Dean Baskin

Today we finally introduced semiclassical defect measures. I am hoping to start talking about the damped wave equation on Wednesday

We’re doing general coordinate changes and about to introduce the covariant derivative

I want to get to semiclassical measures so I just presented most of the rest of the calculus as a list of properties. I think we will probably use Hörmander’s trick for L2 boundedness, though.

Changing coordinates, tangent and cotangent spaces! It’s getting fun!

Today we did the composition law for general symbols. I think starting Friday we will restrict our symbol classes quite a bit to make our lives easier.

Today we differentiated tensors in inertial frames in special relativity and then spent the rest of class arguing that inertial frames cannot exist in the presence of gravity. Onward!

Working toward composition for general symbols.

Yesterday we talked about more general tensors, which included a he identification of a finite dimensional vector space with its double dual.

Today we discussed one-forms as arising (locally) from differentials of functions and the one-form/vector pairing as directional derivatives. I think we have one more day of general tensors before we finally get to the equivalence principle.

Today we proved the Borel lemma and started working toward various composition formulas for symbol classes.

This is in sharp contrast to the analytic case, where having a nonzero radius of convergence imposes a pretty significant bound on how fast the sequence of derivatives at zero can grow. In contrast, for smooth functions, you can specify them and they can grow arbitrarily quickly.

That problem: given any sequence of numbers a_k, there is a smooth (C^\infty) function with the k-th derivative of f at zero being equal to a_k.

Today was day 11. We proved the composition formula for left-quantized operators and introduced symbol classes. Next time we will prove Borel’s lemma as it relates to symbols. This theorem has essentially the same content as one of my favorite graduate analysis problems.

Today was day 11: we started talking about covectors and one-forms and really tried to pin down the pairing between them and vectors. Students often struggle with this, so I spend a lot of time on it.

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Day 10: 4-momenta of photons and the beginning of our discussion of tensors

On day 10 we proved the composition formula and found the relationship between the commutator of the quantized operators and the Poisson bracket of the symbols.

Today was day 9 and we stated and almost proved the composition formula for pseudodifferential operators.

On day 9, we talked about computing 4-velocities, proper time, and measuring energy.

Day 8: the scalar product, 4-velocities, 4-momentum and the famous formula

We are following Zworski and his proof of composition is really slick. On day 8 we quantized linear and quadratic exponentials, which will figure prominently in the proof of composition.

I’m doing some fun research and teaching two really fun classes this semester. My relativity class is over full!

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Today: day seven. I realized we needed to pay some symplectic debts and so we had a whirlwind introduction/review of the symplectic geometry of R^{2n} (really a tangent bundle)

Today was day seven. We introduced vectors in special relativity. I try to take the point of view (helpful somewhat for motivating tangent spaces later) that there is an underlying four dimensional vector space and observers come with their own “standard basis vectors”, ie isomorphism with R^4

My kids love it

I also learned that the last concert is on Apple Music video

That song is soooo good

My five year old has recently gotten into this album. I hadn’t listened to it in twenty years. El Chico del apartamento 512 is a true banger

Even though you can’t really get around “vectors are things that transform like a vector” in an undergraduate course, I try to convince them to think that inertial observers come with their preferred basis for an abstract vector space and then write the components in terms of that basis.

Day six (today): we look at some classical results in special relativity (time dilation and length contraction), then start motivating vectors in special relativity.