How many mathematics do we need?

[Quantum River]

I am always worried about the forum moderators will move my thread to other sections such as math section, classical physics section, while I am posting a thread to Quantum physicists (I will call them physicists afterwards in the thread) or Quantum physics students. Although, some threads may lack the direct connection to Quantum physics, it is most relevant to Quantum physics. This thread is written for people who are doing Quantum physics, making Quantum physics, and creating Quantum physics.

Let me ask, How many mathematics do a theoretical physicist need? There are two kinds of physicists, experimentalists and theoreticians. I am only interested in the last case. So how many mathematics do a theoretician need to do Quantum physics?

Today the academia atmosphere is different from 19th century and even 20th century. There are absolutely no Gauss, Riemann and Poincare any more. I am not saying there are no scientists as great as Gauss or Riemann. In fact I means there are no man who is both a physicist and a mathematician. Someone may refute me by citing Edwards Witten, a Fields prize winner. I am not familiar with the string theory, so I can't say much about it. But the problem is the present split between mathematics and physics.

Nowadays it is so difficult for a Quantum physics student to study the necessary (?) mathematics. I have some books on Lie group, Riemann geometry, Partial differential equations, Automorphic form, Langlands program and the like. I have wasted much time to read them and I just could not understand them. I am afraid that many physics students may have the same feeling. But I can't dump such "rubbish", because I need them to understand and do Quantum physics. In order to understand the Gutzwiller trace formula in semiclassical Quantum physics, I have to know Calculus of variations in the large and Selberg trace formula. In order to understand the Selberg trace formula, I have to read Langlands program. I am horrified by mathematics.

I think the present mathematics is not very healthy. There is no such great mathematician as Gauss, Riemann any more (just my personal opinion). For example, in order to verify the Russian mathematician Perelman has proved the Poincare Conjecture, the mathematical community need two special teams to read Perelman's papers. I wonder whether there were some teams to judge Riemann's paper On the Hypotheses which lie at the Bases of Geometry. Of course I have only little knowledge about mathematics and especially today's mathematics, but I can't stop wondering why. I have also read the article in New Yorker [1] and feel a little pity and sadness for the great mathematician and physicist Henri Poincare.

What is mathematics to a physicist? A tool, a language. Right! But not that simple. A simple answer is a dangerous one, because it stops us to see deep.

I tried to find some math forum just like physicsforums.com, but only find some kids on the math forum. The competition between mathematicians and mathematics students should be much more intensive than our physicists.

So at last, I ask physicists to share with us some secrets. What parts of mathematics should we learn? How to read mathematics books? Is there some physicist who really understands e.g. Gauss-Bonnet formula? Or our physicists are just copycats and repeat mathematicians' abstract words. Perhaps someday the name of Mathematical Physics will change into Physical Mathematics.

[1]: http://www.newyorker.com/fact/content/articles/060828fa_fact2

 

[Haelfix]

There are many physicists who are world class mathematicians and viceversa.

As to your question how much math should a physicist know? Well in general the answer is, the more the better. As for the absolute basics for a proffessional theorist, well it varies per field, but i'd venture to guess at least the knowledge of what a first year grad student in mathematics would know, though perhaps oriented into physics language.

 

[FunkyDwarf]

Concepts can largely made without complex mathematics, its manipulating and exploring these concepts that's harder without it.

 

[J77]

Physcists need specific mathematical ideas as much as mathematicians need them; you find, learn and use mathematical concepts when you need them.

I think it's dangerous for a researcher in the sciences to sit down and learn a complete "list" of mathematical concepts without any intention of applying them; sitting down and learning a bunch of ideas does not make a good physicist (scientest/researcher) - it's more about being able to learn and understand a concept when it becomes clear that you need it.

 

[Quantum River]

There are many physicists who are world class mathematicians and viceversa.

Haelfix, could you give some list? I can read their papers.

 

[Quantum River]

Physcists need specific mathematical ideas as much as mathematicians need them; you find, learn and use mathematical concepts when you need them.

I think it's dangerous for a researcher in the sciences to sit down and learn a complete "list" of mathematical concepts without any intention of applying them; sitting down and learning a bunch of ideas does not make a good physicist (scientest/researcher) - it's more about being able to learn and understand a concept when it becomes clear that you need it.

Sometimes one really needs to sit down and learn the mathematics. The mathematics we need are not just some specific functions and we can refer to every formula in the math table. Sometimes, one really needs to understand mathematics. Of course it is difficult.

 

[J77]

Sometimes one really needs to sit down and learn the mathematics. The mathematics we need are not just some specific functions and we can refer to every formula in the math table. Sometimes, one really needs to understand mathematics. Of course it is difficult.

You shouldn't take the attitude of wanting to learn X's book from cover to cover.

Having a photographic knowledge of particular methods does not make a good scientist.

It's much better to immerse yourself in a field and pick up/learn the techniques as you go along, imo/ime.

 

[mathwonk]

I am a mathematician so I don't know what a physicist needs to know. But I can agree with some other statements here that you can get what you need best not by reading books but by talking and listening to people who have the knowledge you want.

For that reason, in my opinion, efforts such as the much ballyhooed "free courses online" from MIT are not quite as Earth shaking as they are claimed to be, at least not as long as they are mostly typed class notes.

E.g. I have studied and even taught measure theory and integration off andf on over the last 40 years, but never really grasped some key points until I sat in on the first introductory lecture by my friend Edward Azoff, a real analyst, last semester. It only took one lecture by a master to get a new understanding of why the lebesgue integral is defined the way it is.

I still know very little about algebraic cycles and K theory, but yesterday I learned a little more from a survey talk by a bright young mathematician, who just remarked that K(0) is a universal target for chern class constructions on vector bundle, and K(1) is a universal target for determinants of endomorphisms of vector bundles. That statement of where they come from and why, is very helpful in understanding why people care about these things. This is much better than some lengthy book definition of how to construct them, but not what they are good for.

Try to sit in on surveys and introductions to topics, to get a quick feel for them. Then you can more profitably read and benefit from reading more detailed treatments. or jut ask people here. I, and others here, can probably give you a short explanation of the gauss bonnet formula, or understandable references for it.

A simple remark is that it says that if you bend a surface inward somewhere, it has to bend outward somewhere else. I.e. the average over the whole surface, of the curvature never changes, and is in fact dependent only on the topology of the surface.

take a sphere and bend it into a u shape. you have created a minimum and two maxima instead of only one of each, but you also have a saddle point now which cancels out one of the minima, so the average or sum is the same.

there is a simplicial approach to the gauss bonnet formula that seems especially simple.

the formula implies that a surface with handles can never be given a metric woith positive curvature, because then the average curvature would also be positive, but the handles count negatively on the topology side of the formula. conversely, a surface of poisitive curvature is a sphere.

This theorem for three manifolds, was only proven in the last 30 years or so by Richard Hamilton, and may play a role in the poincare conjecture proved by Perelman. You do not have to be able to vet Perelmns proof to know enough riemannian geometry to use it in physics.

A friend of mine is currently teaching a cousre in riemannian geometry from the book of do carmo, and i recommend it, but much better would be to simultaneously sit in on the course. I hope to hear a few lectures myself, but professors have less time for this than do students.

 

[mathwonk]

here is alittle more on the gauss bonnet theorem. first of all the simplicial version is an exercise. consider a polyhedron whose faces are triangles, and define the curvature at a vertex as 2pi minus the sum of the angles of the triangles meeting at that vertex.

then check that the curvature summed over all vertices does not change when you subdivide the triangles into more triangles, i.e. even though this may change the curvature at each vertex.

then compute that the sum of the curvatures at all vertices equals V-E+F times maybe 2pi. this is the gauss bonnet formula.

in differentiable terms it reduces to the poincare hopf theorem that the number of zeroes of a vector field equals the euler characteristic, i.e. is a topological number namely V-E+F.

this last result also goes back to riemann, as you might have predicted, who showed that the number of zeroes of a differential form on a surface of genus g is 2g-2, a topological invariant.

(I have not read Gauss.)

now wasn't that easier than plowing through hundreds of pages of mathematical text? I hope so anyway.

 

[mathwonk]

actually i thought this was going to be a joke thread, like how many mathematicians does it take to change a lightbulb?

 

[phun]

http://superstringtheory.com/math/math1.html

The guides to math I, II, III are nice listings of math topics needed for string theorists. I think string theory is the most mathematically advanced subfield of physics as of now.

 

[mathwonk]

gosh that's seems like a lot of math.

 

[Hurkyl]

Concepts can largely made without complex mathematics, its manipulating and exploring these concepts that's harder without it.

Well, yes and no... that piece of "complex" mathematics usually is exactly that concept you are trying to discuss. The only reason it seems like the concept can be made without the complexity is because the explainer is waving his hands and brushing aside little details.

In fact, some (many?) mathematicians get discouraged from studying physics because too much time is spent explaining "concepts" and they never get around to saying what it is we're really doing mathematically.

 

[leon1127]

http://superstringtheory.com/math/math1.html

The guides to math I, II, III are nice listings of math topics needed for string theorists. I think string theory is the most mathematically advanced subfield of physics as of now.

I have primany knowledge only upto differential forms...
This makes me a maths major instead of phys major.:yuck:
Just the book "calculus of variation in large" by Morse alone, I have spent a year on perparatory reading and I am still not at the level of the book's audiance.

I strongly agree what mathwonk have suggested. Reading books with theorems and proofs don't help an ordinary personal like me too much. I rather want a person ( or a book, resp.) who (which, resp.) has analogies about the ideas. For exmple, in the ch. 1 of "an introduction to morse theory" by Yukio Matsumoto, he gave a very clear analogy of morse lemma in 2D. Just that particular analogy gave me a whole new aspect in sufficient condition of extremal of lagrangian. The book by morse never given me that impression.

 

[Quantum River]

I have primany knowledge only upto differential forms...
This makes me a maths major instead of phys major.:yuck:
Just the book "calculus of variation in large" by Morse alone, I have spent a year on perparatory reading and I am still not at the level of the book's audiance.

I strongly agree what mathwonk have suggested. Reading books with theorems and proofs don't help an ordinary personal like me too much. I rather want a person ( or a book, resp.) who (which, resp.) has analogies about the ideas. For exmple, in the ch. 1 of "an introduction to morse theory" by Yukio Matsumoto, he gave a very clear analogy of morse lemma in 2D. Just that particular analogy gave me a whole new aspect in sufficient condition of extremal of lagrangian. The book by morse never given me that impression.

"Morse theory has received much attention in the last two decades as a result of the paper by Witten (1982) which relates Morse theory to quantum field theory and also directly connects the stationary points of a smooth function to differential forms on the manifold." [1]
[1]: http://mathworld.wolfram.com/MorseTheory.html

 

[mathwonk]

the book by milnor on morse theory is usually recommended.

 

[mathwonk]

About all the morse theory most people need is summarized in the picture in the early aprt of milnopr's book of a torus (doughnut) resting on end. the height function thus ahs 4 critical points, a max a min and 2 saddle points.

the index of these critical points, measured by the eigenvalues of the symmetric matrix of second partials, are 1,1, and -1,-1, (?hence they add to zero, the euler characteristic of the torus?).

The homotopy type, i.e. a rough topological approximation, of the torus, is also obtained from this data as follows. start with a disc, representing the bowl near the bottom of the torus, and note that as the height rises, the aprt below a givenb height has the same topology until the ehight apsses a critical level.

When we pass the first saddle point, we add a handle to the bowl making it homotopy- like an easter basket. when we pass the next saddle point we add another loop, like a 2 loop daisy chain with a bowl at the bottom.

Finally when we reach the top, we paste on another "disc", or rectangle, with boundary sides equal to the two loops, getting the full torus.

at each critical level the index of the critical point tells us what dimension cell to add on.


briefly, morse theory tells you how to build a rough topological (up to homotopy) approximation of your manifold, just from the finite set of critical pionts of any general height function, and their indices.


The other basic results of diff top/geom are the poincare hopf index thweorem and the gauss bonnet theorem.

poincare hopf says that the sum of the indices of any vector field compoutes the euler characteristic, i.e. gives a tiny bit of information about the alternatinbg sum of the number of cells of avrious dimensions making up the manifold. the vector field associated to a flow of water on the torus above has 4 zeroes, on at each critical point, also of indices 1,1, and -1,-1.


the gauss bonnet theorem say the average curvature over an even dimensional oriented manifold equals (2pi times?) the euler characteristic.

since for a hypersurface, curvature can be measured by the jacobian derivative of the gauss map translating the outward unit normal vector at each point to the corresponding unit radius vector to the sphere at the origin, one can "pull back" the vector field of a north - south flow on the sphere, to the hypersurface, and deduce the gauss bonnet theorem from the hopf theorem.


all these thigns also go abck ultimately no doubt to the basic theorem on amnifolds, the stokes theorem.

there you have a course in diff/top-geom, in a nutshell.

 

[Ki Man]

Isnt it true that the farther you get into advanced physics and advanced mathematics, the more blured the lines between the two subjects becomes.

 

[Mororvia]

It would have to depend on what you mean by "advanced." I don't think there are too many mathematicians working in condensed matter physics but there are certainly advanced topics there.

 

[morphism]

Isnt it true that the farther you get into advanced physics and advanced mathematics, the more blured the lines between the two subjects becomes.

That's not true. Mathematics becomes very different from physics.

 

[mathwonk]

if you want a brief intro to morse theory from the great john milnor himself, for free, the first 4 pages are apparently viewable on amazon:

https://www.amazon.com/gp/sitbv3/reader/102-8428200-5434549?asin=0691080089&pageID=S00A&checkSum=DHgzdhDwYp4fl0KG2/iEgdrSvgkfiYhSPdX1bfCwh08=&tag=pfamazon01-20

 

[Quantum River]

Recently I read Michael Atiyah's article "Mathematics in the 20th Century". Atiyah said "21st century might be the era of quantum mathematics or, if you like, of infinite-dimensional mathematics". So mathematicians should ask how many physics mathematicians need?

 

[mathwonk]

we never get enough physicists. they come up with good important questions, and they often have no clue how to solve them!

 

[J77]

That's not true. Mathematics becomes very different from physics.

The division is there because mathematicians and physicists don't tend to talk to one another!

I'm speaking here from experience in three different institutes where I've worked in both math and physics departments...

...plus, recently I was at the "national" physics conference, here in the Netherlands. The majority of talks I went to were to do with fluid dynamics - a field which I associate with my student days in maths. Also, this thread takes me back to a talk I saw by David Gross - he outlined the future fields of physics. He included things like fast-slow systems and networks - two subjects which have been hot-topics in applied maths for at least 6 or more years.