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Jan
28
asked What is a local parameter in algebraic geometry?
Jan
25
comment Why is the tensor product important when we already have direct and semidirect products?
One way one could naturally be led to wonder about this coproduct comes from classical algebraic geometry, where the product of two affine algebraic varieties over an algebraically closed field $k$ can be computed if you know the coproduct in the dual category of finitely-generated reduced $k$-algebras. (That's a mouthful, but the equivalence of these two categories is the essential geometric statement of the Nullstellensatz.) This allows one to immediately deduce that the product of two varieties is the maximal spectrum of the tensor product of their coordinate rings.
Jan
24
accepted An example of computing Ext
Dec
30
comment Books to learn physics, being a math major
I recommend avoiding University Physics by Young et al. It's not a bad book, but it's very much a heavily marketed textbook, meaning it lacks elegance and beauty and is overshadowed by other books for self-study (e.g., Landau and Lifshitz). University Physics is analogous to those massive tomes used in freshman calculus classes; L&L is analogous to Spivak's Calculus books.
Dec
30
comment Books to learn physics, being a math major
These are some of the most elegant books ever written on the subject of physics.
Dec
24
comment Prove the existence of a largest integer less than or equal to a rational number
I like this answer because it doesn't make use of the extra properties which $\mathbb{R}$ has. It seems preferable to me to avoid special properties of $\mathbb{R}$ to prove this fact which certainly remains true if we totally ignore the existence of $\mathbb{R}$.
Dec
17
asked An example of computing Ext
Dec
13
comment Please demonstrate decomposition of a torus into two cells
A blind calculation that gives you the number of cells you need: Take a cell decomposition of $S^1$ (one 0-cell and one 1-cell). Then the torus $S^1\times S^1$ has as cells the products of the cells of the two $S^1$ factors, so we get one 0-cell, two 1-cells, and one 2-cell.
Dec
13
answered What's Combinatorial Proof/Object/etc.?
Dec
11
revised Motivating Cohomology
added 4 characters in body; added 1 characters in body
Dec
11
revised Motivating Cohomology
added 92 characters in body; added 152 characters in body
Dec
11
comment Motivating Cohomology
"And you get so much information for free just by knowing that certain maps are ring homomorphisms rather than graded abelian group homomorphisms. I can't think of a good example off the top of my head, but I know there is one." One example might be that any diffeomorphism of $\mathbb{CP}^2$ preserves the orientation of that manifold. The cohomology ring is $\mathbb{Z}[u]/u^3$ and the top cohomology group is generated by $u^2$.
Dec
11
revised Motivating Cohomology
added 255 characters in body
Dec
11
answered Motivating Cohomology
Dec
11
comment Motivating Cohomology
By the way, are you looking for motivations of cohomology specifically or just the entire gamut of homology, cohomology theories? Your title suggests the former, but in your question you mention "topology is fine until we get to homology," which seems to suggest the latter.
Dec
11
comment Motivating Cohomology
Bott and Tu's Differential Forms in Algebraic Topology provides a great introduction to de Rham cohomology, which I've always found intuitive and concrete.
Dec
10
accepted Homology and Euler characteristics of the classical Lie groups
Dec
8
comment Is there a general formula for solving 4th degree equations?
"However, if you allow special theta values (a new operation, not among the standard ones!) then yes, you can actually write down the solutions of arbitrary polynomials this way." Do you know of a reference which expands on this point?
Dec
6
accepted Examples of advanced results and ideas explained in a down-to-earth way
Dec
6
asked Homology and Euler characteristics of the classical Lie groups