Doing Mathematics

My personal Substack

Why do I do this? I’d enjoy if more people did this. I’d enjoy if the experience of doing math were both more social and asynchronous, so people can linger on concepts they want to linger on. Unfortunately, I haven’t actually found many webpages where other math students chat math.

I’m also a fan of externalized cognition and exobrains that actually sit on the outside of the body. My conception of the interior / exterior divide is that Obsidian/local storage/etc. are all ‘interior’, and the public interwebs are ‘exterior’. I think it’s good to externalize cognition to help contribute to LM training data — it will give LMs another source of content on ways students/humans get stuck and how to help me in particular get unstuck.

https://ceselder.substack.com/p/i-am-the-open-source-woman

"The most unremarkable of events. Jerome Morrow, Navigator First Class, is about to embark on a one-year manned mission to Titan, the fourteenth moon of Saturn. A highly prestigious assignment, although for Jerome, selection was virtually guaranteed at birth. He's blessed with all the gifts required for such an undertaking — a genetic quotient second to none. No, there is truly nothing remarkable about the progress of Jerome Morrow — except that I am not Jerome Morrow."2

Here are three notes I have on the movie.

1. The pettiness of Vincent’s ambition

2. Conditioned on imperfect technology,

3. Vincent vs. Cluely

The pettiness of Vincent’s ambition

Vincent has a heart condition that pegs his death at 30. When he leaves Earth, he’s 31. There’s a decent chance he dies on the one-year mission, leaving his crewmates with an unexpected, traumatic corpse to blast into the ether. Chilling.

But this doesn’t really matter, because there a lot of astronauts on his mission — presumably slightly more than needed?

Had Vincent’s role actually mattered — had he taken up a role critical to some project’s success — it would be much harder for us to sympathize with him as he hoovers up space center resources and assumes the role.

There are two types of ambition / aspiration. The first is to alter one’s position in the rocket ship / change the role one plays in society. The second is to alter the rocket ships that get launched / change what gets done and accomplished overall by society.

Because astronauts are relatively fungible and redundant (there’s no short supply in this technotopia), Vincent’s ambition is relatively petty, like a modern New Yorker’s:

New Yorkers are ambitious in a very solipsistic way–like, people there have dreams, but in a very “this is my dream for me” way. I think the people who have dreams for the world are more so in the Bay Area […].

A ‘shaping’ role — like that of Gattaca’s director, really, who cares very much about getting more spaceships launched — might’ve been more appropriate for someone in Vincent’s position. He could have a good run shifting resource allocation and ‘growing the pie’ while alive, then die abruptly without leaving anyone in the lurch.

God forbid his mission actually rely on him.

Conditioned on imperfect technology,

The movie relies on imperfect genomic prediction as a plot device:

• Director: “You won’t find a violent bone in my body” — yet murders.

• Vincent out-swims his brother Anton — twice.

• Vincent passes the intense ‘Navigator’ training regimen.3

• Vincent — supposed to have died — but he hasn’t.

• The civil servant isn’t shocked when Jerome shouts at him for testing him.

I think the significantly more interesting question is this:

What kind of interesting movie could you make ‘conditioned on perfect genetic-prediction technology’?4

Vincent vs. Cluely

Cluely (got to note it down now, because the company might not exist when some read this) is an early ‘AI meeting assistant’ that helps people ‘cheat in interviews’.

Vincent cheats in his interview by submitting a sample of Jerome’s urine for genetic screening. And ever after by submitting samples of Jerome’s blood, hair, irises, etc..

For a long time, we’ve been using screening tests and hoping they generalize off-distribution, i.e. in the job. Cluely is essentially adversarial pressure towards using more in-distribution tests — tests that better reflect tasks done on the job.

Vincent would not have needed Cluely: he was behaviorally identical to the science-children, merely genetically distinct.5

I think Cluely only makes any sense at all in a world where two people can be distinct on a screening gate but behaviorally indistinguishable on the job.6 That’s the world Vincent lives in —

modulo his life expectancy. The issue in Vincent’s case is he’s not actually obviously unfairly discriminated against — the ‘expected death at 30yo’ thing is a bit damning for an astronaut, and implies he prooobably shouldn’t have done what he did. But had it really solely been a case of society underestimating and discriminating against him, it would be almost trivial to cheer for him.

We of course want to fix screening gates in those worlds.

5

He lives in a world of ‘non-identifiability’: one where two or more genomes (Jerome Morrow, Vincent Freeman) are observationally equivalent.

Block One (130 mins): Classification, Canonical Form, General Solution

I think, when I first saw Q3(a), I was confused. What were all those first order terms doing on the RHS? Now I note I only needed to worry about the LHS — the principal part. But I wasn’t clocking that at the time.

Also, I was frightened, because I’d never needed to find regions of the plane where the equation was hyperbolic / parabolic / elliptic. But armed with a better understanding of the principal part (the a(x,y)u_xx + 2b(x,y)u_xy + c(x,y)u_yy formula, this has been easy. :))

Now all that remains is to learn what a ‘characteristic equation’ is…

…OK, done. We replace u_xx → (dy/dx)^2, u_xy → (dy/dx), u_yy → 1.

We set dy/dx = λ, solve the characteristic equation for λ, then solve the resultant ODEs to get expressions linking y, x, and constants. We set our new ‘characteristic variables’ equal to those constants.

Finding u_x and u_y was easy enough with the chain rule. When finding u_xx, u_xy, u_yy, I needed to remember to apply the product rule, so we’ll end up with eight terms, including some lower order terms.

By chasing through with substitution, we eventually end up with u_φψ = (RHS consisting of lower-order terms).

When it’s u_φψ = 0, we say the general solution u(x, y) = g(φ) + h(ψ).

This is all fairly methodical and a lot. I’ll need to look later at cases where u_φψ != 0, plus maybe parabolic and elliptic situations.

I must’ve done the chain rule method previously, because it came fairly naturally to me. It’s definitely computationally heavy, though — the kind of thing to regard as ‘free marks’ but consider doing later in the exam.

It’s quite clear that when it comes to Differential Equations, there’s no substitute for solving a problem of each type (Picard-methods, plane autonomous systems, method of characteristics, canonical/normal form) in the days leading up to the exam. The questions have fairly clean ‘type-checks’ with corresponding methods.

Next session:

• Targeted practice: u_φψ != 0, parabolic + elliptic questions

Block Two (110 mins): Fitting general solution to Cauchy data

Part (c). Felt really quite convoluted. I got expressions for g and h, g’ and h’, then differentiated the expression for g and h to get a linear system of equations over g’ and h’ that I could solve. I had to initially parametrise the data curve, and I applied a new mode of thinking: ‘if f_1(x) + f_2(y) = const.’, x increasing must cause y to do what to maintain the constancy / level set?

Fortunately, I’ll have a block towards the end of the day where I try redoing Q3 in full, so we’ll see what stuck. I might try redoing Q1 in full, too, to see if anything’s stuck from the previous session on Q1 (which I didn’t write up).

Banked 4 hours by 16:10…that’s new.

Block Three (60 mins): Data on characteristics

Well, I reviewed the principle that when a data curve equals a characteristic curve (i.e. a curve where φ or ψ is constant), we’ll end up with a situation where we can query some g or h only at one value, and either the function will freely vary off that value (giving infinite possible solutions — the data does not uniquely determine the solution), or the data will make incompatible demands of the function (giving zero possible solutions).

Meta:

• need to be covering more ground for pattern recognition / building the gestalt to do its work, not getting bogged down...one question in one day is utterly unacceptable.

• by the end of Week 2, I should have studied all content in the courses and switched wholly into revision. DE1 gapes on the ‘To Do’ list, followed by a few lingering sections of Complex Analysis.

Study Plan (fungible timings, TBD):

Sunday morning — MS qs for Alex

Sunday afternoon — Testimony/K+R, True Names blogpost, TAIGR review

Monday morning — MS+CA collection review

Tuesday + Wednesday — all unseen DE1 content (Maximum Principle, Gronwall’s Inequality, Picard iterates, continuous dependence on the initial data, Comparison Principle)

Thursday morning + Friday — all unseen CA content (Argument Principle + Rouché, Jordan’s Lemma, cot trick, sector contours, Morera’s theorem)

Saturday — SVD + any unseen / highest-priority LinAlg

Block One: MGFs! (2 hours)

I spent a bit of time with the ideas from this problem. Thankfully, finding MGF of Exp(λ) comes v naturally to me now. But I hadn’t done much ‘CLT-rearranging’ in the style of (c), and the only MGFs I’d found had been via integral, not discrete sigma in the style of (d). It was valuable to look through examples of (c) and (d); what remains to be seen is whether I can apply the same principles on other problems from the same distribution.

Next session:

• Markov / Chernoff inequalities & bounds

• CLT via MGFs; Normal MGF

• 2024 Q1(a)(ii) — same trick, geometric → exponential, slightly different scaling. Cleanest second example.

• 2017 Q2(b)(ii) — log of MGF, expand to o(t²), sum. The CLT version of the same move.

• 2024 Q1(c) — log MGF expansion, summed with 1/i weights.

Block 2: Markov chains! (45 mins)

I think I was able to solve most of this problem in exam conditions. Now I’m going to look at the parts I struggled with.

…I worked through it all again, and I was able to solve everything except a tiny latter part of Q3(b). I think it’s because Alex reminded us that “when you’re doing the latter parts of a question, you can use the earlier parts as though you’ve done them in full” — and that helped me identify the commonality between the situation in (c)(iii) and part (b). :)

Drilling ‘recurrence relations’ (through random-walk contexts they appear in) feels like a very easy, tiny task compared to the rest of Probability. Nice! It was lovely exploiting symmetry lots in this problem. Markov chain problems always feel like fun brain-teasers that barely require mathematical maturity, just like plane autonomous systems.

Block 3: 100 minutes

I worked through this. I hadn’t looked much at joint / marginal / conditional distributions before, or really covered / looked at the bivariate / multivariate Normal material in my notes. I also didn’t internalize covariance super well in Prelims. This was a lot of content, and I’ve found myself pretty tired. I’ll see how much I remember / how much is re-applicable on similar problems.

Next session:

• Learn Gamma distribution properly, properly get drilled on distributions!

• Bivariate / multivariate normal — important to know properly. Special property: uncorrelated implies independent.

• Multinomial distributions

I went down to Cambridge for three years to study law. […] It was melancholic in a way, because all my friends became corporate lawyers and were banished off the face of the earth into corporate law firms. I don’t speak with them a huge amount.

I’ve been rattling around Oxford for a while. So has my friend Malaika. We’re both going to San Francisco afterwards.

I argue too many people focus on ‘People in the same place as you’ when really they should spend more time with ‘People headed in the same direction as you’.

I think it’s important to look around whichever environment you’re in, work out if there are people headed in the same direction, and stay only lightly invested in the environment if not1.

I hardly speak to anyone from my two years in Hong Kong due to heterodirectioneity.

There are obviously caveats (some bounded programs are good), but generally I think that neglecting ‘People headed in the same direction as you’2 leads to sadness, especially after multiple moves.

Good luck, young Riverians!

Block 1 (105 mins): Cousin of Liouville, FTA

I worked through Q4(a) and (b) with Claude. Q4(a) was pretty easy, because it mirrored the proof of Liouville’s Theorem, using Cauchy’s Integral formula with ML inequality to give an upper bound on some quantity. I quickly reviewed the fact of γ(t) = re^2πit, t ∈[0,1] giving a circle-path — I need to be able to reproduce that myself.

Part (b) was a bit trickier, because I never learned Triangle Inequality / Reverse Triangle Inequality super well the first time round (in Real Analysis), and haven’t looked at Jordan’s Lemma properly yet (which apparently also uses the Dominant Term - Error pattern). But I got through it!

Next session:

• Using Cauchy estimates to bound polynomial-growth entire functions

Block 2 (75 mins): Logarithmic derivative + half-plane geometry

I worked through Q4(c). The d/dx (log(f(x)) = f’(x) / f(x) felt good to review, as did some basic conjugate identities, like (a + b)-conjugate equalling a-conjugate + b-conjugate, and complex number times its conjugate equalling modulus squared, a real number. The analytic-geometric back-and-forth was good to study too, even if the UHP-intersection-affine-transformation stuff seemed niche for this course. I’m glad I worked through this, though I needed a bunch of Claude’s assistance.

Block 3 (2 hours)

Keyhole contour. Exhausting and confusing and great. Lots of use of Reverse Triangle Inequality to get tidy bounds (for when R → 0 or ε → 0) involving denominators. Lots of use of polar form. Properly learned how to use branches / principal branches / etc., which was great. The only way to see what’s stuck will be to try another of these in a future session.

Meta

Five hours today, presumably because I stopped for a great Epistemology tute mid-afternoon. Should factor in, perhaps, that hours spent in tutes (cognitively intensive?) detract from the ‘6h = full day of math’ plan.

I’m trying for 6h of math (split roughly into three 2h blocks) on default deep-work days, with at least 2h on any given day.

Next session:

• Argument Principle + Rouché

• Jordan’s Lemma

• Drill contour integration via residue computation

  • Keyhole contours

Block: 120 mins

I encountered this problem in an exam, and I was stumped by (c) and (d).

I knew my Lecture Notes had a section on ‘Conformal Maps’ I’d long been confused by years ago. In fact, the main kinds of conformal maps they focus on are Möbius maps, which map C∪{∞} to itself by using Ψ(z) := az+b/cz+d, taking (a b / c d) from GL_2(C). This is already confusing because Ψ looks like a menorah and I never studied Group Theory super intensively. I do remember that groups have identity, inverses, and associativity, though.

I spent some time reading over the Lecture Notes, guided by Claude, and then re-attempted this problem.

I feel like I understand the definition, the compositionality / homomorphism property, but there’s clearly a lot here to pick up in terms of images, bijectivity, etc..

The only thing for it is to try more problems from more papers :)

1st block (100 mins): Picard’s Theorem

OK, OK, Claude, maybe you’re right: I can’t ignore Picard’s Theorem any longer.

But, ach, it’s presented so badly in Lecture Notes. That’s OK: I’ll apply a lossless encoding.

I love the ‘cone presentation’ of the Lipschitz property.

And now, we have a ‘bow-tie presentation’ of condition P(i)(b): Mh ≤ k!

It helps to keep in mind what question Picard’s Theorem is an answer to: the question of whether IVP y’(x) = f(x, y(x)), y(a) = b has a unique solution on some interval surrounding x = a.

I’m happy to be building multivariate intuition.

I then invested some serious time understanding how to show ‘Lipschitz in y’ vs. ‘not Lipschitz in y’. This was some of the most straightforwardly fun math I’ve done in a long time.

Next session:

• Picard iterates

  • 2025 Q1(a) and (b)(i)–(ii) — the y^(1/3) example plus the modified Picard iteration for monotone non-Lipschitz f. Part (a) alone is worth 5 marks and is very doable. Parts (b)(i)–(ii) are the monotone-convergence argument; if (b)(iii)–(iv) are squeezing you, just read the statements and sketch the ideas rather than executing them fully.

• State Gronwall cleanly

• Handle the “show f isn’t Lipschitz, construct non-unique solutions” setup

  • 2018 Q1(a) as warmup — state Picard, show f(x,y) = x√(y+1) isn’t Lipschitz on |y+1| < k, construct two solutions through (0,−1). This is the canonical pattern.

2nd block (90 mins): Plane autonomous systems

My notes tell me I solved 2024 Q2(a) previously, but looking at 2024 Q2(a), I don’t remember this…must’ve been a different problem IMO. Today’s problem (2025 Q2a) should’ve been simple, but I had to work hard at it, maybe because I’m tired. I studied sketching the direction along nullclines, how not every nullcline-intersection is a critical point, and so on.

I remember stable/unstable nodes and saddle points pretty well, but I have to refresh my memory re stars and inflected nodes (depending on whether M = Iλ) and stable/unstable spirals and centers (depending on a in a+bi).

All the ‘Next session’ points from my last study session (polar coordinates, Bendixson-Dulac, periodic orbits…) still stand.

Meta

I also wanted to review method of characteristics today. But I think I procrastinated a little too long — I had trouble launching into work after lunch and after dinner (14:00-16:00, 18:30-20:30 (+ 20:30-21:00 call → decompress). Really, I worked 16:00-17:40 and 21:30-23:00 in the end.

If I treat my study days more like Markov chains, with 3-4 non-negotiable study blocks / 6 hours of material to get through, it seems more likely I’ll get through the entire plan. Relegating phone calls to weekends / no checking.

NB I sat Probability exam in the morning.

I’ll review method of characteristics 8:00-9:30.

Contour Integral Classification (2 hours + 15mins)

I kicked off with a round of contour integral classification — not solving, yet, just classification.

Takeaways:

1. I want to be applying Euler’s formula all over the place. E.g.:

   1. cos(2z) = Re(e^2iz), cos(z) = Re(e^iz), …

      1. This is needed to ensure decay on the arc.

         1. cos(2z) = (e^2iz + e^-2iz)/2 contains e^-2iz, which is problematic when we sub z=x+iy. But e^2iz behaves nicely.

2. Seriously, I even want to be applying Euler’s formula to convert back from ‘sin/cos in terms of exp’ to z-notation.

It’s not concerning that the denominator fails to vanish in trig cases — if it did, the integrand would blow up on the real line and the integral wouldn’t exist.

Finding singularities in the trig case is a lot more involved. We need to apply the full substitution, then rearrange and we’ll end up with some function of z in the denominator that we can compute the roots/zeros of. This gives singularities. I worked through Q5 and identified singularities at 3i and i/3 — of course, just i/3 would lie within the unit circle.

I then had to find the residue at i/3, and felt a bit stumped. ‘Laurent expansion?’ I wrote. But this is overkill. I should in fact return to Method 1, limit definition for residue computation — it is not so hard. This allowed me to find a residue of -i/4 :-)

3. cos is an even function, i.e. cos(θ) = cos(2π−θ). This is very convenient!

   1. It means that when I see some integration from 0 to π involving cos, I can just integrate from 0 to 2π and halve the result (provided the rest of the function is even as well).

   2. Or when I see some integration from 0 to ∞ involving cos, I can just integrate from -∞ to ∞ and halve the result, by evenness of cos(-x) = cos(x) (again, got to check rest of the function is even as well).

   3. This doesn’t work with sin because sin is odd.

4. When I see a multi-valued function like sqrt(x) or log(x) or x^1/3, I need a branch cut, which means keyhole.

   1. Wait, z^1/n is multi-valued. Why isn’t branch cut / keyhole contour an issue for z^1/2? → because we hide the branch cut in the lower half-plane and never confront it!

5. Computing roots of unity — I comprehended the e^(iθ + 2πik) part, and needed to recall that theta=0 in the case of 1 (for 1 = 2πik), and we’re solving for theta in the case where e^iθn = e^2πik, hence θ=2πk/n.

6. Principal branches

1. 7. Sector contours exist.

   1. They’re used for rotational symmetry of the integrand, i.e. ‘rotating z→(e^iα)z leaves the denominator invariant for some nice α’.

1. 8. In semicircular contours, our objective is to kill the arc. First, we reach for ML inequality. If that doesn’t work, we reach for Jordan’s lemma — in cases where the integrand only decays like 1/∣z∣ but has an oscillatory factor e^iaz with a>0.

   1. When applying ML inequality, we’re going to sub z → R.

   2. ML inequality will fail in cases like 1/R, so we’ll need Jordan’s Lemma.

Next session:

• 1.5h cleaning up contour integrals (review cases 9-12, understand multi-valued function/branch cut/keyhole contour cases, understand sector contours, understand use of ML inequality/estimation lemma for semicircular contour, and do the sector contour problem, write out decision tree pipeline for what method/technique to use upon seeing various contour integral types)

Compactness <=> Function Spaces (2 hours)

Learnings:

1. Norms: I reviewed that sup-norm is the same as ℓ∞-norm on function spaces. I realized there are a lot of kinds of norm I’ll need to get comfortable / practice with:

2. Uniform continuity & continuity: I reviewed the definitions of uniform continuity and continuity, recalling that ‘continuity’ is defined at a point, with a ‘merely continuous’ function continuous at each point.

   1. Continuous image of a compact metric space in R is uniformly continuous. (We prove this through proof by contradiction on a sequentially compact metric space, i.e. ‘consider a function continuous but not uniformly continuous on a compact metric space…’

      1. I also need to review ‘uniform convergence’ (separate concept).

3. In function space, functions are ‘points’ in the metric space! You can literally have a ball centered on a function, containing functions within some distance of it. This was a viewquake.

The ‘point’ is primitive and many things can be points.

‘Ball’ is legacy from spaces like R^n where this makes sense. ‘Balls’ may be considered relatively ~formless/shapeless. What’s important is that it’s the set of ‘points’ at most ‘distance’ from our ‘center’ point.

In the case of ‘totally bounded’ metric spaces, we can define/specify the cover. Then, for any point (function) in the space, there exists some function it’s in the ball of!

All this machinery allowed me to tackle this problem from TT22:

It definitely stretched my brain, which is good — principle ‘hit the parts that hurt’. NB we only use uniform continuity of the ‘anchor functions’ from the specified set of balls that totally bound the space, not the arbitrary function itself! — I understand this now…

4. ‘Bounded’ only implies ‘totally bounded’ in finite-dimensional spaces

Be careful not to port the wrong intuitions over! C[0,1] is infinite-dimensional!

Claude was absolutely spot-on that I needed to review function spaces!

Next session:

• 1h cleaning up uniform continuity + homeomorphism non-existence arguments via topological invariants

Claudejitsu

• I ask questions like “What do you think is the highest-priority Metric Spaces material I'm missing right now?”. The answers often surprise and help me.

• I feel so supported and encouraged when I work with Claude.

  • i literally weep with joy that every motivated child in the world gets a companion who grows alongside them today

• One huge blessing of GPT/Claude is an end to marking, forever. This is…blissful and incredible for educators. I feel almost breathless considering the delta and how many humans have marked my and others’ work over the years (thank you)

First block (120 mins): Liouville’s Theorem & Identity Theorem

Liouville’s Theorem

Statement

I should’ve written ‘Let f : C → C be an entire function’ rather than ‘Suppose f is holomorphic/complex differentiable on all C’. Entire is the technical term, and I need to always specify range as well as domain.

Proof

I remembered that it routes through CIF, and even that I’d need to take the distance between two points |f(w_1) - f(w_2)|, and bound above using Estimation Lemma/ML-inequality with an expression tending to 0!

Here’s what I needed to add:

• Specify the circular path centered at the origin with radius 0

• Use |f(w) - f(0)| i.e. set the second point as 0

• For ‘M’ in the ML-inequality, bound above with the explicit supremum on the circle where |z| = R

  • This permits w → |w|, f(z) → M, z → R

This yields an upper bound that tends to 0 as R → ∞. Pretty cool :)

Identity Theorem

Statement

I hadn’t seen the Identity Theorem before, but its statement was fairly intuitive to me: under conditions of holomorphicity, equal somewhere means equal everywhere.1 I ended up glancing at it, then writing this:

Suppose there are two functions, f_1: U → C and f_2: U → C, that are each holomorphic on all U. Consider S: {z in U : f_1(z) = f_2(z)}. If S has a limit point in U, then S = U.

The only additional thing I needed to specify was that U be a domain, i.e. open and connected.

I also need to know the local structure of zeros of holomorphic functions [on open sets]. i.e. that these are either isolated points or surrounded by zero-neighborhoods.

Proof

I wrote up the Identity Theorem’s proof in my preferred structure, and afterwards was able to reproduce it from memory. :)

Second block (90 mins): Cauchy-Riemann, Casorati-Weierstrass, Residue Theorem

Casorati-Weierstrass Theorem

Application

We explored the case of f(z) = e^1/z a bit. For arbitrary w, I rewrote s.t. z = 1 / ln(w), but then I got stuck — I don’t understand complex logarithms very well.

Rewriting in terms of complex logarithm, we get:

from definition of complex logarithm:

Now, as we let n tend to infinity, we get values of z arbitrarily close to 0.

Statement

After reading the Casorati-Weierstrass Theorem, I was able to reproduce it as follows (bold corrections added by Claude):

Suppose f: U\{a} → C is holomorphic on open set U with an essential singularity at a. Then for any r > 0 with B(a,r)⊆U, f(B(a, r)\{a}) is dense in C: i.e. for any ε > 0 and z_1 in C, there exists z_0 in B(a, r)\{a} ⊆ U s.t. |f(z_0) - z_1| < ε. (The closure of the image of B(a, r)\{a} is C.)

Proof

I’m told to try proving by contradiction.

OK. Suppose the image is not dense near a, i.e. there’s some value z_1 ∈ C that’s not attained, some r > 0 s.t. z_1 is not a limit point of f(B(a,r)\{a}). It’s hard to see how that breaks holomorphic property of f or the existence of an essential singularity at a…

I need to know Riemann’s removable singularity theorem, i.e. that when U is open and z_0 ∈ U and f: U\{z_0} → C is holomorphic, z_0 is a removable singularity iff f is bounded near z_0. And I need to apply that to a different function, not f itself but rather g(z) = 1/(f(z)-z_1)…

I was able to reproduce the proof after writing it out in this my preferred format, though I migh further specify that f(z) could have a removable singularity or pole at a, contradicting the hypothesis that a is an essential singularity. And I always need to point out that B(a,r) belongs to U in the statement.

Third block (75 mins): Residue Theorem

After writing out Residue Theorem statement + proof in my preferred format, I was able to return to these and reproduce them in fourth block. I just needed to note the last part ‘by definition of the winding number’, and how we write Res_a(f) rather than Res_a(f)(z), since this is not a function of z, and clarify that the singularities are isolated. :)

Fourth block (75 mins): Residue, Cauchy-Riemann

I was able to reproduce Residue Theorem statement + proof.

Cauchy-Riemann equations came back to me quickly, as expected. I just need to recall that u, v are functions on R^2 and write u(x_0 + h, y_0)…accordingly.

We did a quick synthesis where we realized I lean on Laurent series a lot.

And that was it. That was my six-hour proof-day. I’ve started soft-capping study-days at six hours for sustainability, with mandatory rest. This is because you need rest & recovery in the evening to do well in the morning.

Next sprint

1. Jordan vs unit circle discrimination: Mixed batch of 12 integrals — some real-line with cos(ax)/sin(ax), some [0, 2π] with cos θ/sin θ, a few sector/keyhole disguised to test the diagnostic. For each, you classify before computing: which contour, which technique, why the others fail. Then fully solve four of them. Goal: diagnostic becomes automatic.

2. Techniques medley: Reps on the contour techniques covered but not yet automatised, plus the argument-principle family. Structure: 2 keyhole contour problems (branch cut handling), 1 sector contour (you justify the angle from symmetry, not memory), 1 cot(πz) summation, 2 Rouché / argument principle problems (zero-counting). Quiz-mark format.

It feels pretty trippy that a bunch of these proofs I’ve just learned are bookwork 5-7-markers in exams. There’s a lot of residue and integral computation, and that’ll be the focus of any future Complex Analysis sprint.

Hours 1-2: Method of Characteristics

Solved a basic problem & investigated thoroughly where things break (blow-up in z at some t, discontinuity in data curve at some s, vanishing Jacobian for some s at t=0 or at some t)

Next session:

• Work through 2024 Q3 in full — hits every step including well-posedness

• Handle the case where the data curve has multiple segments with different Cauchy status (2016 Q3)

• Handle PDEs where the nonlinearity is in z (like z⁴ in the 2023 paper)

• Questions asking about discontinuities in first derivatives (2021 Q1(b)(iv)

• Drill the 3-variable adaptation from 2019 Q2(b)

• (Low-priority) Learn the general solution approach (finding an arbitrary function F and then using data to determine it)

Hours 3-4: Plane Autonomous Systems

I reviewed the ‘plane autonomous systems’ pipeline.

I reviewed how you can just read eigenvalues off the diagonal of any diagonal / upper triangular / lower triangular matrix.

I also needed to review solving (M−λI)v=0 to find the eigenvector (which encodes direction) for a given eigenvalue.

It’s also fine to scale a matrix by A, find eigenvalues of scaled matrix, and divide by A to get original eigenvalues. And scaling doesn’t affect eigenvectors!

I solved 2024 Q2(a).

Next session:

• Polar coordinate transformations for systems with circular symmetry — compute ṙ and θ̇, show that circles r = const are trajectories, determine stability of limit cycles (2022 Q3, 2025 Q2(b))

• Conserved quantities — when they ask "determine a function h(x,y) so that trajectories are level sets of h", you derive the ODE dy/dx = Y/X and look for an integrating factor or exact equation (2021 Q3(a)(ii), 2017 Q3).

• Bendixson-Dulac — state it cleanly (if ∂(φX)/∂x + ∂(φY)/∂y has fixed sign in a simply connected region, no closed trajectories exist), and know how to find φ. The corollary with φ = 1 (just checking ∂X/∂x + ∂Y/∂y) should be first attempt. Work through 2017 Q1(b)(iii) which requires constructing φ from a hint.

• 2024 Q2(b), which has a parametric family

• (Low-priority) Review & understand ‘linearization’ rigorously

False start: Picard’S Theorem & ODEs

I spent 25 minutes looking at Picard’s Theorem & its applications.

Plane autonomous systems: Picard guarantees trajectories never cross. But this is a one-line invocation (“by uniqueness from Picard’s theorem”), never a computation.

Method of characteristics: Picard guarantees the characteristic ODEs have unique solutions. Again, just cited, never computed.

Second-order PDEs: No real connection.

I then decided to pivot and spend the remaining 1.5 hours looking at second-order PDEs and the maximum principle. I’ll save Picard’s Theorem for May.

Hour 5: Second-Order Semi-Linear PDEs

Type A: “Classify this PDE” (2-3 marks). Read off a, b, c, compute b^2−ac, state the type. Appears in 2021 Q1(c)(i), 2018 Q3(a). Completely free marks.

(I can do this!)

Type B: “Find new coordinates (ϕ,ψ)” (3-5 marks). Solve the characteristic quadratic, integrate the ODEs. Appears in 2021 Q1(c)(ii), 2018 Q3(b). Mechanical once you’ve practised it.

(I can do this!)

Type C: “What’s the link between this first-order and second-order PDE?” (2-3 marks). The answer is almost always: if v solves the second-order PDE, then u = v_x (or v_y​) solves the first-order one. The characteristic projections of the first-order PDE are the characteristics of the second-order PDE. Appears in 2021 Q1(c)(iii).

Type D: Maximum principle application (5-15 marks). Prove uniqueness or continuous dependence for Dirichlet/heat equation. The template is: let u = u_1 - u_2, show Δu = 0 (or u_t − u_xx = 0), apply max principle to get u ≤ 0, apply to -u to get u ≥ 0, conclude u = 0. Appears in 2023 Q3(a) with a twist.

Type E: “Show the general solution is...” (5-8 marks). The full substitution into normal form, then solve. 2018 Q3(b) does this.

The central question is: given a second-order PDE and some boundary data, does a unique solution exist? That’s the well-posedness question.

The answer depends on the type of PDE, which is why you classify first. Different types need fundamentally different kinds of data — elliptic needs BVP, hyperbolic needs IVP, parabolic needs IVP forward in time. Give the wrong kind of data and the problem is ill-posed, no matter how hard you try.

The change of variables to normal form is how you prove the classification is meaningful — it shows that every hyperbolic equation locally looks like the wave equation, every elliptic one looks like Laplace, every parabolic one looks like the heat equation. So studying those three model equations tells you about all second-order PDEs.

The maximum principle is then the tool you use to actually prove well-posedness (uniqueness + continuous dependence) for the elliptic and parabolic cases. It’s not a separate trick — it’s the payoff of the classification.

So the logic is: classify → know what data is appropriate → prove the problem with that data is well-posed via the maximum principle.

IVP (Initial Value Problem): You’re given u and u_t ut​ at t=0, everywhere in space. Then the solution evolves forward. Think: you know the state of the whole system at one moment. This is what the wave equation needs — position and velocity of the entire string at t=0.

BVP (Boundary Value Problem): You’re given u on a closed curve surrounding a region, and you want to find u inside. No time variable at all — it’s a static problem. Think: you know the temperature on the walls of a room, and you want the temperature everywhere inside. This is what Laplace’s equation needs.

IBVP (Initial-Boundary Value Problem): A mix — you’re given u at t=0 (initial) and u at the endpoints for all t (boundary). Think: a finite bar where you know the initial temperature profile and the temperature at both ends is held fixed. This is what the heat equation on a finite interval needs.

The intuition for the table: elliptic equations describe equilibrium (no time direction), so they need boundary data all around. Hyperbolic and parabolic equations describe evolution (there’s a time direction), so they need initial data to get started, plus boundary data if the spatial domain is finite.

Hour 6: Second-order PDEs and the maximum principle. This is lower frequency as a standalone (≈4-5 papers) but often appears as a sub-part worth 5-9 marks, so it's efficient insurance. Know three things cold: the classification (ac > b² elliptic, ac < b² hyperbolic, ac = b² parabolic) and how to find the change of variables to normal form via the characteristic ODE dy/dx = (b ± √(b² - ac))/a; the maximum principle for ∆u ≥ 0 (statement and proof — the ε-perturbation trick with v = u + (ε/4)(x² + y²)); and the maximum principle for the heat equation u_t - u_{xx} ≤ 0 (same perturbation idea, with v = u + (ε/2)x²). From these, you derive uniqueness and continuous dependence for the Dirichlet problem and the heat equation IBVP — these are pure marks if you know the template. The 2018 Q3 energy method for the heat equation (multiply by w, integrate, use boundary conditions) is also a recurring pattern. Review the well-posedness table: which problem types (IVP/IBVP/BVP) are well-posed for which equation types.

Hour 1: MGFs

Moments: E[X], E[X^2], E[X^3], …

MGF: E[e^tX], because e^tX = 1 + tX + (tX)^2/2! + …,

so by linearity of expectation (since series of absolute values has finite expectation), we get 1 + tE[X] + t^2 E[X^2] / 2! + … . Can differentiate, set t = 0, and peel off moments.

I reviewed how to find MGF of Exp(λ), using the ‘integral of PDF = 1’ trick. I need to recall that Exp(λ) starts at 0.

I also need to get more used to Gamma(r,λ), which is just Exp(λ) multiplied by λ^r-1 / Γ(r) * x^r-1. Γ(r) provides normalization, and ‘r-1’ accounts for how much space there’s been for (r-1) events before the rth. Also runs from 0 to ∞.

I was able to find MGF of Gamma just fine.

I need to remember that in both Exp(λ) and Gamma(r,λ) cases, this only works if t < λ so that integrand decays and converges appropriately.

Next session:

• Drill: N(μ,σ²) → exp(μt + σ²t²/2), Uniform[0,1] → (e^t − 1)/t

• Know the uniqueness theorem statement cold (finite on [−t₀, t₀]) and the continuity theorem statement (M_{Xₙ}(t) → M_Y(t) implies X_n →^d Y).

• Then practise the trick: computing the MGF of a linear combination aU + bV of independents via M_U(at)M_V(bt). This appears on 2018 Q1, 2021 Q2, 2022 Q1, 2024 Q1, and 2025 Q1 — it's nearly guaranteed marks.

• Also know Theorem 3.4 from notes: M'(0) = E(X), M''(0) = E(X²), and the Taylor expansion log M(t) = μt + σ²t²/2 + o(t²) that drives the CLT proof.

Hour 2: CLT + Convergence

I reviewed the CLT proof, and put it into my preferred multi-step form.

I also reviewed the Uniqueness Theorem (if two random variables have the same MGF finite on [-t_0, t_0] for t_0>0, they have the same distribution) and Continuity Theorem (if a sequence of random variables’ MGFs converges to some random variable’s MGF, the sequence of random variables converge in distribution to that random variable, provided all MGFs finite on [-t_0, t_0] for t_0>0)

Next session:

• The underlying technique with log — taking log M_{S_n}(t/√n) and showing it converges to t²/2 — appears in disguise on roughly half the papers (2017 Q2(b)(iii), 2021 Q1(c), 2024 Q1(c), 2025 Q1(c)). Write out the proof once from scratch: standardise, expand log M_Y to o(t²/n), sum n copies.

• Drill the three convergence modes: almost sure ⇒ in probability ⇒ in distribution. Know the key facts: convergence in distribution to a constant is equivalent to convergence in probability (2022 Q1(a) asks exactly this).

• State the WLLN and SLLN precisely (2018 Q1(b), 2019 Q2(a)).

• Practise the Markov/Chebyshev inequality proofs — Markov's inequality and the WLLN proof appear as "state and prove" on 2018 Q1 worth 11 marks.

• Finally, know the Chernoff bound technique: P(S_n > na) ≤ exp(−na²/2) via MGF + Markov's inequality. The 2025 Q1(c) and 2018 Q1(b)(iii) both use this exact method.

Hour 3: Joint Distributions & Change of Variables

I reviewed the ‘change of variables’ result and broke it into three parts:

1. Define the map

2. Define the Jacobian

3. State the PDF (thanks to (U,V) being jointly continuous

I reviewed the "independent exponentials, let U = X/(X+Y), V = X+Y" trick that gives U ~ Uniform, V ~ Gamma independently (Example 4.2 in notes)” application also. I noticed how U[0,1] = U(0,1) because the probability at any given point is 0, and how when random variables match in distribution + (in)dependence, we can manipulate their distributions and say things about both, and so on. How joint continuity is required for RVs to have a joint PDF. Also had to spot Gamma PDF in the wild.

Next session:

• (2) the standard Exp(λ) pair → polar coordinates computation (2018 Q2(b))

• Drill the conditional density formula f_{Y|X=x}(y) = f_{X,Y}(x,y)/f_X(x) — it's asked directly on 2019 Q1(a)(iv), 2021 Q2(d), 2025 Q1(b)(iii)

• Watch the support carefully: the 2025 paper's X = UV, Y = U/V problem and Example 4.3 (V = X+Y, W = X−Y) both have non-rectangular support that breaks apparent factorisation. You already know independence requires rectangular support — make sure you can spot it under pressure.

At this stage, I noticed Poisson processes were no longer on the syllabus, and I’d need to drill Markov chains intensively, since they feature heavily in Q2 & Q3.

Hour 4: Markov Chains — Definitions & Hitting Probabilities

I spent a bit of time with the formal definitions.

Transition matrix — it’s a stochastic matrix, so not only does each row sum to 1 but also all entries are non-negative. Remember the quantifier i,j∈I by the side of P = (p_ij)_i,j∈I.

Markov chain / Markov property actually has a fairly elaborate formal definition, which I should internalize. It’s about a sequence of random variables and the values they take. We start at X_0. We index states by timestep.

‘Communicating classes’ are equivalence classes from a particular symmetric equivalence relation. They can be closed or open.

Communicating classes all have the same period:

To define recurrence and transience, I actually have to define the first return time. It’s infimum of all the returns.

Next session:

• Drill the hitting probability setup: h_i^A is the minimal non-negative solution to h_i = 1 for i ∈ A, h_i = Σ_j p_{ij} h_j for i ∉ A.

• Do the gambler's ruin calculation: h_i = ((q/p)^i − (q/p)^N) / (1 − (q/p)^N) for p ≠ q, h_i = 1 − i/N for p = q.

• Similarly for mean hitting times k_i^A: the recursion k_i = 1 + Σ_j p_{ij} k_j.

• Know the recurrence criterion via Σ p_{00}^{(n)} = ∞ — this is what 2024 Q2(c)(ii) tests, asking you to determine which states of a random walk on Z² are visited infinitely often.

• The Stirling-based argument for Z^d recurrence (p_{00}^{(2m)} ~ cm^{−d/2}, which sums to ∞ iff d ≤ 2) appears in notes Section 5.9 and is directly deployable on 2024 Q2.

• Also do the 2025 Q3(b) hitting probability question: a chain on Z with steps +2 (probability p) and −1 (probability 1−p), finding h_i for all i via the cubic px³ − x + (1−p) = 0. This is the freshest exam specimen and tests whether you can set up the hitting probability recursion for a non-standard step size.

Hour 5: Stationary Distributions & Detailed Balance & Reversibility

The ‘πP = π’ criterion for a stationary distribution is quite straightforward.

The theory of stationary distributions is for irreducible matrices. Just the positive recurrent ones have stationary distributions, which are unique, and each entry is given by reciprocal of mean return time.

If we add aperiodicity to irreducibility, we get that the probability of being in some state at time n — including the probability of being in that state from any previous state — converges to that state’s stationary probability as time runs to ∞.

I studied the Time-Reversal Theorem (aperiodicity no longer required), which requires defining Y_n = X_N-n, the time-reversal, and giving its transition matrix Q’s entries as q_ij = (pi_j / pi_i) p_ji. That’s easy to obtain from how π_j p_ji = π_i q_ij.

The matrix is time-reversible (in equilibrium) iff P = Q — i.e. iff the detailed balance equations hold! Replace q_ij with p_ij in the above, and you’ll get the detailed balance equations over all i, j. When a matrix and distribution are in ‘detailed balance’, that distribution is stationary for the matrix.

So when I’m thrown this question:

(1) Define reversibility in equilibrium: define transition matrix Q (& its entries). RiE iff P = Q, i.e. the chain and its reversal are statistically indistinguishable

(1) Detailed balance is equivalent to reversibility in equilibrium

(3) This is because the detailed balance equation can be re-arranged to give matrix Q’s entries

Next session:

• 2024 Q3(b)

• Retry algebraic approach on Q3(a)(iv)

In future:

• Ergodic theorem + convergence to equilibrium

• Novel Markov chain modelling

Ideas to increase # of study hours each day

1. Sometimes extend blocks from 1h to 1.5h

   1. Retain 0.5h as minimum block length

This will help with reducing context-switching

One strange thing about doing mathematics is that you often don’t know ‘how much you contain’ until the moment of elicitation.

I’ve become very interested in ‘knowing what you know’ — Language Models (Mostly) Know What They Know; luminosity (according to Timothy Williamson in his book Knowledge and Its Limits (2000), a condition is luminous if and only if, whenever an agent is in that condition, they are in a position to know that they are in it), being able to trace the provenance/origins/generators of your beliefs…

But with mathematics, you’re often a blank slate until tried.

Hour 1: Contraction Mapping Theorem

This falls under ‘Completeness’.

I reviewed the Contraction Mapping Theorem proof and was able to split it into three parts. I like this approach.

1. UNIQUENESS

2. EXISTENCE

   1. Iterates converge…

   2. To a fixed point

Next session:

• Drill: the proof that C[0,1] with the sup norm is complete (it appeared on the 2024/25 paper and the 2023/24 paper), making sure to be able to write out the "uniform limit of continuous functions is continuous" argument cleanly

• Practice the CMT application pattern: given an integral operator T, show it's a contraction by estimating d(Tf, Tg) using the sup norm. The 2024/25 Q2(b) is a perfect specimen — work it fully.

• Revise: closed subsets of complete spaces are complete, and Cauchy sequences that have a convergent subsequence converge.

Hour 2: Compactness & Sequential Compactness

Note to self: I actually know the definitions of ‘compactness’ and ‘sequential compactness’ cold. (Just NB: it’s X = the subcover, not <=).

I spent a bit of time with the proof that compactness implies sequential compactness, which I’d seen before. It reminds me of the Contraction Mapping Theorem proof, because it involves inductively constructing a convergent [sub]sequence; (in CMT, we then say something about the limit, namely that it’s a fixed point).

Next session:

• Drill the "is this space sequentially compact?" questions — the Hilbert cube, the space of sequences converging to 0 with sup metric (2022/23 Q2(b)), the Cantor set. The pattern is always: either construct a sequence with no convergent subsequence, or use Bolzano–Weierstrass plus closedness.

Hour 3: Connectedness & Path-Connectedness

This was a fun hour!

After being reminded of the 3 equivalent definitions of connectedness:

1. Not disconnected — i.e. can’t be expressed as the disjoint union of two nonempty open sets

2. X and Ø are the only clopen sets

3. Any continuous f: X → {0, 1} must be constant

I was able to remember right away the proof that path-connectedness implies connectedness, using #3. Yay!

I then spent a bit of time learning the proof that a connected open subset of a normed metric space is path-connected. I can definitely feel the difference between proofs I’ve studied before (where I’m now reviewing) vs. not.

Next session:

• For problem practice, work through the "determine connected/path-connected" classification questions — the 2024/25 Q1 with K(A) and the 2020/21 Q1(a)(ii) are gold. Remember: proving connected needs a universal argument; proving disconnected just needs one partition.

Upcoming Hours

Hour 4: Open/closed sets, closures, continuity, and the “toolbox” results. These aren’t standalone questions but they’re the scaffolding for everything else. Revise: closure K(A) = A̅ characterisations (the 2024/25 Q1(a) tested this directly), the sequential definition of closed sets, dist(x,F) = 0 iff x ∈ F̅ (2022/23 Q1(a)(iii)), continuity in terms of open preimages, and the Lipschitz/contraction/shrinking map hierarchy (know the definitions cold — the 2014/15 paper devoted an entire question to distinguishing these). Also make sure you can handle subspace metrics and identify open/closed sets in subspaces.

Evening Block (2 hours) — Past Papers

Hours 5–6: Timed practice. Pick one full Metric Spaces section from a past paper — I’d suggest 2022/23 or 2024/25 since both have excellent coverage across completeness, compactness, and connectedness. Do the Metric Spaces questions under timed conditions (roughly 45 minutes per 25-mark question). Then mark yourself against the techniques you revised today. Focus on: did you state definitions precisely? Did you use the right characterisation of connectedness/compactness? Did you handle ε-N arguments cleanly?

Quick-reference “must know” list to keep beside you all day:

The three definitions of compactness and their equivalence, CMT statement + proof + application template, completeness of (C[0,1], ‖·‖_∞), three characterisations of connectedness, path-connected ⇒ connected proof, closure/interior definitions, continuity via open preimages, Lipschitz vs contraction vs shrinking definitions, and the fact that continuous image of a compact/connected space is compact/connected.

hit the parts that hurt most

I’m going to outline how Jones’ takes from behavioral/development econ break on contact with The River. As Nate Silver puts it:

The place where I fit in is what I call “the River”. It’s a place for people who are very analytical but also highly competitive. The archetypal activity in the River is poker. […]

There are other communities in the River, though: Silicon Valley, Wall Street, sportsbetting, crypto, even effective altruism, all of which are covered extensively in the book. […] There are traits like decoupling, contrarianism and a high risk tolerance that I share with the River, for better or worse.

The River says to invest in yourself

Jones reports on an academic study into how many IQ points it takes for someone to prefer €100 + ε a year from now over €100 now. He says: “A fifteen-point rise in IQ predicted a shift of about 2.5 euros’ worth of patience”.

This feels absurd. Of course, there are all these prosaic factors that favor taking €100 now:

• The university/study might go bankrupt/not follow through

• National inflation(!)

  • This study happened in Germany: the 2005 CPI inflation rate was ~2%

• Cognitive overhead: it’s one more thing to track1

But then there’s also ‘how much will the marginal euro be worth to you specifically one year from now’. If you’re young and ramping up, you might be earning twice as much in a year. Consider also investment strategy! If you’re smart, you should be able to invest and get returns above 2.5%.

It’s amazing that cosmopolitans cooperate

I’ve been thinking for a while about ‘induction into the cosmopolitan community (River)’: how for me, it started with United World College, which got me out of the community where I’d been active for sixteen years, and now I’ve blinked, been living out of a carry-on for six years, maintain friendships through passive surveillance, and trust sparingly.

Jones concludes the chapter by saying:

Imagine two societies: one in which the future feels like a dim shadow, the other in which the future seems as real as now. […] Which society will have more politicians who turn down bribes because they worry about eventually getting caught? […] the intelligent act more like good citizens.

I’d like to say “it’s a great credit to how we’ve managed globalization that people don’t really defect, despite all the location-switching”. But is it true? I think things might be ‘broken at the top’. From Zvi’s excellent review of ‘Going Infinite’:

[…] Various versions of this sort of thing are going to keep happening […]

Was there a reckoning, a post-mortem, an update, for those who need one? […] There were general recriminations. There were lots of explicit statements that no, of course we did not mean that and of course we do not endorse any of that, no one should be doing any of that. And yes, I think everyone means it. But it’s based on, essentially, unprincipled hacks on top of the system, rather than fixing the root problem, and the smartest kids in the world are going to keep noticing this. We need to instead dig into the root causes, to design systems and find ways of being that do not need such hacks, while still preserving what makes such real efforts to seek truth and change the world for the better special in the first place.

I perceive cosmopolitan cooperation as fairly fragile and complicated. Note that I factor in ‘defection by omission’. I’m going to write a lot more about inter-relatability in the coming weeks — the notion that’s missing from current discussions of redistribution, enhancement, self-improvement, social mobility, utopian abundance, and telescopic altruism2.

Illuminating the future

Since “the future feeling like a dim shadow” promotes the pursuit of near-term gratification, many think we should take steps to ‘illuminate the future’. One way might be upsampling the votes of young people. Others might be slashing inheritance tax, promoting ‘longtermism’, hyperstitioning the 22nd century, appreciating the Long Now, or keeping hereditary peers, depending who you’re talking to. Perhaps the future will, by default, start to feel brighter and more vivid as we approach longevity escape velocity — though I don’t look forward to the period of intense, harrowing anxiety and stampede I expect to accompany that transition.

Takeaways

1. Conventional wisdom breaks down, hard and repeatedly, on contact with The River. The River is OOD; things don’t work the same way.

2. Gosh — all these stylized facts. Some academics are truth-seekers in the main; some believe a hypothesis to be true and try to marshal evidence to convince everyone else. I wonder what Jones would think about the impatience of highly intelligent, cosmopolitan River members. I’ve become motivated to read the rest of the book.

3. Intellectual arbitrage is serious and real; I do think mathematicians could contribute a lot to economics, just as economists have contributed a lot to other fields.

Claude span up a 12-hour study plan for me.

This is what my workspace looks like using split view:

Hour 1: Minimal Polynomials & Diagonalizability

There’s a ~finite set of things they can ask me to do with minimal polynomials.

They can ask me to find a minimal polynomial. This will generally involve finding a characteristic polynomial via det(Ix - A) and then looking at the factors, seeing which degrees annihilate.

Example problems:

2021 Q1(b)

These problems will typically be parametric matrix problems, where e.g. the minimal polynomial depends on the values of lambda and mu. In this case, I dropped 1 mark by failing to remember that lambda * mu = 0 includes cases where one but not both of these variables are zero :sweat-smile:

2016 Q1(c)

Here, I got blocked by computations. I computed that characteristic polynomial is x^3 - (a + 3)x^2 + (3a + 2)x - 2a (happy I went back and corrected a sign error!), but then I stared and wasn’t sure how to factorize, even though just subbing x = 1, x = 2 and using Factor Theorem would’ve worked to yield (x-1)(x-2)(x-a).

I forgot the upper triangularizability criterion, which is that χ split into linear factors (these need not be distinct — distinctness gives diagonalizability).

And so we go through the cases where a = 1, a = 2, and we check whether the degree-two polynomial of distinct linear factors annihilates, which determines whether minimal polynomial is a product of distinct linear factors and thus matrix is diagonalizable.

Postponed: Proof review

• Existence and uniqueness of m_T

• The proof that λ is an eigenvalue iff it's a root of m_T iff it's a root of χ_T

• The diagonalisability criterion (m_T splits into distinct linear factors)

• The triangularizability criterion

• Cayley-Hamilton Theorem

Right now, I can use the machinery; haven’t tested proofs.

Hour 2: Primary Decomposition Theorem

In hour 2, I looked at the PDT proof.

Suppose we have an annihilating polynomial with coprime factors. Then their kernels give a T-invariant direct sum decomposition. And those factors are the minimal polynomials of their kernels if monic.

The Primary Decomposition Theorem comes in five parts.

1. V = Ker(a(T)) + Ker(b(T))

2. That’s a direct sum

3. Those subspaces are T-invariant

4. If a(T) monic, that’s minimal polynomial on its kernel

5. Induction on r

Postponed: Using PDT

• 2023 Q1(a)(ii): explicitly decomposing F₃³ into primary components for a given matrix over F₃

Hour 3: Jordan Normal Form: Construction, computation

I’m given a fact and told it’s important:

The key insight is dim ker T^i - dim ker T^{i-1} gives the number of blocks of size ≥ i.

I RRE A to find that dim(ker(A)) = 2. So the number of blocks sized at least 1x1 is two.

Initially, I try to square the RRE-matrix. This doesn’t work! RRE-matrix is a whole different matrix. So I square A, and I don’t even have to RRE it — it’s already the zero-matrix. Now I know that dim(ker(A^2)) = 4.

And so dim(ker(A^2)) - dim(ker(A)) = 2. So the number of blocks sized at least 2x2 is two. And so we’re done — we have two 2x2 blocks.

I was able to solve this preceding problem by simply analyzing the sequences for each Jordan form case. Every nilpotent map on finite-dimensional vector space has a JNF, so the JNF cases are exhaustive!

I’m told what a Jordan chain is:

Postponed:

• Proving facts about nilpotent chains, 2023 Q2(a)

Back during Inkhaven, I wrote a piece, ‘The cost of mastery’, in which I suggested people avoid mastery because it erodes freedom. When you’re more capable, the opportunity cost of making [public or private] mistakes — including working on the ‘wrong’ things — is more legible.

I care about this because I care about aligning incentives. If incentives contravene publicly getting really good at your craft, we’ll get less public mastery.

In this piece, I’ll talk about a couple more reasons why specializing can be psychologically hard. Then I’ll talk about doing it anyway.

Repetition shortens your subjective lifespan

The mind compresses memories of activities it repeats, so your life ‘feels’ shorter when you live the same day over and over. But I think you have to repeat and drill to ‘get good’. Unfortunately, your life is going to feel shorter if you commit to doing one thing really well.

Sometimes, I can feel the days start to blur; a month passes by in a flash; I look back and notice limited variation. This makes me a significantly worse conversationalist. There’s nothing new to report; I have entered a deliberate state of cryonic suspension, rigor mortis.

And this is…roughly how I want to be? I would rather have a subjectively-short life that accomplishes the things I care about than one that’s ‘long’ and exciting but contributes little new.

‘High context’ begets alienation

We’ve covered two costs of getting good: 1) Missteps feel costlier1; 2) Your subjective lifespan feels shorter.

There’s also 3) Alienation. As you go deeper into some craft or field, you’re going to get higher-context and be work-relatable to less people, because fewer people have invested in the frames and cognitive machinery you care about and rely on.

Solutions include ‘friends that exist in books’, LM scaffolds, the many-to-many academic community / blogosphere / republic of letters, and passive surveillance for context renewal.

Doing it anyway

Specializing will shorten your subjective lifespan, alienate you, and punish missteps. Being aware of these side-effects helps me feel more comfortable with them. It also helps me cherish friends who understand. I think that’s the main reason I write this. I didn’t really understand what was happening when I first entered a specialized community2; it took me a long time to understand who I’d become through its [implicit and explicit] curricula. I emerged with odd generators and learned to communicate these. Then, to make intellectual progress, I decided to specialize all over again3, contorting my mind into more novel shapes and becoming less relatable.

I think a lot of people fail to make progress because they want to stay relatable. I think that to break new ground, you often have to forsake your long subjective lifespan and relatability. Of course, you can be pretty bright and cheerful about this!

In this case, understanding what’s happening to me makes it more bearable. I think the best gift people can give themselves and their friends/family is encouragement to become less relatable on the path to productive intellectual specialization.

Forecast = endorsement

Say there’s some event you care about, E. You want E to happen.

Initially, your default policy is π_0, and you estimate P(E | π_0) < 1. But since you want E to happen, you come up with a new plan, π_1, to make E more likely. Now you calculate P(E | π_1) and check: are you satisfied? If not, you devise a new plan, π_2, and calculate P(E | π_2).

If each plan is better than the last, you’ll obtain a monotone increasing sequence of probability forecasts. By Monotone Convergence Theorem, this sequence will converge to sup(P(E | π_k)).

You can do anything

There’s an intuition going around that sup(P(E | π_k)) = 1.

Just…need…more…dakka…

Deep anti-determinism

I’m not technodeterministic like Nick Land; I’m not sociodeterministic like Chris Lakin. Stay sensitive to the counterfactual.

I’m suspicious of forecasters. If you see something coming you don’t like; why don’t you change it? I model the parts of the world I intend to act on.

As Shane Legg put it:

[Solomonoff induction] only addresses the problem of passive inductive learning […]. Whether the agent’s predictions are correct or not has no effect on the future observed sequence. Thus the agent is passive in the sense that it is unable to influence the future. An example of this might be predicting the movement of the planets across the sky, or maybe the stock market, assuming that one is not wealthy enough to influence the market. In the more general active case the agent is able to take actions which may affect the observed future. For example, an agent playing chess not only observes the other player, it is also able to make moves itself in order to increase its chances of winning the game. This is a very general setting in which seemingly any kind of goal directed problem can be framed.

Back to the drawing board, go!!

Background

Today, I was reading ‘Inconsistent Languages’. In it, Matti Eklund describes the difficulty of assimilating statements that are individually {justified, acceptable, legitimate} but collectively {incoherent, paradoxical, untenable}.

The Sorites Paradox exemplifies {individually justified, collectively untenable}. But I have never enjoyed Eubulide’s ‘heap’ framing. I prefer Eklund’s:

The sorites paradox goes as follows. We have a series of color patches, ranging from red to orange. The first patch looks red (to normal observers under normal conditions), the last orange. Adjacent patches are observationally indistinguishable with respect to color for normal observers in normal circumstances. These stipulations are clearly coherent. But now consider the following argument: First, the first patch looks red, by the stipulations of the thought experiment. Second, if patch k looks red, then patch k + l looks red as well, since adjacent patches are observationally indistinguishable with respect to color. But then, by elementary reasoning, we can conclude that the last patch looks red. But the last patch looks orange, not red.

Eklund also digs into the Liar’s Paradox:

Consider a liar sentence, a sentence λ that says of itself that it is not true:

λ: λ is not true.

Suppose first that λ is not true. Since this is precisely what λ says, λ is then true. Hence, by reductio, λ is true. But if λ is true, then λ is not true: for λ says that it is not true, and if λ is true then what it says is the case. Contradiction.

I also happened to, last week, encounter the notion of Probabilistic Dependency Graphs (Richardson and Halpern, 2021). The way it goes is this: original Bayesian Networks (Pearl, 1985) would represent events, and put probabilities on that event happening / not happening. This is a directed acyclic graph, so each event’s probability is conditioned on the transpiring / non-transpiring of upstream events. You quickly get to the point where you have to evaluate a whole bunch of binary ‘did this happen? did this not happen?’ to get the probability of the root node.

Probabilistic Dependency Graphs are directed graphs with cycles allowed. You can record P(F), P(G), P(F|G), and P(G|F), all separately, and these do not have to agree. To display unconditional P(F), you display this probability on an edge from a ‘1’ node to F. Note how in the example below, 0.9 = P(F) != P(F|G)P(G) + P(F|¬G)P(¬G) = (0.92)(0.05) + (0.08)(0.95) = 0.122. This seems much closer to how human cognition works! We have ‘trapped priors’ from earlier cognitive processes, and sometimes these ‘legacy intuitions’ won’t agree with the priors we’ve developed by conditioning on more recent perceptual experience.

This is really similar to the idea Eklund ends up developing, where he describes how individually compelling sentences generate ‘pull’ that we end up subscribing to on the basis of our semantic competence:

Competence involves being disposed to accept the culprit. Let us say that an unsound argument exerts pull just in case speakers’ semantic competence disposes them to accept the culprit or culprits in the argument.

…but that these individual ‘pulls’ lead us to adopt a set of beliefs that are collectively inconsistent — which cannot all be true.

Upon noting that your semantic intuitions lead you to accept an inconsistent corpus of statements, you can refuse to take these intuitions at face value, rejecting one or more of them as non-veridical, without thereby manifesting lack of semantic competence. (Rather the opposite: you positively manifest logical skills.)

Representing Paradoxes as PDGs

We can actually represent the Sorites Paradox on a dependency graph!

And, if we were to represent this on a Probabilistic Dependency Graph, we would use probabilities of ~0.99 rather than 1.0, say.

This would allow us to measure inconsistency on the graph, using the Inconsistency Measure recently developed by Richardson (2022):

First, let’s try a joint distribution μ where we suppose each patch is red; A_1 = … = A_n = 1, which aligns pretty closely with each edge except that from ‘I perceive last patch as orange’ to ‘Last patch is red’.

We’re going to set the confidence parameter to 1 on each edge for simplicity.

We have our definition of KL-divergence — ‘the expected excess surprisal from using the approximation μ instead of PDG when the actual is PDG’.

We’re able to quantify inconsistency across the chain pretty easily:

If we try a joint distribution where we suppose each patch is orange; A_1 = … = A_n = 0, we get an inconsistency metric that does not grow with n. Rather, we just have a lot of surprisal (ln 100) when we learn the first patch, which we perceived as red, is orange, and a tiny bit (ln 1/0.99) when we learn that the last patch, which we perceived as orange, is indeed orange.

If we set some patch such that ‘this patch and all patches that follow it are orange’, we get a solution that also does not depend on n. You’ll pay the ln(100) penalty when you finally switch from red to orange, plus the ln(1/0.99) at the end, and an additional ln(1/0.99) for each step you dither around waiting to switch over to orange.

One of the cleanest joint distributions is the one where we let P(first patch is red) = 0.99, then set P(next patch is red | patch is red) = s. The value of s that minimizes inconsistency will vary depending on n! We can get to very, very low inconsistency scores by setting s close to 0.99 for large n.

There’s even an ‘exact optimizer’ such that inconsistency tends to 0 as n tends to infinity(!!!). This case is super weird because beyond n = 458, we end up with a situation where P(last patch is red) is actually too low — (remember we set P(last patch is red | I perceive last patch as orange) = 0.01. So we end up adding a tiny bit of probability that the chain might flip from orange to red.

Variations: Symmetric & Deterministic Anchors

Interestingly, Eklund doesn’t distinguish strongly between the asymmetric case:

Second, if patch k looks red, then patch k + l looks red as well, since adjacent patches are observationally indistinguishable with respect to color. (Eklund, 2002)

and the symmetric case (Wright’s ‘tolerance principle’ from which Eklund derives the asymmetric case):

[If patch k looks red, then patch k - 1 and patch k + 1 both look red]

But whether we let the Markov chain be asymmetric or symmetric has serious consequences for what level of consistency we can hope to achieve. I won’t go into all the details here, but this table sums it up (note that we’ve just looked down the ‘asymmetric PDG’ cases together :)):

We could also vary by setting P(first patch is red | I perceive first patch as red) = P(first patch is orange | I perceive first patch as orange) = 1. This makes it a tiny bit harder to achieve good outcomes; note how inconsistency for the symmetric case’s ‘exact optimizer’ is slightly higher now:

Philosophical Implications

What point was Eklund trying to make regarding priors? How does all this machinery connect back to the literature on epistemology: what are its philosophical implications?

One thing Eklund cares about is that standard views of rationality might require excluding inconsistent possibilities entirely — that is, assigning them probability 0.

Instead, Eklund endorses allowing inconsistent worlds into the support of your credence function. We penalize inconsistent worlds instead of getting rid of them. Our priors over inconsistency structures start to matter: how much probability mass we assign to consistent assignments, mildly inconsistent ones, wildly inconsistent ones…

This means two agents can agree on all observational data, but differ purely in how their priors weight inconsistent configurations. They’ll disagree.

When we formalize beliefs in PDGs, inconsistent worlds correspond to PDGs that no joint distribution μ can fully satisfy (symmetric PDGs in the case of the Sorites Paradox). Eklund’s ‘penalty for inconsistency’ becomes KL divergence. ‘Forming a rational belief’ consists of minimizing inconsistency relative to the prior (but also, in my view, refining your priors…).

Some common philosophical views in terms of PDGs

Precisification

Precisification (criticized by Russell, 1923) would correspond to saying “yes, anything less than 20 points away from ‘RGB(255, 0, 0)’ counts as red.” We’d then have deterministic cutoff at a fixed k. Inconsistency converges to infinity in the symmetric case, and a fixed bound in the asymmetric case.

Epistemicism

Williamson and Sorenson argue there is a boundary where ‘red’ turns to ‘orange’: we just don’t know where it is. That corresponds to a distribution over sharp cutoffs / k.

Supervaluationism

Supervaluationists argue there are many admissible precisifications — many possible values of k.

Fuzzy Theory

Quite compatible with the ‘exact optimizer’ approach. Each Ai∈[0,1] is a degree of truth…

Conclusion

I think there’s a lot that could be done to represent paradoxes in terms of Probabilistic Dependency Graphs. Doing so might enable philosophers to understand where/how exactly they disagree, and which constraints are ‘optimal’. I wouldn’t advise humans invest effort in this. This subfield feels like a prime candidate for ‘automating philosophy’.

Part 2: Pick your contrarianism on the right level of abstraction

Part 3: ‘You staked out some barely arable land’

A screed against bikeshedding. Each post is standalone.

Sometimes, I’ll see a researcher who’s clearly found a niche. They’ve looked around, and realize, with a cheer!, that no one else is digging here. They dig in. And — if they can keep getting employed to do so — they keep digging.

I’ll give one example from analytic philosophy, where we generally acknowledge researchers are often led astray into minutiae.

For example, Lizzie Fricker has spent her life studying ‘Testimony’…

…and I do not think this work matters very much, in that I don’t think it’s shaped people’s behavior or proven invaluable to anyone at some critical juncture. (Corrections welcome!).

Beyond analytic philosophy, I think there are also fields of AI research where people look around, realize no one else is digging, and dig in.

For example, I recently saw a project on ‘(classical) music composition as a novel LLM evaluation domain’. I think this sounds extremely fun, and of academic interest, but this was presented as a project within an AI alignment research program. And in that context, I think I think this is a case of ‘staking out barely-arable land’. The project is above some threshold of legibility, but only just. Less novel domains matter more.

Generally, if no one else is digging, there’s a reason!, and you should probably head back to more fruitful seams!, unless you have a very strong reason why digging is valuable anyway. I actually get more excited about a research topic when I see other people have cared about it for a while1.

In more arable fields, being ‘the best’ and making a contribution requires a higher skill level. This is worth aspiring to! It will demand more of you. If you are going to do something solo, off-the-beaten-track, you should have a strong justification for why this really matters. This will often be because it’s an instrumental technology — looking at science horizontally, instead of vertically. That’s how Robin Hanson got away with his intellectual entrepreneurship, pioneering emulations and futarchy: these are technologies that matter for / underlie everything else.

Take your mnestics. A lot of ambition is about saying no to things that aren’t worth your time. Do the ‘obvious’ things, and pursue the ‘obvious’ fields. Join your friends cultivating the most arable land.

It’s time to report on how Lydia-2027, now on Lydia-2026, has been going.

Infrastructure

The Three-Slot Model

23 Feb: in sf u get 3 slots for evolution

where you live, how you spend your daytimes, how you spend your evenings/weekends

I’m still living at the Convent, and also working here now over co-working spaces, where there tended to be too many superficial / distracting conversations. Unfortunately SamA was right

I started meeting people mostly for breakfast; I had a real laugh about joining Twitter bullet list’s aristocracy, but…you would be surprised!

Research

This is part-time through SPAR right now, an org I could not recommend more. Essentially, in terms of intermediate self-forecasting results, myself and team all get that the models are overconfident about their abilities and integrity.

Been looking back on old papers — Language Models (Mostly) Know What They Know was an early Anthropic production, focused mostly on what Williamson (2000) would call luminosity:

a condition is luminous just in case whenever one is in it, one is in a position to know one is in it

So — we want to see if we can teach the models epistemic humility. RL training on rollouts where they ‘fail’. You can imagine there are lots of cases where epistemic humility might matter.

Another update I had, re AI research, is that MATS-like programs are mostly focused on making sure nothing bad gets out of the labs, through auditing etc. — what I in April termed ‘guarding’ — and not making models that can actually pass any audit, including ones more demanding than those we’ve developed — what I in April termed ‘robustness’. I feel best about working on ‘robustness’. Other people thinking about near-term risks and threats from AI: Vie McCoy, Dean Ball, Amanda Askell, Joe Carlsmith; there are several ‘factions’ & if you think for yourself, people will consistently be surprised and confused about what you think as they try to place you.

Things I wish I had time for but don’t

1. ‘SPAR/MATS' for AI-enabled science’. 151 people liked my recent note. I’m inspired by cases like Andrew Rodriguez attempting to cure Amy Deng’s brain tumor. The Fable of the Dragon-Tyrant — yes, specifically the YT vid — caused a viewquake. What can educated, motivated people achieve under 1h/week with an academic seeking to build out their field? SPAR’s found the answer’s ‘non-trivial amounts’ in AI safety. Research slides for efficiency. I’m confident this would be really good, and I don’t think there’s anything like it right now. There are plenty of educated, motivated professionals who are undertaking sabbaticals, career transitions, or who really just enjoy putting hours in after work, who I think could join the scientific effort. If you want to work on this, please get in touch!

Research Interests

• Expand my take on probabilities as moving suprema — forecasting as an iterative, dynamic process given embedded agency

  • I really need to dig into the literature on this. Viewing Bayesian updating as an iterative, diachronic process goes a long way. I’m currently reading Hedden on Time-Slice Rationality. Lots to think about…Anthony DiGiovanni gave me a small viewquake re Prisoner’s Dilemma & one-boxing. My gosh! He’s right when he says:

    • “The broader lesson of this post is that decision theory and anthropics are really complicated. I’m puzzled by the degree of confidence about these topics I often encounter (my past self included), especially in claims that anthropically updating EDT “doesn’t work,” or similar.”

      

• Articulate the process of ‘threading the needle’ — Bostrom’s “We are not trying to predict what will happen. Rather, we are investigating what we can hope will happen if things go well.” Build out the science of articulating constraints and finding desirable futures that navigate between them

• Explore whether a world in which we develop ASI but don’t deploy it widely could go well — how ‘practical alignment’ can help with navigating dual-use technologies — Carlsmith’s “I’ll say that a system is “practically aligned” if it does not engage in misaligned behavior on any of the inputs it will in fact receive” — extending existing work on constitutional classifiers

• Articulate what good/healthy/legitimate/minimal non-state surveillance looks like (sufficient for a ‘vulnerable world‘ w/o imperilling the future through superfluous domestic surveillance)

  • Longer writeup of the problem from me here

• Continuation of work on introspection-based early alert systems / testing models’ self-forecasting abilities. Here’s one summary:

Recent research has shown language models (LMs) to be capable of introspection – acquiring knowledge that originates from internal states (Binder et al., 2024). This previous research has operated primarily on token-level properties of short outputs.

We wanted to see whether LM agents can discern properties of longer outputs across multi-turn settings.

We also wanted to see whether LM agents can introspect in self-defence – i.e. detect settings where they’re likely to drift off-course, and use this awareness to stick to their reflectively endorsed values.

If they can, this could provide the basis for ‘introspection-based early alert systems’, where models use introspection to a) maintain value alignment in adverse settings, and b) flag potential problems to auditors.

Success looks like finding settings where introspection would help with these objectives. Further success looks like extending this to continual learning settings, where LM agents can anticipate and block value drift.

Stack Updates

New items!:

Mac Automator x Arc Browser x Rectangle setup

Cargo skirt

Supplements:

Water bottle

Uninsulated, so I can carry high/low temperatures

Lock box

To combat ‘porch pirates’(!) who enjoy Thistle meal delivery as much as I do

Coming Soon

1. ‘Why slower isn’t necessarily safer’, touching on the ‘optimal timing’ considerations from Full-Stack Futurism and aiming to make these accessible to audiences who haven’t considered them before

2. ‘Table stakes’: on ‘raise your bar for success’, dynamic baselines, the second type of homeostasis, ‘x by default’, ‘'how to be a human’ starter pack’/sanity waterline, default trajectories, inadequate equilibria, charting what's not priced-in, outperforming for investors, pulling the rope sideways, reference class thinking vs. curse of dimensionality

3. What I’m learning doing analytic philosophy as an optimization/prioritization-focused person — about staying oneself, defence against the dark arts, cognitive security / epistemic immunity, going off-script, communicating with aliens, staying sane and original, 80/20ing, avoiding frame capture, orthogonal moves, epistemic humility

Great reads in Q1

There are four standouts:

1. Did Claude 3 Opus align itself via gradient hacking? (2026)

2. The Moore’s Law of Synthetic Gene Circuits (2026)

3. Which Future? (2026)

4. The Division of Cognitive Labor (1990)

I want to get through more blog posts from Michael Nielsen, Ethan Smith, all these wonderful convex thinkers…

Q2-Plan

Until 4th April:

Finish philosophy. Right now, I’m covering induction, understanding e.g. Solomonoff induction and issues with it…writing about that tomorrow, then MATS Symposium, then onto ‘The A Priori’ (maybe fairly light, after having covered Bayesianism/Problem of the Priors and Induction already?) and ‘Causation’ (unclear how light?). Very much 80/20ing these things. A lot of my work is integrating consequentialist worldview with the literature.

7th-8th April: I’ll try five math papers to see how I do. I think I’ll be OK at Metric Spaces & Linear Algebra, but have forgotten a lot of Differential Equations, and Probability Theory / Complex Analysis somewhere in between.

9th-20th April: I’ll work full-time on top-priority math topics, with a view to answering one question in each subject well. e.g. phase plane analysis, Markov chains, residue-based evaluation of integrals. Playing to strengths at this time.

21st April: I’ll travel back to the UK! Land 14:10, sleep 22:00.

22nd April: I’ll recover from jetlag & prepare for exams.

23rd-25th April (morning): I’ll sit some exams.

25th April (afternoon): Hopefully I’ll make Session 3 onwards of EV Unconf.

27th April - 17th May (Weeks 1-3): Mostly math, with some phil, SPAR still officially on. Always 1 day/week rest.

18th May - 14th June (Weeks 4-7): Ramping up math, with less phil, post-SPAR-Demo-Day. Always 1 day/week rest.

15th June - 26th June (Weeks 8-9): I’ll sit some exams.