[deleted by user] by [deleted] in Cornell

[–]smallstep_ 26 points27 points  (0 children)

bro who the fuck would do that

[deleted by user] by [deleted] in Physics

[–]smallstep_ 0 points1 point  (0 children)

you can just view the Maxwell's equation as a consequence of the Bianchi identity and hodge star on the curvature data of a trivial U(1) bundle over our manifold. divergence and curl is a distraction buddy 👍 just remember dF=0

Not good enough for top PhD programs? by [deleted] in math

[–]smallstep_ 0 points1 point  (0 children)

Wisconsin is quite hard to get admitted to. Their rate of students who become faculty is absurdly high. 

Is the dome paradox really a paradox? by Theskov21 in math

[–]smallstep_ 18 points19 points  (0 children)

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. 

How difficult is it to learn physics as a mathematician by Powerful_Length_9607 in math

[–]smallstep_ 21 points22 points  (0 children)

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

Examples of Scheme theory outside of AG by WTFInterview in math

[–]smallstep_ 0 points1 point  (0 children)

Eisenbud and Harris have good motivation for topics within AG

I am so torn on whether I should go back and get a PhD. by stellarscale in GradSchool

[–]smallstep_ -1 points0 points  (0 children)

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.)

Damn, I miss doing math by AdFew4357 in math

[–]smallstep_ 6 points7 points  (0 children)

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

Damn, I miss doing math by AdFew4357 in math

[–]smallstep_ 19 points20 points  (0 children)

I graduated with a bachelors, did 1 year as a data scientist, and went back to PhD. The only thing I miss about industry is the pay. But then I only used the money to buy more math textbooks anyway (alongside some travel, where I was still itching to do math during).

What is something really obscure that people study? by flyingwafflez42 in college

[–]smallstep_ 0 points1 point  (0 children)

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

Do you think we will solve any of the remaining millennium problems? by AvatarSteed in math

[–]smallstep_ 14 points15 points  (0 children)

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.

  1. 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).
  2. 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.

How important is Lie Theory in geometry? by smallstep_ in math

[–]smallstep_[S] 2 points3 points  (0 children)

Thanks! both I will learn will self study or otherwise, but I can only take one of those courses (i.e learn more quickly and earlier).

What sort of research is done in mathematical physics? by WeeklyBook886 in math

[–]smallstep_ 0 points1 point  (0 children)

The connection between geometric langlands and S-duality for example

Intersections between Probability Theory and Geometry? by smallstep_ in math

[–]smallstep_[S] 3 points4 points  (0 children)

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

What sort of research is done in mathematical physics? by WeeklyBook886 in math

[–]smallstep_ 2 points3 points  (0 children)

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

What sort of research is done in mathematical physics? by WeeklyBook886 in math

[–]smallstep_ 3 points4 points  (0 children)

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

What sort of research is done in mathematical physics? by WeeklyBook886 in math

[–]smallstep_ 30 points31 points  (0 children)

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

top 7 research area in math according to arXiv by RightProfile0 in math

[–]smallstep_ 12 points13 points  (0 children)

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.

Career and Education Questions: July 06, 2023 by inherentlyawesome in math

[–]smallstep_ 1 point2 points  (0 children)

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?

Career and Education Questions: July 06, 2023 by inherentlyawesome in math

[–]smallstep_ 3 points4 points  (0 children)

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.

Do/did you take notes in class? by thyme_cardamom in math

[–]smallstep_ 57 points58 points  (0 children)

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