Meeting Notes 2024-04-09

This Week’s Finds

We had quite a few interesting finds this week.

Greens Functions Two Ways

Joshua and Joel discussed two different approaches to solving for Green’s functions. Let’s first recall the basic setup. Given a differential equation \(L\ u(x) = f(x)\) with boundary conditions \(\boldsymbol{D}\ u(x) = \boldsymbol{0}\), the Greens function \(G(x, s)\) satisfies the equation \(L\ G(x, s) = \delta(x – s)\), giving the natural solution  \(u(x) = \int G(x, s) f(s) ds\).

The first approach to finding the Greens function, deemed constructive, was presented by Joel and is what is taught in the physics department math methods courses. The constructive approach “builds” the Greens function by applying these criteria roughly in order: (1) Greens function solves the given equation \(L\ G = \delta\), (2) Greens function obeys the boundary conditions \(\boldsymbol{D}\ G = \boldsymbol{0}\), (3) Greens function is continuous in \(x\), (4) Greens function has precise “jump” in derivative at \(x = s\), and (5) Greens function is symmetric in arguments \(x, s\).

The second approach to finding Greens functions, deemed corrective, was presented by Joshua and has computational advantages in many situations, though can incur additional effort to “correct” the given answer for analytic continuity via advanced/retarded integrals. This method essentially exploits Fourier transforms \(\mathcal{F}: f(t) \mapsto \tilde{f}(\omega)\) to provide an analytic expression for the simple Greens function, \(G = \mathcal{F}^{-1}[1 / \mathcal{F}(f)]\)

 

Riemann, Lebesgue, and Borwein

The above discussion prompted a discussion of integration techniques, integration by exhaustion, as well as Riemann’s rearrangement theorem. In particular, an example from QFT was studied: \(\int_0^\infty ]sin(x) / x dx = \pi / 2\). This example was recognized as a specific case of a more general type of integral, namely the Borwein integrals, whose integrands are products of the form \(\sin (a_n x) / a_n x\) where \(a_n\) is odd. An interesting fact is that many of these integrals are equal to \(\pi / 2\), up until \(a_n = 15\).

 

Focus Session:

Recap of reading in Lee Ch. 2.

 

Smooth Functions and Smooth Maps

Having defined smooth structures on manifolds, we were able to look at the first class of smooth objects: smooth functions. We defined a function to be smooth on the manifold if its pushforward in the chart is smooth in the Euclidean sense. Similarly, we defined maps between smooth manifolds to be smooth if the implicit map between their two local Euclidean spaces is smooth. We then reviewed the conceptual current running through smooth objects starting from 1-dimensional Euclidean functions, then vector valued/arg functions, then extensions to manifolds and manifolds with boundary.

 

Partitions of Unity

We briefly reviewed the notion of a partition of unity, which is an essential technique in smooth manifold theory that is more flexible than the gluing Lemma for smooth functions, allowing for a more flexible way of defining local constructs.

 

Up Next:

Interesting or Relevant Talks:

• Colloquia were discussed

References:

1. Greens Functions: https://en.wikipedia.org/wiki/Green%27s_function
2. Borwein Integrals: https://en.wikipedia.org/wiki/Borwein_integral
3. Rearrangement Theorem: https://en.wikipedia.org/wiki/Riemann_series_theorem