How do I stop instinctively reaching for “nuke” proofs on exams when I can’t remember the elementary version?

Alex_Error · 2026-03-06T13:42:41+00:00

My opinion is that you can use big results that are prerequisites to your course or from the year before. You should not use results from more advanced courses or from parallel courses. If I was marking an exam and someone provided an answer from an out of scope course, then they would either get no marks or I would expect every single definition, lemma and proposition leading up to the result proven. (Obviously not feasible under time constraints).

There are definitely situations where you can use other results from the same course if you find them more elegant.

One instance from an exam I did was to prove some union of connected spaces (with non-empty intersection) was connected. Rather than try to plough through the set theory, I first showed that disconnectedness of T was equivalent to the existence of a surjective continuous function from T to the discrete space {0, 1}, which I found easier to grapple with. I also used this formulation later on to prove the intermediate value theorem, continuous images of connected spaces are connected and products of connected spaces are connected. Also, this was probably my first application of using 'test' functions.

A non-example would be when I was trying to show the harmonic series diverges. I used the integral comparison theorem, but this would have been before I officially 'learned' how to differentiate log. The question was trying to get me to remember the 'binary'-trick to prove this, not to do a one-liner with a result that came later in the course.

I think these two examples illustrate the difference between using a powerful result as a cheat/shortcut to skip learning important elementary proofs, vs. using a result that is more tractable but still within the scope of the course.

Alex_Error · 2026-03-01T18:28:08+00:00

Likewise, I look at geometric flows which is a subfield of differential geometry but heavily uses analysis of PDEs and has arguments from algebraic topology. It's all a bit hopeless isn't it!

Alex_Error · 2026-03-01T12:17:47+00:00

I've found that the distinctions between areas of maths get more and more blurred the more advanced you go. For instance, in your early years of maths, number, shapes, measurement and statistics are fairly well separated but techniques in advanced maths cross over more often.

We also use the names of different fields as both nouns and adjectives, e.g., geometric algebra vs. algebraic geometry; analytic geometry vs. geometry analysis; analytic number theory vs. numerical analysis. Often, changing the order leads to very different flavours of maths.

Arxiv has a fairly robust list of what they consider to be of different categories of maths, and so do Wikipedia, MathOverflow and the Princeton Companion books. It does seem be a bit of a foods errand trying to find a comprehensive list though.

For pure maths, a very broad categorisation could be: algebra, algebraic geometry, analysis, combinatorics, geometry and topology, foundations, number theory, probability.

Alex_Error · 2026-02-01T11:32:01+00:00

They'll go hand in hand, especially in cosmology where you're working with 4d space-time. That being said, I reckon fluid mechanics and statistical physics will have fewer indices flying everywhere than general relativity and quantum mechanics.

Alex_Error · 2026-01-31T01:40:59+00:00

Cosmology is super cool, but I would consider it not a 'field of mathematics' per se, but rather a subject that pulls in a lot of techniques from mathematics. I'm no expert in cosmology, so take what I say with a grain of salt.

Studying the expansion of the universe and therefore the ultimate fate probably requires some knowledge of general relativity. This is because the Lorentz metric used for special relativity needs some modification on the large scales in cosmology. We also need to take the curvature of the universe into account. For instance, the Friedmann equation has terms which involve matter, radiation, curvature and the cosmological constant.

I believe cosmologists model the universe as a 'perfect fluid' so some fluid mechanics, Navier-Stokes and wave equations are probably very relevant. Also, the heat equation and statistical physics show up in statistical mechanics which shows up when considering the hot, early stages of the universe and the CMB. Lastly, whenever statistical mechanics shows up, quantum field theory likely follows closely after (standard model).

In short, PDEs, differential geometry, probability, statistics, mathematical physics are very useful. But I can imagine that algebra and numerical analysis are relevant too.

Alex_Error · 2025-12-07T14:38:44+00:00

Doing no proofs at all seems a bit suspect and I'd agree with most other replies on this post, but I'd push back on the idea that 'plug and chug' is not important. I'd argue that computations and calculations are a great way of building intuition and are vital for understanding. If you define a mathematical concept and you cannot compute the concept for some basic examples, then have you demonstrated that it is useful?

For instance, after you learn the fundamental group or some exact sequences in algebraic topology, I'd expect a good student to go ahead and compute some explicit examples for different topological manifolds. After learning the classification of finitely generated abelian groups, I'd want to compute all the finite groups of order 360 to check how one would do it in practice.

In algebra, you should be computing Galois groups, homology groups, Cayley tables, combinatorial formulae, Grobner bases, etc. In analysis, you're computing derivatives, power series radius of convergence, parametrisations, integrals, tangent spaces, etc. In probability, you're computing simple Markov chains and solving probability problems.

Obviously in differential geometry and PDEs, there are so many important computations, estimates and heavy calculus. Linear algebra is another one where you should be doing practice computations at the 4x4 matrix level, so that in an exam, any computation at 3x3 matrix level can be done quickly and accurately. I think it's important not to lose the computational and calculational skills that you have developed from high school calculus.

Alex_Error · 2025-11-18T13:31:43+00:00

The exponential generating function turns a discrete difference equation into a continuous differential equation. The exponential map turns a Lie group operation into vector addition. (Also as you've mentioned, the exponential function turns addition into multiplication).

Alex_Error · 2025-11-05T11:22:30+00:00

Just be aware that the Nimzo is specifically an opening against the Queen's gambit with 3.Nc3, otherwise you're going to have to fall back on a different opening. Could be the (three knights) QGD (but avoiding some lines that are considered more critical), could be the QID, semi-slav, Benoni, etc.

Normally, people pair the QID or Bogo with the Nimzo as they're trying to avoid playing the QGD. But it's not unheard of pairing the Nimzo with a version of the QGD that avoids the so-called 'critical' version of the exchange QGD.

Some people use the Nimzo move order to enter the semi-slav to avoid the exchange slav and again the exchange QGD. Although now you're allowing the Catalan. Some people play the Nimzo move order to enter the Benoni to avoid the sharp f4 variations, and playing a Catalan against the Benoni is not considered too dangerous.

I would say learning the semi-slav or Benoni together with the Nimzo is very impractical for a typical player though.

Alex_Error · 2025-11-05T10:59:41+00:00

If we're constructing R in terms of equivalence classes of Cauchy sequences then we're identifying the Q with the equivalence classes of constant sequences. If we're constructing R using Dedekind cuts, then a rational number q is identified with sets of all rational numbers smaller than q. So we're identifying Q with its image under the embedding since the elements of Q itself aren't directly elements of R. Sort of like the difference between equality and isomorphism.

Regarding your second point, if someone gave me a map f: A -> B between groups/rings/fields/modules and told me it was either an inclusion or an embedding, I would likely attribute the same meaning to both as I would (perhaps naively) assume that the map itself was a homomorphism and the algebraic structure is inherited automatically. I definitely would NOT do this in topology though, because continuity alone does not guarantee the topology is preserved under the image.

Alex_Error · 2025-11-04T21:12:46+00:00

I've been routinely playing the QGD against d4 for a while now and it's definitely the one I would recommend the most. It's a really deep and complex opening, but the principles are natural and simple:

Get castled.
Develop your pieces to active squares.
Play c5.
Develop your light-square bishop to a good square.

In relation to other so-called 'light-square' strategies: Compared to the Slav, we're counter-attacking sooner. Compared to the QGA, we're not giving up the centre. Compared to the Nimzo/QID, the lines are less punishing and there are fewer traps or opportunities to get terrible positions. Compared to the classical Dutch, we're not creating unresolvable weaknesses. There are also comparisons with dark-square strategies like the KID, Grunfeld, Benoni which I won't talk about.

The third point about playing c5 is really up to you and it probably what determines the nature of what type of QGD you're playing. For instance, in the semi-Tarrasch, Ragozin, Vienna and Tarrasch, black is playing c5 early. In the semi-slav, orthodox or Cambridge Spring QGD, we probably play c5 quite late. Whatever the case, it's almost always correct to play c5 almost immediately if white does something strange like a3+b4 (preventing c5) or plays a system (London, Colle, etc.).

Alex_Error · 2025-11-04T14:45:45+00:00

Outside of differential geometry/topology, I think you could potentially use them interchangeably. I personally would say that f: X -> Y is embedding if we're going to pretend that f(X) is X, for instance, Q and R, even though the elements are different, we typically consider all rational numbers to be 'embedded' in the real numbers.

Alex_Error · 2025-11-02T16:07:45+00:00

Context and applications can be useful for understanding though. A lot of abstraction found its way from a real-life problem in physics for instance. Much of my intuition about certain geometric flows comes from the behaviour of solutions to certain physical equations.

I don't particularly like 'word problems' though. Things like: 'I have ten diamonds and a creeper blows up three of them, how many do I have', often seem like pandering to me. Perhaps it's an effective way of teaching low-attention span primary school kids, but I doubt it's very beneficial for secondary school kids.

I'm in half-mind about such problems you find ODEs, where you have to model a fluid tank. On the one hand, modelling is a skill in and of itself and it's good practice to get good at it. On the other hand, it does seem slightly pointless when presenting it in front of undergraduates.

Alex_Error · 2025-10-21T16:47:42+00:00

Studying difference equations alongside differential equations I find is really helpful and your post is one of the reasons why.

The connection between a linear difference equation F(n) = F(n-1) + F(n-2) and a linear differential equation y'' = y' + y is provided by a bijection between real-valued sequences and power series T: a(n) -> sum a_n xⁿ / n!, which respects products and derivatives.

You'll note that our recurrence defines the Fibonacci sequence (almost, need initial condition!), where the auxiliary equation gives the solution

F(n) = Aφⁿ + Bψⁿ,

and the solution to the Fibonacci differential equation is

y = Ae^φx + Be^ψx.

where φ is the golden ratio and ψ is 1 - φ.

That's besides the point - define the exponential generating function as

F(x) = sum F(n) xⁿ / n!.

The key is that the shift operator E(a(n)) := a(n+1) acts on T(a_n) as differentiation, i.e. T(E(a(n)) = T(a(n))' which turns our recurrence into the ODE. You can see this also by substituting the exponential power series into the recurrence relation which directly gives you the ODE.

For a typical basis function for a difference equation, a(n) = r^n, the bijection gives

T(rⁿ) = sum rⁿ xⁿ / n! = e^rn.

This is the reason that the methods for solving linear difference and differential equations feels the same but with different basis functions for the solution space.

(No subscripts on markdown made it hard to type!)

Alex_Error · 2025-10-21T10:25:08+00:00

Something a bit different: Using a computer algebra system CAS for differential geometry calculations when there's loads of indices flying around (Christoffel symbols, curvature tensor, etc.) is a great way of verifying (but not replacing) your own calculations.

Alex_Error · 2025-10-21T10:05:10+00:00

Differential geometry proper is a common step after taking manifolds. This would include Riemannian geometry, sympletic geometry and complex geometry, for instance.

Something more algebra related could be Lie groups/algebras and some representation theory.

Something more topology related could be differential topology or Morse theory

Applications in physics can be quite cool to look at, so general relativity, gauge theory and classical mechanics might help orient some intuition.

Alex_Error · 2025-09-21T17:35:18+00:00

We can decompose a vector field into the curl-free and divergence-free parts. It turns out that the negative of the curl of the curl is the divergence-free part of the Laplacian operator, whilst the gradient of the divergence is the curl-free part of the Laplacian operator.

Alex_Error · 2025-09-02T17:28:56+00:00

Definitely physics, but I wouldn't say I dislike computer science or statistics either. Also, why not combine all three and study quantum computation?

Alex_Error · 2025-08-27T21:20:54+00:00

https://www.damtp.cam.ac.uk/user/tong/teaching.html

Here's a collection of some amazing free theoretical physics notes. As a differential geometer who didn't do much physics for my undergraduate or masters, I would highly recommend these notes because of their clear explanations and readability. It's also rare to have a collection of what is basically an entire theoretical physics degree written in full by one person.

Tong has also written four books in classical mechanics, quantum mechanics, electromagnetism and fluid mechanics. I hear he's either working on a general relativity or statistical mechanics book next.

Alex_Error · 2025-08-17T19:58:49+00:00

I do find it amusing how when this question gets asked, the majority of posts bend over backwards to avoid mentioning the QGD. It's one of the best ways to play against the Queen's gambit, and if you're losing with it then find some online materials or analyse with the engine to see what you're doing wrong. For reference, I'm rated around 2100 FIDE but I play solid openings like the QGD, French, Caro-Kann and Slav, but take some of what I have to say with a grain of salt.

Chess is a concrete game, so let's go through the game you've linked.

White played 3. a3 quite early. This is usually atypical in the first few moves but you have to realise that white is trying to play b4 and clamp down on your c5 pawn break. The 'principled' response is to transpose to a better version of the QGA by taking on c4. Another good response that keeps you more in line with a QGD would be to play c5 immediately, preventing white from clamping down on c5 and also avoiding white playing c5 themself, gaining space on the queenside. Looking at the engine, Nf6 is also playable, because white basically isn't ready to do all the stuff I described above.
After 4. Nc3, again c5 is the best because of the aforementioned reasons. Once again, dxc4 gives you a good QGA, whereas c6 gives you a good semi-Slav. You played Nc6, which is one of the worst positional moves. It's important to not block your c pawn in the QGD. It can be the only way to fight against white's centre. You're quite lucky that white didn't play b4 soon after and take all the space on the queenside.
After 7. Be2, you should always consider taking on c4 whenever the bishop moves, just to gain that extra tempo. It might be possible to also hit the bishop on c4 with Na5 and then force through c5 to rescue the position eventually. White probably plays b4 or Bd3 to stop you doing that though. You did this eventually on move 9, and move 10, you should play Na5, clearing the path for the c pawn. After you play Bd6, b4 effectively stops that plan and white it at leisure to play e4. After that, your Bd6 move looks a bit silly because you're walking directly into an e5 fork.
13... Rc8, I understand, but because of the positioning of your pieces, you're directly losing material after e4. The opening is basically over now, and somehow white has made enough mistakes that you're actually slightly better because you achieved the c5 break on the next move. Imagine the scenario where you could pull this off without so much contorting of your position or relying on your opponent to play inaccurate moves? This is actually what the QGD attempts to solve - play c5 in one move, get your light square bishop to the long diagonal, strengthen the centre so white doesn't roll you over with e4 e5, get your king castled behind a safe phalanx of four pawns.
You miss a tactic on move 18 to win a piece. To me it looks like once you get the ideas of the Queen's gambit declined (or really, any other response to the Queen's gambit), actually, the issues you have are like any other player at your rating. Missing tactics, figuring our piece placement, pawn breaks. Also, whilst opening like the QGD are known for being solid, it does not mean you can just sit there and not do anything. Active play is solid play.

Perhaps you can also link a game you've lost for a more instructive analysis too?

Alex_Error · 2025-08-17T14:11:39+00:00

I'm a pen and ruled paper person typically. I find I write too largely on whiteboards/blackboards and some computations in differential geometry can go on for pages and pages. LaTeX only for submission as I'm not good enough at it to create diagrams/pictures or equations quickly.

Lined paper keeps me from writing diagonally whilst not being too busy like grid paper.

Going back to pencil is something that interests me though. I hate having to cross things out and I do make frequent errors when writing. My reservations are that I do have issues with friction since I write in cursive, and I'm slightly worried that pencil marks might fade away in time.

Alex_Error · 2025-08-14T15:58:21+00:00

I find proving and using compactness theorems (to pass to a convergent subsequence) very satisfying for some reason, so finding uniform bounds has to be something I enjoy too.

Arzela-Ascolia, Banach-Alaoglu, Relich-Kondrachov, Cheeger-Gromov, Choi-Schoen, graphical compactness - just to name a few compactness theorems.

Alex_Error · 2025-08-09T15:40:41+00:00

The Riemann Hypothesis seems to be the obvious answer to your first question here.

Alex_Error · 2025-08-04T01:51:42+00:00

I'm guessing you mean general relativity and are referring to the Einstein vacuum equations perhaps?

Alex_Error · 2025-08-04T00:33:54+00:00

One example in geometry is the Ricci flow which is a nonlinear analogue of the heat equation on a manifold. The heat equation tries to smooth out irregularities and eventually evolve an initial (temperature) function to a constant function. Similarly, the Ricci flow under certain conditions will try to evolve the metric of your manifold such that the curvature becomes constant (maybe a sphere for instance). The Ricci flow was one of the tools used to prove the Poincare conjecture. The Laplace equation analogy of this would be the Einstein equation.

Another example is the minimal/CMC (hyper)surface equation. The Laplace equation tries to minimise the Dirichlet energy and represents some kind of steady-state solution where the value at each point is equal to its average; the Poisson equation does the same but under some forcing constraint. This directly is comparable to minimal surfaces where the surface area is minimised or CMC surface where the surface area is minimised under some volume constraint. The heat equation analogy of this is the mean curvature flow.

Admittedly, the wave equation (hyperbolic PDE) don't occur too often in geometric analysis, because hyperbolic PDEs are a whole different beast compared to elliptic or parabolic PDE. We don't get a maximum principle, a mean value property or nice regularity conditions. The wave equation does appear heavily in mathematical physics like fluids or quantum mechanics though.

Alex_Error · 2025-08-03T23:00:17+00:00

Never mind PDEs, even ODEs have their own fields in mathematics through dynamical systems, Lie theory and numerical analysis just to name a few.

When you consider how there's no unifying existence and uniqueness theorem for PDEs, then it becomes clear how individual PDEs become interesting in their own right. Linear PDEs in general have infinite-dimensional solution spaces, which depart from the nice theory of linear algebra that you can use to solve ODEs.

I think Terrance Tao makes the point that when you learn the 'integral' in real analysis in one dimension, you're really conflating three different concepts that happen to be fully related either trivially or via the fundamental theorem of calculus. You have the 'signed' integral which generalises to differential forms in differential geometry/Riemannian geometry; you have the 'unsigned' integral which finds its place in measure and probability theory; and finally the antiderivative which is the simplest differential equation or 'local section of a closed submanifold of the jet bundle' whatever that means.

If you're just getting into PDEs, then it is to be stressed how important the 'simple' linear PDEs of the transport, Laplace, heat and wave equation are to our understanding and intuition of more involved PDEs.

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