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10h
answered What are fun elementary subjects in probability?
2d
awarded  Nice Answer
Jan
31
comment Construction of invariants of 4-manifolds with the Kirby calculus
Most of the invariants of $4$-manifolds satisfy various surgery formulas showing that they are in some sense compatible with Kirby calculus. However, there is one issue in dimension $4$ that is not present in lower dimensions: many smooth $4$-manifolds admit several non-diffeomorphic smooth structures. This is ahrder to see using Kirby calculus alone.
Jan
30
comment Trace theorem for magnetic Sobolev spaces
The answer depends on the behavior of $A$ at $\infty$.
Jan
30
comment Trace theorem for magnetic Sobolev spaces
The space depends heavily on the behavior of $A$ at $\infty$. With no assumption on $A$ you cannot expect a precise answer concerning $H_A^1$. The curvature of $A$ may also introduce more subtle constraints.
Jan
29
awarded  Nice Answer
Jan
29
comment Derivative of eigenvectors of a matrix with respect to its components
In math $\otimes$ signifies tensor product. To understand the last equality observe that in (2) the dot denotes the derivative with respect to the parameter $t$. If we let $t=b_{k\ell}$ then $\dot{n}_i=\partial_t n_i=\partial_{b_{k\ell}}n_i$, $\dot{B}=\partial_t B=\partial_{b_{k\ell}} B$ and you get the last formula.
Jan
29
revised Derivative of eigenvectors of a matrix with respect to its components
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Jan
28
revised Derivative of eigenvectors of a matrix with respect to its components
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Jan
28
revised Derivative of eigenvectors of a matrix with respect to its components
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Jan
27
comment Bott-Samelson construction of a perfect Morse function on G/T
Maybe Sect. 3.4 of these notes www3.nd.edu/~lnicolae/Morse2nd.pdf might be more appealing to your student. $G/T$ is a coadjoint orbit and as such it is equipped with many perfect Morse functions. In these notes I also discuss special coadjoint orbits such as Grassmannians and flag manifolds.
Jan
27
answered Derivative of eigenvectors of a matrix with respect to its components
Jan
26
comment Transferring connection information to associated bundles and back
The definition of the associated connection that you use is the most elegant but the least practical as far as concrete computations are concerned. If you choose to work with a local description as in Prop 3.3.5 the computations become more transparent. Equivalently, think that the principal bundle $P$ is trivial, but your connection is not. Do the computations in such a case.
Jan
25
comment Transferring connection information to associated bundles and back
Check Prop. 3.3.5 and Sec 8.1.2 of the lectures.
Jan
25
comment Transferring connection information to associated bundles and back
Sec. 8.1.1 of the lectures deals with principal bundles. However, the result being true for any connection on a vector bundle, it will be true for a connection induced from a connection on a principal bundle via a representation of the structure group. The key formulas are (3.3.12)-(3.3.14)
Jan
25
comment Transferring connection information to associated bundles and back
See Section 3.3.3. of these lectures www3.nd.edu/~lnicolae/Lectures.pdf
Jan
24
comment Atiyah-Guillemin-Sternberg convexity theorem
See Sec. 3.5 of these notes www3.nd.edu/~lnicolae/Morse2nd.pdf
Jan
21
awarded  Necromancer
Jan
18
answered Surgery of $S^3$
Jan
18
revised Reference Request: Variational Problem
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