Topology vs. Differential Geometry: "Ricci Flow and the Poincaré Conjecture" by John W. Morgan, Gang Tian

The essence of the Poincare conjecture: he conjectured that every simply connected 3D-manifold is homeomorphic to the sphere.

In non-maths language, Poincaré's conjecture is that if you have a shape which everywhere looks like 3D-space (much like our universe seems to) [this is what makes it a "3D-manifold"], and this space is "finite" in some sense and without boundary [this is the "closed" part, for manifolds] in a way such that whenever you draw a loop in your space you can pull it to a point (like you can pull a rubber band on a regular sphere to a point, no matter how you wrap it, whereas there are non-trivial loops on the surface of a doughnut, or a "torus) then this space, up to a bit of deformation, must be a 3D-sphere. The language is a little misleading here - "simplest shape in 3 dimensions" isn't really what is meant. I can immerse a normal sphere "in" 3 dimensions, what is important is the internal dimension of the sphere, as described above. A 3D-sphere isn't the surface of a normal ball that you'd be used to, it is its analogue which can be immersed in 4D-space, but if you lived on one it would "feel" like you lived in 3D-space. You couldn't draw it "in" 3D-space.

The reason for the 3D-sphere, and not n-sphere for any n, isn't that the same statement doesn't work for any n, it's just that historically the 3D case was the hardest to solve. It's an interesting phenomenon in topology that some things become easier to prove in high enough dimensions (in some sense, you get more room to work with - there is a standard trick called the Whitney trick which works in dimensions 5 and above which is often the reason for some things in topology being solved in high enough dimensions). Of course, it's not as simple as saying "everything gets simpler". It's possible to show, for example, that an effective classification of manifolds of dimension 4 and above is impossible.

Probably the most famous event in mathematics in the last 25 years was Wiles' proof of Fermat's Last Theorem, for which Wiles received numerous honours. But the complete proof of Fermat's last theorem depends on a result proposed by J. P. Serre and proved by Ken Ribet, a conjecture by Taniyama and Shimura suggested the path to follow, too many mathematicians to list here made other contributions, and finally Wiles' "proof" contained a mistake which was pointed out by Richard Taylor (who helped Wiles fix the mistake).

People read about celebrities lives, what they do on a day to day basis and care about just everything they do. This is entirely the opposite in science, with a few notable exceptions. In science Fermat's last theorem is famous, Andrew Wiles is not. The law of cosines is famous, Francois Viete is not. Electromagnetism is famous, Heaviside is not. Gauge invariance is famous, Hermann Weyl is not. It is with very few exceptions the theories and discoveries that are remembered, not the names. The only exceptions I can think of are Einstein, Newton, Hawking and maybe Aristotle.

Hamilton was kind of frustrated and jealous of the fact that Perelman solved Poincaré using Ricci Flow as the basis, but he could not do it for almost 2 decades even after pioneering Ricci Flow. There is a book "Perfect Rigor" by Gessen (It's rather unfortunate Gessen does not attempt a more balanced biography; read it for the facts not for Gessen's opinions). I would suggest all math aficionados to read it to understand the man who solved Poincaré. There are inevitably arbitrary variables in life that make it "unfair." Richard Hamilton fell victim to one of these "unfair" circumstances: I think he was "too old" when Grigory Perelman was given the Fields Medal recognition, a recognition that Hamilton, too, should have been included, due to the former's foundation work upon which Perleman used to accomplish his own work. The arbitrary cut-off age that qualifies a person for the Fields Medal is 40. If you turn 41 a few hours or days or weeks before the announcement, you don't qualify, regardless of the quality of your mathematical contribution. But, as the philosophy of the medal puts it:  the Fields Medal is to "encourage" the younger and promising mathematicians, NOT to "reward" some ultimate or crowning accomplishment in mathematics. Once you accept such an arbitrary rule, it is an arbitrary rule that is easy to live with, however you feel toward it personally. If the Riemann Hypothesis was solved tomorrow, it would be known about whether or not it was a known name or someone from obscurity that solved it. And either way, no one would care about the person, no one would want to know how they live their life, what events they go to, or what they enjoy.

Wow, I'm really amazed at how many false statements people make about this kind of stuff, and if you're new to this field then you're likely to be very confused by learning incorrect things. First of all, this area of mathematics is not topology; it's differential geometry. That is like saying that calculus is in the field of arithmetic. Also, the normal sphere we think about is not a "3-dimensional sphere," but a 2-dimensional sphere sitting in 3-dimensional space. Calling that object a "3-dimensional sphere" is just confusing because then people get the idea that you're talking about the entire ball, so including the inside of it. You are only talking about the surface of that ball, which is a 2-dimensional surface, hence it is a 2D-sphere. A 3D-sphere is the surface of a 4-dimensional ball, and these concepts are important when you want to abstract these objects from the Euclidean spaces they are living in and consider the differential structure of them without a reference space."

Morgan and Tian's is still one of the best treatments out there when it comes to understanding Ricci Flow.

Lagrangean Systems: "Levels of Infinity - Selected Writings on Mathematics and Philosophy" by Hermann Weyl, Peter Pesic

“It is a well-known anecdote that Hilbert supported her [Emmy Noether] application by declaring at the faculty meeting, ‘I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university and not a bathing establishment.´”

In the memorial address “Emmy Noether (1935)” delivered in Goodheart Hall, Bryn Mawr College, 26 April 1935, and included in “Levels of Infinity - Selected Writings on Mathematics and Philosophy” by Hermann Weyl, Peter Pesic

Mathematics is, in a sense, profoundly anarchistic - you can't use authority to change or control its progress, and nothing is ruled in our out without proof agreed by the collective of practitioners, and Weyl was one of our most distinguished practitioners of the art of doing beautiful mathematics and physics. Sometimes practitioners have a brave and frankly generous stab at letting the layman get a feel for some of the broader concepts, but ultimately this is an intellectual edifice that's been built by thousands of people over the last five centuries or so and there's no reason whatsoever that we should be able to understand it at all without putting in the hard yards - the problem is not with math, it's with us and our arrogance in assuming that's possible. Weyl, as this homage book testifies, was able to put math into language people could understand and it's absolutely essential for a general audience. Language needs to be a vehicle of understanding and not an obstacle to it.

What amused me as an engineer is how engineers are taught many mathematically valid shortcuts that they use to solve many problems, while mathematicians are not taught them. Then again, how engineers and mathematicians interpret the ideas expressed in the mathematics that they use is obviously different, so perhaps although I find it amusing it is not particularly important in the greater scheme of things, (if there is a scheme). Of course, we do get taught be shortcuts, but only in the context of understanding exactly where they break down. We engineers get to live in a world of 'nice' functions where we can do things like differentiate under the integral or assume sin theta equals theta without getting too antsy about it...

I'm glad both Hilbert, Einstein and Weyl made a top shout out to Emmy Noether! She proved one of the most important and foundational results in modern physics - in a just world she'd be as well-known as Einstein, but (a) she was a woman and (b) there's no easy way to explain what she did with a glib pop science metaphor...but after having read Weyl's kind of mathematical eulogy for her, and because today is woman's day (8th March), I'll just have to give my two cents... 

Noether proved it as a theorem specifically about physical systems. It only works because the physics is fully determined by a Lagrangean which is minimised. And if that Lagrangean is covariant under a continuous symmetry (e.g. spatial translation) it leads to a conserved quantity (e.g. momentum). If the system cannot be described by a Lagrangean whose action is minimised then Noether's Theorem does not necessarily hold. Noether showed that physics being the same whatever time it is leads to Conservation of Energy. Being the same regardless of your position leads to Conservation of Momentum and being the same no matter what direction you look at leads to Conservation of Angular Momentum. All of which are examples of a symmetry which results in a conserved quantity. I'm not sure it really requires the usual glib metaphors to explain, most people have heard of Conservation of Energy and Momentum. You can explain Conservation of Angular Momentum by the usual example of a skater rotating faster as they pull their arms in. And the idea that physics is the same at all times and places and whatever direction you look at should be straightforward to understand with a small amount of thought. The extraordinary thing is that it isn't a particularly complicated proof and isn't really about physics particularly. What is surprising is no one discovered it earlier. Even Newton had the mathematical tools to do so. That he and none of the succeeding two centuries of mathematicians did suggests she had a special talent. Maybe because she was really a mathematician where she is famous for solving much more difficult problems. But it is strange nevertheless that Noether's Theorem isn't more famous. Certainly up there with Einstein's Special Theory of Relativity. And of course is widely used in theoretical physics today.

It is still important today because the basis for any theory of physics such as particle physics is also a Lagrangean whose action is minimised. If that Lagrangean is covariant under a continuous group then there is an associated conserved quantity called the Noetherian current. Another conserved quantity which can be explained by Noether's theorem is conservation of electric current as a result of phase symmetry in the wave function of quantum mechanics.

As always the ghost of Emmy Noether, one of the greatest mathematical physicist of the 20th century for her work on symmetry and conservation of quantities (energy, momentum, angular momentum), presides over all. It is a pity she was never awarded a Nobel Prize of her own. I would describe Noether's work as (a) mathematical physics for her work on symmetry and conservation and (b) pure mathematics, for everything else. For her work on symmetry alone she deserves to stand in the pantheon of great mathematical physicists. Both for its insight and subsequent centrality to modern particle physics and quantum mechanics.

Thanks Hermann Weyl for doing what you did at the time.

NB: The essay on Noether, along with the essays “The Mathematical Way of Thinking” (1940), and “Why is the World Four-Dimensional?” (1955, the year Weyl died), on their own, are worth the price of admission.