## How to study Math while the world teeters on the brink of war

The following blog post was inspired by two things: first, by a Youtube video by the author John Green craftily titled “How to Make Potatoes While Dread Presses In from Every Direction”, and second, by a fifteen minute ordeal I’ve only just now had with a particularly challenging notation from a textbook I’m reading on Stochastic Processes. I figured it would be worthwhile making my own version of Green’s anxiously escapist how-to but for the niche audience I imagine found my blog through my academic endeavors.

Green’s video premiered back in February 25, two days into Russia’s invasion of Ukraine. The war is neither sudden nor unforeseen, as days before as apparently US intelligence had been warning of the imminent possibility of a provocation happening even months before the actual conflict began. Now with the war continuing to ravage the lives of innocent Ukrainians and throwing the Russian economy in for a loop, the rest of the world spirals into questionable balance. Germany has declared a massive military spending in support of the Ukrainian cause, and the United Nations, after a resolution supported by 141 of 193 member states, has demanded Russia to withdraw from the country’s borders, further cornering an already isolated Russia beleaguered by a flurry of economic sanctions.

The Russia-Ukraine war is a complicated mess haunted by the specter of past imperialism, and a worsened by the ego of a political leader who must, at all cost, cement a legacy before his people and the rest of the world of a fierce and vicious leader. Naturally, the anxiety of where this situation could be headed has rippled out to the world at large. Looking up the search term “World War III” in Google Trends shows a sudden peak in February 24. It would be presumptuous to assume that we (speaking primarily to non-Ukrainian citizens) are anywhere close to dread, especially as the country must end each day counting innocent lives lost in the conflict. But no one can be blamed for nevertheless feeling the looming dread and anxiety as the world teeters on the edge of another (global) war. I say global, because while it’s true we have lived in relatively peaceful times in recent years, it is wrong to think that war wasn’t happening all this time. We only moved it farther away from our homes.

But aside from donating to the Ukrainian cause and keeping our own governments in check, what else can we do in these times when, barely healing from the pandemic, the end already seems to be staring us in the face? For John Green, there’s cooking (and, I hope, writing). And for the graduate students, researchers, and professors out there – well, we study.

So you want to distract yourself from the sickening feeling that you are in the comfort of your living room, facing a textbook of graduate-level mathematics while out in the world another civilian, probably of the same age and qualifications as you, is facing bombs and rifles and dying loved ones. You want to get through this textbook, but you know that previous attempts to actually sit down and study have all been in vain. Every time you tried, something – an urgent matter at work, TikTok, a new Korean drama premiering on Netflix – came in the way and the textbook always ended up unread. Plus, there’s that quiet comprehending that none of this may matter, in the end.

When dread and distractions keep coming to you, an effective strategy is to time yourself. This has the effect of limiting, at least psychologically, the amount of work you have to do while still giving you the opportunity to put in some good work. It may not make much sense on paper, but there’s a considerable difference between “study a textbook” and “study a textbook for an hour”, in that the latter seems more manageable, and promises a foreseeable end to the travail of work. You’ll open up a timer, set it to an hour, during which do what you have to do, and then afterwards you can watch any number of Tiktoks you want, or see another episode of Succession.

This is a huge factor that makes the Pomodoro technique effective. Unstructured, work seems daunting: there’s just too many pages to read, and so much Math to digest. Not to mention, without a taskmaster egging you on from one task to the next, there is a tendency to sacrifice time for distractions. Something has to keep you glued to the pages, or otherwise your mind will dwell to thoughts of why is it than when politicians and wealthy people wage their wars, it’s always the citizens and the poor that have to sacrifice their lives. With the Pomodoro method, you structure your work in sprints of 20 to 25 minutes, with the aim of completing as much work as you can within each sprint, and then in between a rest period of 5 to 10 minutes during which you are allowed to do whatever (such as watch Russian oligarchs escape the crumbling Russian economy on their private jets). It’s an organized dosage of work and distraction that keeps one from ruining the integrity and enjoyment of the other.

Personally, the Pomodoro technique doesn’t work for me when it comes to studying. It works for tasks like writing code, writing stories, and reading novels, but for studying Mathematics the sprints are too short, and sometimes I’ll come to a theorem or a remark that requires a long period of concentration, and breaking that concentration for a ten minute break is only likely to ruin, rather than improve, my productivity. So instead, for Math, I do a larger sprint instead, with no breaks. I’ll set up a timer for an hour and 30 minutes (the usual length of my Math classes back in college) with the goal of finishing as much work as I can before the time ends. Breaks can happen in the middle, for small doses of 5 minutes each, but in general the goal is to get down to business for 90 minutes, and then call it a day (for that task, at least). And then I can be free to think about how ultimately, the freedom I think I enjoy is ultimately an illusion, when my government can – when it so chooses – anytime decide to send me off to die in war, seize all my assets, or have me gunned down at random on the street.

Another important note when studying Math: it’s important to have pen and paper ready at all times. While it may be possible to study Biology, or Literature, or Philosophy while only reading the text and jotting down only the occasional notes, so much of Math is written in notation whose meaning may not be revealed without a laborious breaking apart and re-construction of its symbols. Today I was brushing up on my knowledge of stochastic processes for a research I’m writing, when I came across a particularly baffling notation from Petar D. Todorovic’s An Introduction to Stochastic Processes And Its Applications. Baffling because I can see that the author is showing a result of some importance here, but I could hardly make out what he was trying to say. For the sake of discussion, the suspect notation was the following: $\{ \omega; X(t, \omega) \ne Y(t, \omega) \} = \{\omega; \omega = t\} = \{t\}$ and $P\{t\} = 0$.

If I haven’t lost all my readers already at this point, I’ll now proceed to explaining how I came to understand this notation. This particular part is crucial to the proof that Todorovic was presenting at this point of the textbook, so it was important I understand its contribution to the argument. First, I reviewed the rest of the book’s notation and realized that $\{ \omega; X(t, \omega) \ne Y(t, \omega) \}$ was referring to a rule-based definition of a set: it’s a set of all points $\omega$ such that the two functions $X(t, \omega)$ and $Y(t, \omega)$ will result in unequal values. Earlier in the definition, it is stated that in general $X(t, \omega) = 0$ for all $t$, while $Y(t, \omega) = 0$ except when $t = \omega$, at which point $Y(t = \omega, \omega) = 1$.

So $X(t, \omega)$ and $Y(t, \omega)$ are always equal (both zero) for any value of $t$ except when $t = \omega$. So the set $\{ \omega; X(t, \omega) \ne Y(t, \omega) \}$ is effectively the same as the set $\{\omega; \omega = t\}$. And, if we list down all the values of $\omega$ for this set, then that would be the set containing only one element: $t$, or in notation, $\{ t \}$. It then dawned on me that with $P$, Todorovic was referring to the Lebesgue measure, and coming from the real line, a singleton set like $\{ t \}$ will have a Lebesgue measure of zero, and hence $P\{ t \} = 0$. Notation cracked.

In reality, this notation is about as easy as it gets for Todorovic’s textbook, or any graduate-level Math textbook, which is why this ordeal was pretty embarrassing for me. But it goes to show that even in simple cases, mathematical notation won’t always yield easily when one is only looking at them and not actively engaging in their operations. When studying Math, or reading research paper in a specialized field of it, it helps to piece out the puzzle along with the writer’s discussion. Or else you’re bound to be left behind.

Ultimately, none of this will amount to much when the world plunges into another widespread conflict. But having something to do does keep the mind from buckling under the existential dread of a life this close to being interrupted (knowing that for the citizens of Ukraine, life has already been interrupted). We stand with Ukraine. We condemn the Russian government for making innocent civilians pay the price of his project of reanimating an empire long dead. While the world marches on to an uncertain future (from an already uncertain present), we must all somehow push our boulders and at least attempt a show of calm.