Dear Folks:
Suppose \Gamma is a discrete subgroup of SL2(R), which acts on the upper half complex plane as Mobius transformation. F is its fundamental domain. If z is a vertex of F which does not lie on the extended real line ( that is R\bigcup\infty ) ,then must x be an elliptic point...
Many Thanks! You are right about my confusion on what to learn next after I've learned representation theory on lie algebras of finite dimensions. I know there is a branch of representation theory which relates to number theory - representation theory of p-adic group, but I want to learn...
I'm a bit ashamed to say I've just learned a little about representation theory. I've only read part of the books on representation theory of lie algebras of finite dimentions - Lie groups beyond an introduction by Knapp. There is so much left to learn and a long way to go. I plan to read more...
Where could I know something about the active mathematician of a certain field??
Dear Folks:
Half year later, it will be the application season. I intend to choose representation theory for my major. Where can I find some information about the active mathematicians in this field?? Many...
Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries...
How to calculate Riemannian Metric by distance function??
Dear Folks:
Here is the problem: in |z|<1, we difine a distance between any two points z1 and z2 by d(z1 , z2) = ln((z1 - b)(z2 - a)/((z1 - a)(z2 - b))) ,where a is the intersection of line z1z2 and the circle which is nearer to...
The relation between two terminology "cusp" (group & algebraic curve)
Dear Folks:
I come across the word "cusp" in two different fields and I think they are related. Could anyone specify their relationship for me?? Many thanks!
the cusp of an algebraic curve: for example: (0,0)...
Sorry, I did not read those proofs and I'm sure it is beyond my current mathematical level. Maybe it is the theory that wiki criticize not the language abstract homotopy uses. Thanks for the reference.
Well, what I mean by abstract nonsense is something that purely logic generaliztion without essential application. I've heard of the abstract homotopy theory has solved many essential problems in algebraic geometry, using category language. Thus, sometimes, we could hastely call category theory...
Sorry, I misunderstood the original problem. It should be four, but not up to homeomorphism. To be more precisely, there are four kinds of covering not four kinds of covering space.
How to determine whether the preimage of a point is a imbedding submanifold??
Dear Folks:
It is well known that the preimage of a regular point is a imbedding submanifold, but is it possible that the preimage of a critical point is also a imbedding submanifold?? More generally, is there...
How to construct the two sheet cover of Klein Bottle??
Dear Folks:
There are four kinds of two sheet cover (up to homeomorhism) of Klein Bottle, it is easy to check torus is one of them, what are the others?? Many thanks!
First, the differential geometry method still works when we talk about lie groups with infinite dimensions, while we could not calculate infinite dimension matrix seriously. Matrix of infinite dimension is actually a operator and is usually discussed in functional analysis. This generalization...