Sam Lisi
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 Jun16 comment Gradient nonzero extensions of a vector field on the circle @Leonid Kovalev: thanks for reopening this discussion... I had forgotten about it, and see that one of the points I didn't discuss in enough detail is actually false. May15 awarded Nice Answer Apr10 comment Spivak's “Differential Geometry” Volume 1, Chapter 1 ,Problem #20 part (b) Once you've figured this out, using these decompositions of (A) and (C) will allow you to construct the homeomorphism pretty easily. Apr10 comment Spivak's “Differential Geometry” Volume 1, Chapter 1 ,Problem #20 part (b) Unfortunately, I don't understand the description of what you have done... I apologize if I am about to tell you to do the same thing. I believe Spivak is suggesting that you try to build a homeomorphism "inductively". To do this, think of (A) as obtained by starting with an annulus (cylinder) and then gluing infinitely many copies of a genus 0 surface with 4 boundary components (call this piece X). In the surface (C), he has given you the hint of how to find an annulus. You now want to find a way of exhausting (C) by a countable collection of X's, boundary glued to each other correctly. Apr9 comment Understanding Proof About an Immersion You have a typo: you wrote that you may assume the first $m$ rows are non-zero. You meant "linearly independent" (as is clear in the next sentence). As you remark, the non-degeneracy is satisfied at all $(x_0, y)$, but the observation for $y\ne 0$ doesn't give us anything we care about. Apr6 comment Second order partial derivatives - notation @ThomasM, Gustavo: I would personally avoid writing $\frac{\partial^2}{\partial x \partial y}$ if the function were not $C^2$. For a less regular function, I would personally write $\partial_x \partial_y$ or $\frac{\partial}{\partial x} \frac{\partial}{\partial y}$ to make my meaning absolutely clear. While I agree with Han and Gustavo about what the notation should mean, the slight possibility of confusion makes me want to avoid the danger. Notation should work for the mathematician (the reader of the text you write and you while writing and thinking about it) and not the other way around. Apr5 comment Are integrations on forms “different” from Riemann integrations? I would say that the deepest observation from Stokes' theorem is that in an n dimensional oriented manifold, a k-dimensional oriented submanifold is, in some sense, dual to a k-form (by integrating the k form over the submanifold). You also have duality between k-forms and n-k-forms (by wedging them together and integrating over the total manifold). From this, you are led to consider a form that represents'' a submanifold. This is the basic idea behind Poincaré duality, a key idea in topology and geometry. A beautiful but difficult exposition of some of these ideas is in Bott & Tu. Apr5 comment Are integrations on forms “different” from Riemann integrations? @Martin, Ivan: there is indeed a theory built on Lebesgue integration, called the theory of currents. The Riemann integral and the Lebesgue integral are the same when you consider bounded continuous functions... and in a large part of differential geometry, we consider smooth objects. Nevertheless, once we do analysis on manifolds and need to consider a weaker notion of convergence, we are led to the ideas of currents, singular signed measures and things like this coming from the theory of Lebesgue integrals. Apr1 comment Differentiation of the integral in the parameter @QiaochuYuan and Aspirin: math.stackexchange.com/questions/12909 Apr1 comment Differentation under an integral Take a look at the question as addressed here: math.stackexchange.com/questions/12909 ; wikipedia also has some information en.wikipedia.org/wiki/Differentiation_under_the_integral_sign Apr1 comment Notation pedantry (integration by substitution)? @PSellaz: I think you should speak with your instructor and/or the marker. They are setting the marking scheme for this class, and you need to have a better sense of their expectations than you have. Our opinions here are not terribly relevant. (I haven't looked at your specific work, so this is far from an answer.) Mar29 answered One-to-one correspondence between the flow of an autonomous ODE an its solutions Mar26 comment Is it possible to show that $\operatorname{Arg}z$ is not an analytic function by using the Cauchy–Riemann differential equations? What is your definition of $arg(z)$? does it take values in $S^1$ or in $\mathbb{R}$ or in $\mathbb{R}\subset \mathbb{C}$? The way I think of the Cauchy-Riemann equations, you need to be able to apply them to functions with values in $\mathbb{C}$. On the other hand, I think of $arg(z)$ as taking values in $S^1$. Once this definition problem has been settled, I would sidestep this formula nonsense and compute Cauchy-Riemann on the composition $arg( \operatorname{e}^{s+it} ) = t$ or $=t (mod 2\pi)$. Mar24 comment A matrix and its transpose have the same set of eigenvalues Here's one possible simpler problem that will get you started on the right path. If $A$ is an n by n singular matrix, can you show that $A^T$ is also singular? Mar23 comment What's so special about a homotopy $15$-sphere? @Matt, dimension $4k-1$ is indeed special. Heuristically speaking, exotic spheres are exotic either b/c they don't bound parallelizable manifolds or they bound parallelizable manifolds that aren't contractible. A manifold of dimension $4k$ has room to have very rich signature obstructions to being contractible. Mar23 comment Maslov Index product property. This is a little more delicate to prove if you take a different construction of the Maslov index -- for instance, an alternative definition involves counting intersections with the Maslov cycle. The good news is that any index that satisfies the axioms is the Maslov index, so once you have proved that the Maslov index exists, you can use the most convenient one for the calculation you want to do. Mar23 comment Maslov Index product property. It doesn't follow from the homotopy property -- it follows from a direct computation from the definition they give. Here, they define the Maslov index of the loop to be the degree of the composition of their map $\rho$ with $\Lambda$. The question is then to understand how $\rho$ behaves when you compose with a loop of symplectic matrices... but they have already defined the Maslov index for symplectic matrices almost the same way. It's just a question of chasing through these definitions. Let me know and I can walk you through some more details if this doesn't get you unstuck. Mar16 awarded Yearling Mar10 comment Yet another complex analysis problem @AntonioVargas: I think you should write the answer since your comment really is the key idea. Mar7 answered what does following matrix says geometrically