Zach Conn
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 Jan28 asked What is a local parameter in algebraic geometry? Jan25 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. Jan24 accepted An example of computing Ext Dec30 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. Dec30 comment Books to learn physics, being a math major These are some of the most elegant books ever written on the subject of physics. Dec24 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 $\mathbb{R}$. Dec17 asked An example of computing Ext Dec13 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. Dec13 answered What's Combinatorial Proof/Object/etc.? Dec11 revised Motivating Cohomology added 4 characters in body; added 1 characters in body Dec11 revised Motivating Cohomology added 92 characters in body; added 152 characters in body Dec11 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 $u^2$. Dec11 revised Motivating Cohomology added 255 characters in body Dec11 answered Motivating Cohomology Dec11 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. Dec11 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. Dec10 accepted Homology and Euler characteristics of the classical Lie groups Dec8 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? Dec6 accepted Examples of advanced results and ideas explained in a down-to-earth way Dec6 asked Homology and Euler characteristics of the classical Lie groups