by personal insight - i.e.
someone is likely to figure these things out after the things that came
before. Remember in particular that the idea of four-dimensional
spacetime surely came before in the form of quaternions.
Guthrie
Asked the question that would become the four colour conjecture,
eventually it seems to be the four colour theorem (there is controversy
around this result as it was "proven" by computers in a way that is not
checkable by humans). The question is quite simple - can every
map, like the one of the US - be coloured in 4 colours so that no two
adjacent states share a colour?
Möbius
Known for transformation in complex analysis, and non-orientable strips in topology.
Hardy-Littlewood, Ramanujan
Great strides in number theory. In particular, many steps
addressing the Riemann hypothesis. Suzuki's presentations of
Ramanujan's suspicious-looking results do him a disservice, as
Ramanujan surely contributed great and diverse things in number theory,
all from basically self-taught foundations - a remarkable
accomplishment at this point in mathematics history (much more so than
100 years earlier when other were doing the same thing).
Thank you for your time and patience - I will not desert you
again. On Monday we will discuss 11.2 and I will give you some
preview thoughts on your final exam. Here's It's