Notes and Remarks on Unstable Motivic Homotopy Theory

I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.

Affine representability results: 

For remarks on section 2.3 and Lemma 2.3.2 of [2] (also quoted as Proposition 5.5 and Proposition 5.6 in [1]), we take the approach of stack and descent.  By relating hyperdescent condition for simplicial presheaves with descent for stacks [3], we show that how this implies classifying space of a stack satisfies hyperdescent.

After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.

Exercises on the construction of motivic homotopy theory: 

I also have some informal notes about the exercises in [1]. The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.

 

 

[1] Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry95, 305-370.

[2] Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology22(2), 1181-1225.

[3] Jardine, J. F. (2015). Local homotopy theory. Springer.

Thoughts on Physics, Maths and Reality (Part I)

Here are some casual thoughts related to my old passion for physics and different mindsets between people doing maths and physics. Most of the things discussed here are open to interpretations.

At the early age of physics, physical laws tended to fit into some sort of common sense or intuitions. As physics developed further, people found the existence of “reality” more elusive , particularly for quantum mechanics which has been thought to be counterintuitive. Physics uses maths to construct a “physical reality” that is in some sense highly subjective. Only a tiny part of the presumed existence of an outside world interacts with us by reflecting itself on our perceptions. The rest of the physics story is filled out with mathematical fabrications. How much is fabricated depends on how much we can perceive, to measure or to build a physical picture for it. Lack of direct observations and perceptions in quantum mechanics makes its story less intuitive so that sometimes people feel the need to find a more sensible interpretation which fit in their philosophical views better, like the alternative Bohmian interpretation for Copenhagen interpretation.

 I was ever driven to quest for an “interpretation” of quantum mechanics by studying a few alternative quantum theories including quantum Baysianism, and Bohmian Mechanics (the pilot-wave model). These theories are almost equivalently good at predictions. But the orthodox quantum theory has remained in textbooks as it came first in history. These alternatives are all good candidates of quantum mechanics. There might be some areas in which some of them work slightly better than the others. But none of them stands out to resolve the inconsistency problem in general relativity. People in favour of one of them is in favour of a kind of interpretation or worldview they are happy to accept.

Indeed, all these quantum theories are mathematical theories with their own beauty. That’s how I like pure maths: it stands out as a subject with a lot more freedom creating an abstract reality which does not explicitly depends on the external world.

[Reading Notes] Mechanics by Landau and Lifshitz

The following is an excerpt from [\S 43 The action as a function of coordinate]. I would like to comment on the formal derivation of Hamilton’s equations and the problems of independence of variations.

Aside. Interpretation of the action in classical mechanics

捕获

 

Note that in the derivation above, the variations \delta p and \delta q are regarded as independent.* Actually, \delta q is arbitrary but \delta p is not, even though p, q are both independent variables. Since p in connected with \dot{q} and  \delta p and \delta \dot{q} are not independent.

Notice that before (43.8) is derived, we have applied Ledrendre  Transformation which requires that

\dot{q}=\frac{\partial H}{\partial p}.                                               (1)

So the coefficient of \delta p is  0,  and \delta q is arbitrary, so its coefficient must be 0, Hence we get another Hamilton’s  equation

\dot{p}=-\frac{\partial H}{\partial q}.                                              (2)

Notice that we only derive half of Hamilton’s equations from the procedure above.

Since we can not say that we derive Hamilton’s equations by applying Hamilton’s equations. In order to make this induction above complete, we have to give the proof of another half of Hamilton’s equations, that is (1).

(1) is related to the definition of  p=\frac{\partial L}{\partial \dot{q}}. From the definition of Hamiltonian

H=\Sigma \dot{q}p-L

and \dot{q}=\dot{q}(p,q,t), then

捕获

With the definition of p, p=\frac{\partial L}{\partial \dot{q}}, we have

\dot{q}=\frac{\partial H}{\partial p}.

Hence the half part of Hamilton’sequations is derived.

 

In this way of deriving Hamilton’s equation, strictly, we first derive (1) from the definition of p, and then by applying (1) in \delta S, (2) can be derived.

*We should notice that variations here are simultaneous variations and it’s for a complete system.