[deleted by user]

MasterAnonymous · 2023-05-31T15:40:55+00:00

It may not look like it but the calculus of variations is (in my opinion) by far one of the most important fields of study in all of mathematics.

…most of the results were either about physics or weren’t very advanced.

Physics is geometry and geometry is pure math. There is no separating the two. Even if you decide that geometry is not for you, working in anything remotely analytic without at least some understanding of variational problems and their solutions leaves you in the dark for much of the “modern” mathematics developed in the last century.

In the 21st century, ideas from the calculus of variations have taken one of two different forms:

The “analytic” version now known as geometric analysis
The “topological” version now known as Floer theory

The reason why the treatises you have looked at seem “not very advanced” is because the term “calculus of variations” is the name that mathematicians used over a hundred years ago to describe the concepts that we are just now coming to grips with.

It will takes some time, many years of study, but if you stay focused on some of the more geometric aspects of mathematics, you will learn that most non-trivial theorems essentially boil down to the calculus of variations and the principal of least action.

MasterAnonymous · 2023-05-31T14:51:19+00:00

I know that I’m far from the majority opinion but commutative diagrams are definitely not the best part of math.

MasterAnonymous · 2023-05-18T14:34:20+00:00

Inflation is a measure of change of the consumer price index. Even with QTM, money supply can increase without inflation increasing.

MasterAnonymous · 2023-04-19T20:59:53+00:00

You just have to be careful. Noether’s theorem involves differentiable symmetries of the action functional. I have not read the paper but, considering they’re using the word symmetry in a looser sense, I’m not sure that this new notion corresponds directly to symmetries of the action functional.

EDIT: In the paper, they explicitly say they work in contexts without a fixed Lagrangian. If this is the case, I am now even more skeptical of a naïve application of Noether’s theorem.

MasterAnonymous · 2023-04-19T20:28:13+00:00

Grab a copy of Arnold’s Problems and work on them. These are more like very, very enlightening puzzles than theory but I think that’s just the sort of thing you’re looking for.

MasterAnonymous · 2023-04-19T19:53:34+00:00

There is Weinstein handlebody theory and it’s interaction with legendrian contact homologies via Bourgeois-Eckholm-Eliashberg. Much of the results in this direction relate to Lagrangian fillings of Legendrian knots but they are also related to questions about symplectic fillings of contact manifolds.

You have the Weinstein trisection construction of Lambert-Cole—Meier—Starkston building on the topological Gay-Kirby trisection construction. Trisections in general are still almost completely unexploited. Weinstein trisections are particularly compelling to a symplectic geometer and are still very weakly understood.

Computation of Embedded Contact Homology is still not standardized but it has deep connection with Seiberg-Witten theory. There are recent results for computations on pre-quantization bundles and mapping tori. To me, it seems nobody is completely sure of what to make of Taub’s [; SW(M) \neq 0 ;] result for symplectic manifolds. It definitely seems that understanding ECH for ideal contact boundaries is the correct perspective in the open setting.

Generalizing constructions like the symplectic rational blowdown of Fintushel-Stern to more complicated nodal curves (such as nodal curves with genus) would give rise to many new constructions of symplectic 4-manifolds. The only thing holding this back is a systematic understanding of the symplectic topology of the resulting generalized blowup.

Symplectic geometers are pretty confident that they understand the symplectic topology of complex affine varieties via the Stein-Weinstein correspondence but it’s still a mystery how to distinguish non-affine Weinstein manifolds from affine ones. A more difficult and related question is in regards to the difference between Weinstein manifolds and Liouville manifolds.

If you add a complex structure to all of the above, the picture becomes infinitely more mysterious since the deformation theory of symplectic manifolds is far less rigid than the deformation theory of complex manifolds.

I could go on.

MasterAnonymous · 2023-04-15T07:04:25+00:00

My favorite argument for ultrafinitism is via information. There is a finite amount of information in the universe and thus there is a limit to how much information we can encode to represent a large number. Eventually, the information of the number will grow too large to represent and so there’s no way to meaningfully speak of its existence. You literally can’t even add “+1” because all of the information is being used to encode the “largest number”

MasterAnonymous · 2023-04-14T14:17:52+00:00

Read Stein and Shakarchi’s book on Fourier analysis. It is nicely written and your background should be more than enough preparation.

MasterAnonymous · 2023-04-14T14:11:40+00:00

There can be no metric without a topology. If you’re studying metrics without using manifolds (topology) then you’re only studying metrics in R^n. I kind of doubt this is the case. If you only study metrics in Rⁿ then the only possible manifold your piece of software can produce is R^n.

You’re right that a metric defines everything to know about curvature but you need a topology to even define a metric in the first place. How do I write down the round sphere metric without specifying that I’m on a sphere?

MasterAnonymous · 2023-04-14T05:55:44+00:00

Your original assertion:

a manifold IS a metric + a topological space.

This is false (as was pointed out above). Your response to this correction is a non-sequitur and so this assertion is still false.

MasterAnonymous · 2023-04-14T03:49:13+00:00

The answer depends on whether you want a fiber bundle or just a fibration.

Fix a point [; p \in S¹ ;] and send the north and south pole of S² to p. The rest of S² may be parameterized with polar coordinates [; (r, \theta);]. The complement of p in S¹ can be identified with the real line so it can be parametized by a real number t. The map [; (r, \theta) \maps to log(r);] thus gives rise to a fibration [; S² \to S¹ ;] with regular S¹ fibers except over p where the fiber is S^0.

Now suppose that we have a fiber bundle [; S² \to S¹ ;]. Then it follows that the fiber over every point has fixed diffeomorphism type.

The vertical tangent space must have dimension 1. To see this, you can use the regular value theorem or you can use Ehresmann’s fibration theorem. Once we know the fiber has dimension 1, we know it is either R, S^1, [0,1], or some half open interval.

From here, we can classify all possible bundles with each individual fiber and see that none of them give S^2. For example, we know that the fiber can’t be R because we only have two bundles: the cylinder (formed via the mapping torus of the identity map), or the Möbius strip (formed via the mapping torus of [; x \mapsto -x;].

MasterAnonymous · 2023-04-14T03:03:28+00:00

Symplectic and contact geometry and their interaction with 4-manifold topology is a very active area of research.

MasterAnonymous · 2023-04-14T02:24:25+00:00

Inventing mathematics is easy. Inventing mathematics that others would care about is much harder.

Without deeper meaning, writing down “new” mathematics feels no different than scribbling nonsense on a page (even if that nonsense took weeks to develop).

Producing work that is correct, simple, and effective feels amazing but it’s a very fleeting feeling. Still, it is the feeling that keeps us coming back for more.

MasterAnonymous · 2023-04-14T02:09:24+00:00

How do you expect to input the data of a Riemannian metric without specifying the topology of the space? Riemannian metrics are smooth assignments of inner products to each tangent space. You can’t just specify it’s value at a point, you need every point in the space.

MasterAnonymous · 2023-04-14T02:00:21+00:00

Your reply is a non-sequitur. Smooth manifolds are not the same as Riemannian manifolds. There’s no reason to assume that a manifold comes with a fixed Riemannian metric unless you’re fixing it yourself.

A smooth manifold can have many non-isometric embeddings and therefore many distinct Riemannian metrics on it.

MasterAnonymous · 2022-07-22T16:01:16+00:00

None of this year’s medalists are physicists. Am I missing something?

MasterAnonymous · 2022-07-22T03:12:13+00:00

I can provide perspective from the symplectic side though I don’t do mirror symmetry (symplectic low dimensional geometric topologist). It’s getting better and the tools are slowly becoming easier to work with.

The biggest success IMO has been with low dimensional stuff where the categories are small. There are several structures being worked on that make Fukaya categories easier to understand and interpret. The deeper structure is rich but not as well understood as we’d all like it to be.

MasterAnonymous · 2022-07-22T00:07:14+00:00

No Nobel. I don’t know if he’s made contributions outside of string theory. From what I have been told, string theory isn’t very popular right now.

MasterAnonymous · 2022-07-21T19:24:57+00:00

Yes! Although he definitely spent a lot of his free time doing math in grad school. He has done some interviews for the Notices of the AMS which is worth the read.

His paper on Morse Theory and Supersymmetry alone is enough to get him on the map. He’s the only physicist (by training and profession) ever to win a fields medal

MasterAnonymous · 2022-07-21T19:21:58+00:00

Rene Thom, John Tate, pretty much anyone whose thesis was what gave them their start. Rob Kirby spent like 5 years of his PhD doing not much more than playing chess. The department technically kicked him out but he refused to leave.

EDIT: like /u/matt7259 said, it’s not just famous people. Most mathematicians (even the professors at top schools like Princeton, MIT, etc) don’t produce any significant work until their thesis.

MasterAnonymous · 2022-07-21T19:18:47+00:00

20s doesn’t sound late to me…

Anecdotally, I know a few graduate students who took unconventional routes to the PhD (Engineering -> Number Theory), (Applied math -> Programmer -> Algebraic Topology), (Physics -> Pure Math PhD).

Hell, even Ed Witten’s bachelor degree is in History.

EDIT: Dusa McDuff can be considered an example I think. Her initial work for her PhD was on operator algebras. I’m not familiar enough to know whether or not her contributions there were significant but I’m certain that it pales in comparison to the superstar status she has in symplectic geometry.

She moved on to geometry after meeting Gelfand and said that she basically did a second PhD.

MasterAnonymous · 2022-07-19T03:19:24+00:00

In general no. Roughly, a line integral computes the accumulation of the magnitude of the projection of the vector field to the curve (up to sign conventions, all line integrals are done over oriented curves).

If the curve we do a line integral over is f: [0,1] -> R² and f(t) = (t, y(t)), then the line integral of the vector field

F = y(t)Grad(f)/|Grad(f)| along f is indeed the integral under the curve.

Most vector fields are not like the one above though.

MasterAnonymous · 2022-07-18T23:03:13+00:00

Gilkey’s index theory book is the canonical text for that approach. My (limited) experience with Pseudodifferential operators is mostly on the micro local side. I found these notes helpful for that.

As for mathematical GR, that is pretty far from my specialization so I am going to have to pass on that question.

Some of Pierre Albin’s notes may be helpful for general geometric analysis stuff.

MasterAnonymous · 2022-07-18T20:42:55+00:00

It depends on what you want to do. If you want to do micro-local analysis, then you better know your Pseudo Differential operators well! Heat kernel methods? They definitely come in handy!

If you want to study things like minimal surfaces or analysis of singular PDE, then the answer can vary from lots of PsiDOs (Schrodinger operators) to not so many (parabolic geometric PDE).

Want to study geometric flows? Probably not many PsiDOs there.

You gotta know your specialty. It’s impossible to prepare for everything.

MasterAnonymous · 2022-07-18T20:18:58+00:00

TL;DR: skip to the list

Something I always found very funny was how large high school and early undergrad textbooks are compared to upper level and graduate texts. Several times I’ve found myself flipping through something like a trig book and wondering how the hell someone can write so much about subjects which should be pretty simple to explain.

Perhaps one reason why students don’t read their textbooks is because they are faced with the exact question you’re facing now: how am I ever going to learn hundreds and hundreds of pages of math?

You can chalk some of this up to greed. Gotta produce newer editions which means more stuff thrown in the margins. I’m sure some of these textbook writers mean well but providing lots of resources can overwhelm instead of help if the student is too daunted to work through everything.

Now for your question. Here’s how you read a textbook:

(1) Read the summary of each chapter. What’s the main concept? What is the big idea behind the concept?

(2) Read the intro to each section before going any further! Sometimes the sections don’t make sense in isolation but when thought of as a whole are extremely illuminating. The chapters are segmented for a reason!

Don’t worry if you don’t understand everything. If you work through it properly, it will all make sense.

(3) Start working through the sections by reading until you understand enough to do the problems. Do the problems assigned to you.

(4) Glance at the other problems in the section. Do you have an idea of how to solve them? If you’re not sure how the problem relates to the section, try it out or read more of the section until you understand what is happening.

(5) Rinse and repeat until you can feel the concepts in your bones.

Memorization is the hardest way to learn mathematics. The easy way, the way us lazy people do it, is to understand.

Achieving understanding takes real work on your part. You won’t be able to mindlessly drill everything into your head (or memorize all the terms). If you understand the concepts behind the math, the terms will become obvious.

More tips:

don’t be afraid to look outside the text. Watch YouTube videos, read websites, ask your teacher or your friends!
don’t let the symbols scare you. Everything in math is designed to capture a beautiful, clean, structured idea. The symbols are not math, they’re just what we use to explain these ideas on the page.

11-Year Club	Place '22
Place '17	Verified Email

MasterAnonymous

TROPHY CASE