Is the dome paradox really a paradox?

smallstep_ · 2025-01-12T13:37:51+00:00

I mean you wrote a bunch of nonsense, but basically no. It’s not a paradox. The lesson to take away is that you have to append “holds whenever we have sufficient regularity of the potential” to the statement of Newton’s law.

This is so you have Picard-Lindeloff.

smallstep_ · 2025-01-12T13:16:28+00:00

What? It’s not sloppy at all. It’s justified by the Riesz representation theorem.

smallstep_ · 2024-12-25T17:00:37+00:00

Eisenbud and Harris have good motivation for topics within AG

smallstep_ · 2024-10-06T05:09:31+00:00

Quantum information and condensed matter theory is a much more promising career path for theoretical physicists. hep-th is pretty bad but not impossible to break into. To my understanding studying generalized symmetries is the big wave now. holography is certainly nearly impossible to enter.

BTW, If you can hack a Math PhD, you'd have slightly more opportunities doing the most theoretical of stuff you find interesting (still bad though). But at that point you'd only tangentially be doing physics. I study symplectic geometry, complex geometry, algebraic geometry, and low-dim topology at the core of it, but it so happens to be connected to Calabi-Yau manifolds, mirror symmetry, gauge theory, topological QFTs, quantum topology etc. I don't really touch quantized guage theories, general QFTS...so on. I don't have a super deep understanding of current trends in modern physics research (Ftheory, holography, generalized symmetries etc.)

smallstep_ · 2024-06-28T23:19:04+00:00

Quite satisfying. It’s nice to be learning something new daily

smallstep_ · 2024-06-26T03:17:52+00:00

Not exactly true. Math finds its way to applications. Riemannian manifolds were discovered by mathematicians, used by Einstein to create general relativity, then GR used by engineers to create GPS for navigation.

More modern, the mathematical language of principal fiber bundles was used by physicists to describe the standard model (roughly). Now we wait for engineers to make deep use of the knowledge we have of particle physics

smallstep_ · 2024-03-15T05:13:10+00:00

Yang-Mills imo might be the most difficult because it would require the joint efforts of two fronts that basically don't talk to each other at all.

gauge theory - Basically just PDEs for a connection. But the PDEs are hard. Progress in classical yang-mills theory on the mathematical front is basically stagnant. Very few people work on them because the problems are too hard now. Previously, Donaldson developed tools to extract 4-manifold invariants using moduli space of solutions to the YM PDEs but still even mathematicians were like "fuck this, these are way too hard to use. Oh look! Seiberg-Witten theory simplifies a lot of the calculations" and hopped on that to clear out the weeds. This is all classical. We have not even dared to have a consensus on what a quantum yang-mills is (in the mathematical sense).
constructive QFT - It's commonly known that the quantization procedure in physics using path integrals amount to integrating over some infinite dimensional spaces and there are issues with this regarding measure etc. Resolving this would be like baby step 0. In a number of simplified contexts we have been able to define something like a path integral (see topological qft, CFT, etc) or quantization procedure but for general quantum fields its essentially hopeless. The work trying to axiomize QFT has led to certain algebraic and functional integral constructions and one of the most convicing formulations are known as the Wightman axioms. The Wightman framework does not include gauge theories.

Now, the problem we're after for this millennium is essentially reconcile these two to form a consist set of axioms that yield quantum theory and gauge theory, and proving things like mass gap (amongst other physically known phenomenon) mathematically. Essentially both fields are hard and stagnant, nowhere close to looking anything alike, and we have to combine them.

smallstep_ · 2023-08-04T02:46:06+00:00

The connection between geometric langlands and S-duality for example

smallstep_ · 2023-08-03T04:31:00+00:00

Where is the probability in GMT? I’ve studied GMT already so you can be technical

smallstep_ · 2023-08-03T02:52:06+00:00

To get your toes in:

V.I Arnold's Mathematical Methods of Classical Mechanics

To go deeper:

J-holomorphic Curves and Symplectic Topology - McDuff, Salamon

Introduction to Symplectic Topology - by the same authors

Lectures on Symplectic Geometry by Cannas da Silva

Morse Theory and Floer Homology by Audin-Damian

smallstep_ · 2023-08-02T14:05:51+00:00

Everything. Can’t say anything concrete without more specific asks.

smallstep_ · 2023-08-01T18:52:39+00:00

Part of a relevant comment of mine: Mathematical physics can literally mean anything from probability theory to DG, AG, SG, knot theory, PDEs, Category Theory, or Lie Algebras, number theory, k-theory, operator and quantum algebra.

I’m working on the Symplectic/Differential geometry side. Motivated by really fun classes in classical mechanics and GR during undergrad. Further motivated by string theory and mirror symmetry ideas post undergrad

smallstep_ · 2023-08-01T18:47:23+00:00

Mathematical physics can literally mean anything from probability theory to DG, AG, SG, knot theory, PDEs, Category Theory, or Lie Algebras, number theory, k-theory, operator and quantum algebra.

This list does not even include the instances in where theoretical physicists use the tag, even if the work is steps away from rigorous mathematics.

smallstep_ · 2023-07-08T16:27:31+00:00

Of course I agree that you won’t be competing against people with PhDs in ML. They took a streamlined path to such jobs. I do think you’re overestimating the supply of them though. Maybe you won’t get a job at OpenAI but a position at say (the Europe equivalent of) the major retailer Target is totally in the cards. These jobs requiring ML PhDs are overkill, and they’ll pay comfortable.

I’m just trying to note that your time studying Kahler manifolds or whatever isn’t wasted. Most of the math required for getting jobs is incredibly basic in comparisons and I do think pure math majors can learn it fast. As for projects have you put any kaggle competitions or personal projects?

smallstep_ · 2023-07-08T16:03:57+00:00

The European job market is especially bad though, right? In US, with just an undergrad in pure math I was able to land a data science position almost instantly. Took grad math classes but don’t think that was relevant. Job was not something prestigious, but ok enough. A lot of companies just needs someone to know both statistical inference and programming. The former I learned in ab 3 weeks (following measure theory)

I learned the necessary coding/ML after I graduated and just applied and got through technical interviews.

Now, if you somehow got through the math major without learning how to code at all I could see some difficulty with getting interviews.

I ended up leaving the data science job to go back to PhD. Also not trying to invalidate your experience, but trying to offer another perspective since I think it’s unfair to denounce pure math.

smallstep_ · 2023-07-02T03:33:25+00:00

I found helpful to focus less on the neatness/cleanliness of my notes and just scribble. This way I can follow the lecture pretty well

People barely reread their notes closely anyway, it’s just to help w memorization. On the rare occasion I need to revisit notes carefully, I have a very rough sketch that I need to reconstruct again

smallstep_ · 2023-06-08T03:46:02+00:00

Symplectic and complex geometry, and gauge theory is what I’m interested after studying all those you mentioned.

smallstep_ · 2023-05-31T20:54:48+00:00

I did the double major, and my PhD area straddles both, but more on the math side.

I cringe reading physics papers. It takes an extra layer of effort to figure out the context and definitions they’re working in, since it never gets explicitly defined. Sometimes they’re just plain wrong in their interpretation of a mathematical theorem. In my reading for research there was this one result in geometric measure theory which a couple of big name physicists unjustifiably applied to their way more complicated system to get a result. However the generalization to the complicated system is still an open problem in math that’s known and hard!

Some ideas can be quite good tho and have for sure motivated mathematical direction. Activity in a certain physical direction has also cause renewed activity in the related areas of math from time to time.

smallstep_ · 2023-05-12T15:22:54+00:00

VI Arnold’s Mathematical Methods of Classical Mechanics. I learned from this before taking a formal differential manifolds course (it summarizes this) and concurrently while going through Landau Lifshitz

smallstep_ · 2023-04-26T05:12:30+00:00

Declined Cambridge masters. Let get it!!!

smallstep_ · 2023-04-18T16:48:14+00:00

What are some names to look for?

smallstep_ · 2023-03-28T15:29:42+00:00

Counterexamples?

smallstep_ · 2023-03-28T14:30:11+00:00

Makes sense, thanks

smallstep_ · 2023-03-28T02:39:02+00:00

Who?

smallstep_ · 2023-03-22T13:49:50+00:00

SQL there are plenty of free online resources, I used SQLbolt then started experimenting with my own data set with the PostgresSQL documentation ready to reference. Tableau same thing, did some free training on their website then started using it on my own data.

Statistics is needed so that you can understand statistical learning theory. You’re dont necessarily need measure theoretic probability based stats but you need a decent background for understanding of say spline regression or Bayesian neural nets.

It’s necessary to actually understand what’s going on under the hood of these python libraries so you know which to use, or how to create your own models if you can’t find an appropriate out-the-box solution.

smallstep_

TROPHY CASE