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 Sep14 comment homeomorphism question relating to the topological 3-sphere You are talking about the usual construction of $S^{3}$ the one asked is bit unusual. In general there are many ways to triangulate a surface (and here break up a solid into polyherdral parts: essentially a CW complex). If you keep track of the smaller parts and their intersections you can decompose the solid into more manageable parts. The question is showing the sphere is two solid tori. Sep13 comment The matrix exponential: Any good books? For finite dimensional matrices you are essentially talking about exponential maps on lie groups. So look up any elementary nook on lie algebras and lie groups (may be it is worthwhile to consider compact lie groups first). Off hand I can think of M.Artin's Algebra, Bump's Lie Groups. Sep13 comment What branch of mathematics is most needed in the industry or how one can make living with mathematics (apart from teaching)? I think you need to define relevant job more clearly. If you wish to maximize your chances at a financial institution studying economics or perhaps mathematical economics is more useful. Studying mathematics for that purpose is not the most efficient route. Sep13 comment Proving that there are infinitely many primes with remainder of 2 when divided by 3 Use Dirichlet's theorem on primes in Arithmetic progressions. Well it is an overkill but you don't have to do anything.en.wikipedia.org/wiki/Primes_in_arithmetic_progression Sep13 comment I've heard that some kinds of mathematics are so abstract, what's the point in it? I do not understand if you mean abstraction of programing languages in a concrete technical sense or a rather loose figurative sense. But, mathematics is no more abstract to a mathematician than playing a (difficult) piece of music to a pianist. Of course to an outside r it looks strange and esoteric. Sep13 comment Connectedness of centralizer exercise Look at any discussion on Spec of a commutative ring (like exercise to the fist chapter of Atiyah Macdonald). Sep13 comment Connectedness of centralizer exercise As you computation shows, the points lying in the centralizer of t in Sp(4) is an algebraic variety over $K$ (same in the Gl4 case but slightly more complicated). The relevant topology topology is the Zariski topology. More concretely, look at the ring $K[x_{1}. \ldots, x_{8}]$ and quotient it by the ideal $(x_{1}x_{7} -x_{5}x_{3}, x_{1}x_{8} + x_{2}x_{7} - x_{3}x_{6} -X_5}x_{4})$ and show that the resulting quotient has idempotents. Sep13 answered Some question on filtrations Sep12 answered What's a good place to learn Lie groups? Sep12 answered Trivilisations of Vector Bundles Sep12 answered Suggestions for further topics in Commutative Algebra Sep12 answered How many subsets are there in a set of size $n$? No combinatorics Sep12 awarded Editor Sep12 comment Derivative of $f(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$ sorry ! typo fixed. Sep12 revised Derivative of $f(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$ added 26 characters in body Sep12 answered Local Isomorphism on Topological Groups Sep12 awarded Teacher Sep12 answered Derivative of $f(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$ Sep12 answered Looking for a Calculus Textbook Sep12 answered A question about the definition of a neighborhood in topology