There are those who like to poke holes in things. There are those who, when confronted with a new idea or technology, like to enumerate all its weaknesses and reasons it might not work. They shoot down suggestions. They’re uncomfortable with an upbeat last word.

Others feel more solution-oriented. They stare down all the reasons something might not work, dissatisfied — they keep trying, keep trying, throwing more Dakka at the problem and doing heroic goal-factoring until they come up with ways it might, conceivably, just about work.

In cybersecurity, there’s this notion of the ‘red team’ and the ‘blue team’. A red team plays offense by simulating real-world attacks, while a blue team plays defense by protecting and responding to those threats.

People see me playing on the blue team, and get confused. They think maybe I’m not preoccupied by AI risk? The fact of the matter is just that we need people on both teams. As Bostrom says —

⁠“We are not trying to predict what will happen. Rather, we are investigating what we can hope will happen if things go well.”

You too can join the blue team :)

It involves thinking about what criteria would need to be fulfilled, what would need to go right for some technology to work well. You go out and make sure those criteria are met. Then repeat.

Relevant:

Against commentary as a full-time occupation

Casablanca (1942) I first watched in Dec. ‘24. Citizen Kane (1940) I first tried to watch on a United flight earlier this 2026, but to be honest, my P100 mask was under the headphones, and there weren’t any subtitles. I’ve started putting P100 over the headphones, and finally got blown away by this film :)

Conviction

Citizen Kane is pretty clichéd / stereotypical in its preoccupation over whether spineless Kane has any morals. It’s Burr from Hamilton and Burr — the consequences of “standing for nothing”. We get these sorts of reports:

“He had a generous mind; I don’t suppose anyone had so many opinions. But he never believed in anything except Charlie Kane; he never had a conviction except Charlie Kane and his life. I suppose he died without one. Of course, a lot of us check out without any special convictions about death, but we do know what we’re leaving; we do believe in something.”

In Casablanca, having no convictions is a point of pride. 🤭

“I have no conviction, you see. I blow with the wind, and the prevailing wind happens to be from Vichy.”

Rick’s coworkers, shocked and ecstatic: — “Boss, you’ve done a beautiful thing!”; Rick, stony-faced: “Get outta here, you crazy Russian!”

“I’m not fighting for anything anymore except myself. I’m the only cause I’m interested in.”

In Casablanca, convictions and sentimentalities are to be disavowed — IMO because their ambient presence is greater; it can be assumed. I’d rather live where people have convictions but disavow them than the converse.

Good People

Casablanca > Citizen Kane because it has better characters. Conviction is so present, so assumed that it has to be repressed, disavowed — counter-signaling of the highest degree. Unlike in Kane, characters are not jealous. Laszlo begs Rick to go with Ilsa so she might live. You see, Casablanca is a movie filled with good people, and Citizen Kane is filled with shitty people, and I want to watch movies about good people at the end of the day.

Everyone self-sacrificing. So fucking transcendental

Casablanca

Internal Systems / Duopoly of Self

Rick Blaine: Who did you bribe for your visa? Renault or yourself?

Ugarte: Myself. I found myself much more reasonable.

Renault: “It would never do for the Chief of Police to be caught drinking after hours and have to fine himself.”

Rick Blaine: Well, so long. Don't forget you owe Rick's 100 cartons of American cigarettes.

Ferrari: I shall remember to pay it to myself.

Some Croatian Mikulec also notices this. Blogs make me feel more connected to my fellow audience members :)

Questions

• Rick actually does sell his bar, Rick’s, to Ferrari. So he actually seriously intended to leave on the plane? Or just knew he’d have to leave for good after assisting Laszlo & Ilsa? Where did he plan to go next, if not on the plane to Lisbon? Brazzaville was Renault’s suggestion.

• This badass activist gathering — “the Underground meeting” — it sounds so adolescent. Like the Animal Liberation Front in Okja. Cool, and edgy, but ineffectual. What could really have happened of such importance at Casablanca’s ‘Underground Meet’? (genuine q for people more familiar with historical European resistance to Nazi encroachment).

• How was the airport ready if Renault didn’t call them?

• Why can’t Rick return to America?

The Urgency of Telepathy

“You’d say anything now to get what you want.” Ilsa needs an activation oracle / thought-to-text. Rick cannot trust her self-reports — won’t let himself.

Social Grace

Fun obsequiousness that greases the wheals

1. “I was informed that you are the most beautiful woman ever to visit Casablanca. That was a gross understatement.”

He does not actually believe this. I did not pick this up on first pass. But it’s like charismatic people who go, “ah, two of my favorite people!” to acquaintances, playing up their affection. Prosocial

2. “It was much more than we thought to get here” — these gentle lies I never would’ve caught before. I’m quite confident they set out without enough funds from the start. But when you get yourself in a pickle, people will help you out of it — more so than if you did not leap in the first place. That’s why it’s quite important to “dive in”.

Observations

• Ilsa doesn’t tell Laszlo, but Rick does. lmaooo

• The way Rick responds to Ilsa’s crying — in Paris and in Casablanca — oh.

• “It wasn’t my secret — I knew so much about his work, and if the Gestapo found out — ”

• “You might as well question why we breathe.” Laszlo’s very death-or-liberty

• “Ilsa is Laszlo’s wife. She probably knows things Strasser would like to know.” Ah — the days this was true between spouses — before today’s security-clearances and NDAs :)

• The Germans thumping on the piano with the letters in it, OMG.

  • Putting letters of transit in the piano in front of everyone — damn.

• Lol, initially I interpreted “you are a man” as “you are a mensch” or “you’re the man, thank you”. “And if that girl did a very bad thing [for a man she hadn’t met yet]?” Concrete-abstract bias

• I also know now that one interpretation of ‘the person you love most / best’ is the person you tell everything. And Rick knows everything. But Ilsa does not tell Laszlo everything. To Laszlo she says “be careful”, not “I love you”. I genuinely couldn’t really read how her affections were distributed on first pass, bar “they all three of them love the cause more”

• It’s so campy. Renault is so campy. All of it, the resistance, everything

• “You knew what it would be like running? […] I’m settled now” aa

• Sam is an IC, the most gifted entertainer in town. The others participate in politics, the art of the possible

• The goddamn search lights in the background — how aren’t they seen?

• Ilss comes out in a curfew. She has interpersonal agency.

• You can actually do KQV self-attention over a movie on your second pass. Spoiler-free is great; knowing what’s coming is also great.

One last observation I didn’t make on first pass:

“Why don’t we get married in Marseilles?”
“That’s too far ahead to plan”
“Yes, I suppose it is a little too far ahead.”

(later chronologically, earlier in the movie):

“Where were you last night?”
“That’s so long ago, I don’t remember.”
“Will I see you tonight?”
“I never make plans that far ahead.”

• Masking interpersonal strife in the stress of the times — “I love you so much. And I hate this war so much.” It is not for the war that she cries.

Other epic / classic movies I’ve enjoyed this past year:

•⁠ ⁠⁠Gone With The Wind

•⁠ ⁠⁠The Color Purple

•⁠ ⁠⁠Brokeback Mountain

•⁠ Titanic

& screwballs that were worth watching for other reasons:

•⁠ ⁠Devil Wears Prada (1 & 2) (for inter-relatability more than for myself)

•⁠ ⁠⁠Moulin Rouge! (striking depiction of desperate camaraderie)

I still have to watch more classic Hollywood TC recommended: “Key Largo”, “Bringing up Baby”, “High Noon”, “Night of the Hunter”, “Pillow Talk”.

Claude recommended me some new shows! I loved Atonement in 2024. This whole genre also echoes Casino Royale, Miss Saigon, Evita, etc.. Also GPT’s recs (probs better / more precise).

Citizen Kane

Citizen Kane is just a colder uglier universe than the illustrious Casablanca, so it doesn’t do to dwell on it too much — but a few remarks:

Rights vs. Gifts

“You're not going to like that one little bit when you find out it means that your workingman expects something is his right, not as your gift!”

It reminds me of Peronism:

Eva: But to give all my descamisados
A magical moment or two

Che: To look never further than this week or next week
To have no impossible dream?

Love

Citizen Kane — what the obsessive desire to be loved will do to a person.

“Wasn’t he ever in love with her?” // “He married for love. That’s why he did everything. That’s why he went into politics; it seems we weren’t enough; he wanted all the voters to love him too. That’s all he wanted out of life, was love. That’s Charlie’s story, how he lost it. You see, he just didn’t have any to give. Oh, he loved Charlie Kane of course, very dearly. And his mother, I guess he always loved her.”

“For you and this public thief to take the love of the people of the state away from me?”

“You just want to persuade people that you love them so much that they ought to love you back — only you want love on your own terms. It’s something to be played your way, according to your rules.”

“You don’t love me. You want me to love you.”

Quotes

• “How did I find business conditions in Europe? With great difficulty.”

  “I’m always glad to be back, young man. I’m an American. Always been an American.”

• “I’ve got to make the New York Inquirer as important to New York as the gas in that light!” intelligence / information as utility. 467 employees!

• “There’s always a chance, of course, that they’ll change Mr. Kane — without his knowing it!” (when the Inquirer acquires the Chronicle men)

• “The Spanish-American war” — gosh. Just like the Anglo-French War, Russo-German War, and Sino-American War in Last and First Men. “If we hadn’t had that war, we’d have the Panama Canal.”

• Some basic, tired tropes: “Maybe I wasn’t his friend, but if I wasn’t he never had one”. // “You really like me, though, even though you don’t know who I am.” // “I don’t know what you’ll do. Sail away to a desert island, probably, & lord it over the monkeys.” // “He was disappointed in the world, so he built one of his own, an absolute monarchy.”

• ‘Reality is negotiable’: “That a whole audience just doesn’t want you—”
  “That’s when you gotta fight them.”

• Sovereignty: “There's only one person in the world who's going to decide what I'm going to do and that's me.”

Take me back to Casablanca any day!

Many of my best teachers have ‘played high’ in order to get some kind of point across. Sometimes, this involves putting on a little more confidence than one actually has, to inspire others, in contexts where doing so is harmless. Showmanship!

Sometimes, I ‘play high’, generally to strengthen calls to action I’m confident about. To a hyperverbal recent grad lacking conviction:

“Do I seem like someone who has a lot of conviction to you? I got this by reading and writing on Substack. So sign up to Inkhaven; start posting more :) if you want to acquire conviction”

I really like when teachers are honest with their students. But some teachers take this to a cringing, fawning extreme — disavowing any belief rather than ‘playing level’ with “Here’s my best guess. What’s yours?”.

When teacher plays low, student lacks teacher. It shouldn’t be overdone.

Sometimes I encounter people who are playing high not in service of others. They’re usually insecure about their position — maybe they know less than the people they’re ostensibly serving, but don’t want to let this show. These people, when they get a scrap of information due to their formal position, guard it jealously, because it’s one of the few things that sets them ‘apart’. It’s a temporary band-aid artifice, even though given time, their context and capacity is insufficient to sustain the gap they’re manifesting.

You can tell I am unimpressed by playing high in service of oneself. By contrast, I respect playing high in service of others.

In general, I feel like people are under-steered, under-guided, and under-provided authority figures in our Modern World.1 It’s often really helpful to provide someone an anchor, something they can hold onto. They can later deviate, but having a concrete, clear, crisp initialization as opposed to a mushy one seems to matter a lot and help people. Often, simply being openly decisive matters more than exactly what decision you take.

Decisiveness begets accountability! A lot of people live in fear of being perceived accountable for something, cowed by the Copenhagen Interpretation of Ethics: “when you observe or interact with a problem in any way, you can be blamed for it”. It can lead adults to ignore vulnerable young people. It can lead people to shrink from ever playing high. ‘You have all these rules’, and to be honest, they do tend to make it so you don’t ever need saving.

But I want you to own your agency. Liability avoidance is a scourge. It’s risky to not take risks and be downside-focused. I can’t convince you in this piece to own your agency, how much more exciting and thrilling life is when you do so, take risks with your reputation in service of others, have integrity — the richer trajectory-equilibrium you can put yourself on. But it’s how I live.

Imagine my surprise when I opened ‘The Assistant Axis’ and found this:

I’m quite sure a sage can assist me much more than a consultant can. So I’ll read this research, but like many, I have limited interest in stabilizing an ‘Assistant’ persona. I’d much rather harden the humans interacting with the systems and let the systems play all kinds of roles in service of us.

I want to write to everyone, but it feels like context is fragmenting. I keep having thoughts that would’ve bordered on heretical in the communities I grew up in (2022-25). I’m so tired of the notion of ‘wrong-think’.

Whenever I share how I think about the world, how I think about things shaping up, what I think is best to do, I talk to people who say that I’m wrong, but they can’t explain exactly why because they’re under NDA. Or they just can’t be bothered…they can’t always articulate exactly why. I sort of try to move the conversation along. It does confuse me, though, because I feel like I’d never act like they’re acting in their shoes.1

It’s enough to make me give up on strategy. I feel a sort of ‘defeatedness’, a sort of helplessness; I feel like I can’t get enough information to make good decisions. I’m sad that people talk to me in a way that leaves me feeling this way, because I think if I were in their shoes, I’d try to talk to whoever in a way that doesn’t leave them feeling this way afterwards.

I always try to be as generous as I can with my thoughts, and with trying to help people see my perspective. It occasionally leaves people feeling worse after they talk to me, and it’s not always the best thing for me either. Sometimes people keep asking me questions, and I feel like they’re extracting and extracting and it’s taking a lot out of me, but I try to push through. And then I try to write a blogpost; I try to give people the information they need to make good decisions; I think that’s part of empowering people.

People still ask me what I think is best and most impactful to do, and I answer honestly, but also share that I honestly have so much uncertainty.

Even here, I feel like I can’t talk openly about what I think is important and what I think is best to happen and so on. I really miss talking openly on my blog, like I did when I guess I was squeezing my thoughts into more acceptable boxes. I’ve never really liked it when people take pride in contrarianism, e.g. those loud people focused on culture war issues. I don’t want at all to be contrarian; I just want to be right. And then I want everyone else to be thinking well, also.

This is a ‘vibes’ post that sort of peters out. I wish I could write more openly about my priorities and hopes and dreams and plans, but it doesn’t feel sensible. I’ll try to write ‘Q2 Review, Q3 Plans’. I miss openness.

Security dilemma

• A fly in amber

• In cryonic suspension

  • Suspended animation

• Lobotomized

I’m a shadow, a whisper. I swallow general content with 200 other Math students. The dynamism of Lydia-2026 and Lydia-2027 elude me. I’m comforted that person existed once; I know she’ll exist again, but who is she?

The world doesn’t stop moving because I do. Scaling continues. Friends start new initiatives! I am so, so happy.

Do I stop? I sense ‘the long arc of a life’. And I think of professors I’ve met, situationally aware professors who have one foot in academia, one in the wider world; I like that.

Academia is falling. I express myself through mathematical metaphors. I don’t mind who researches, only that it gets done.

I am like the old lady who swallowed a fly, then swallowed a spider to catch the fly, then swallowed a bird to catch the spider…

I metabolize by blogging. I posted twice in May.

Sometimes I feel more ‘like myself’. Research, friends who elicit well from me:

I have not been much of a person these last two months, not really.

I need to be more of a person again. Blogging, reading, research: these help.

Atelic studies don’t empower; it’s important to remain vaguely empowered.

I am a person. Earlier posts are irrefutable evidence that I was once a person and will be again. Temporarily embarrassed high-entropy individual

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)