Vector and function

So I’ve been thinking about the great talk by Shaowei (great ideas always make us think). Apart from the sheaf, another important thing for me is the fact that vector and function are the same thing. A function is an infinite-dimension vector, while a vector is a function that maps from {1, 2, …, n} to .

I came to understand that a function is an infinite dimension vector while studying Gaussian process (a Gaussian process is a Multivariate Normal Distribution with infinite dimension). After Shaowei’s talk, I came to understand the other way round. Now, I have the complete picture.

But a question I had is that if vector and function are the same thing, how come one is used to represent a point and the other is used to represent a collection of points? Well, we use these abstract concepts to represent specific things, but the concepts are not the things they represent. Furthermore, I reckon the symbols tell us that one point in an infinite-dimension space is the same as an infinite number of points in a one-dimensional space. I couldn’t see this connection between this two geometric objects when a coordinate system was in my head, but abstracting that out, the symbols showed the way. And with that thought, it sort of makes sense to get rid of the coordinate system….

An inspiring math talk

I attended a math talk by Assistant Professor Shaowei Lin yesterday. The talk was about the history of algebraic geometry, about how Descartes came up with the coordinate system 500 years ago to symbolically represent geometric objects, about how mathematicians spent the next 500 years embracing the system only to try eliminating it over the last 100 years. And they succeeded. He talked about Bertrand Russell, Alexander Grothendiek and many great achievements of mathematics before, during and after their times. As always, his talk was very interesting and inspiring.



Here’s a did-it-myself monitor stand. Two shelves and four legs salvaged from old cabinets. I just had to buy the brackets and assemble everything.

Why? Well, first of all, it’s cheaper: the brackets cost $5.80 compared to $15-20+ for a normal monitor stand. Secondly, it fits my space; the ones off the shelves don’t. Thirdly, it’s fun, it’s cool, and it gives me satisfaction. And lastly, I want to teach my son to reuse stuff, to give new use for old things, and to make use of available resources. Plus a bit of physics and engineering, after all.

It’s good wandering, it needs commitment, and it’s good training

Research output is highly nonlinear. It’s unlikely that you can (or at least, I can) consistently progress by the same rate each day. For the past couple of weeks, I’ve just been wandering around looking for a direction (which means I am not moving forward). But I believe that once I found a direction, I can make a jump. The prospect of that day makes the wandering enjoyable. I remember the conversation I had with Sean and another colleague—let’s call him the third guy. Third guy said a research career is risky, as you’ll never know if you’re gonna figure out the answer. Both Sean and I thought that it is the uncertainty that made it interesting.

Although output may not be consistent, input can be, and should be. I’ve decided to commit 12 hours a week to it. And recently, my little one has resumed his “good” sleep timing, which means I can have the two hours in the early mornings again. Great! It’s recess week now, and my teaching assistant duty will start next week. I have more commitments in the second half than in the first, which means I have to be more organized. First semester was totally about coursework. Second semester was 75% research and 25% coursework. This semester is a balancing act. I see it as good training.

I’m taking two days of semi-break: spending the days at home with the little one while fitting work into the non-baby intervals (early morning, late night, when he eats etc.) This is also good training: I have to divide my work into chunks, just as GP advised me previously. By the way, it’s nice that the little one is having his sunshine period right during my recess week. He’s so joyful. His smiles are so adorable (it can melt anyone’s heart). I’m gonna enjoy it while it last—very soon he’s gonna hit another sensation week and will be grumpy again….

Keep pushing.

More advice over breakfast

We had a breakfast talk this morning, and I was one of the two organizers. The topic that I planned was “How to cope with your first PhD semester?” but it the talk soon widened up to coping with a PhD in general. There were interesting ideas around, most of which came from SL, our guest-of-honor for the day.

  • Many useful pieces of advice is contained in this blog post on the SUTD Brain Lab website
    • Have a notebook to write down ideas (still experimenting with different ways to take notes)
    • Have a personal website (I’ve already done this, yay!)
    • Have regular meetings with your advisor (thankfully, I don’t have to push for this as my advisor is super “on”)
    • Write down minutes of the meetings (need to start doing this)
    • LaTeX (I taught myself LaTeX—with helps from Google, of course—and I’ve been submitting homework and reports in LaTeX since the middle of my first semester).
    • GitHub (Sean’s been telling me to do this, and then Zunction wrote a nice tutorial and asked me to try out. I’m gonna start tomorrow).
    • Python and other Python related stuff (during the summer I started learning Python, but I was also learning R at the same time. And I found learning two languages at the same time difficult. So right now, Python’s status is “in the queue”).
  • Every 3 months, sit down in a quiet corner and have a mind map to connect all the research ideas into a big picture. Eventually, you’re gonna have a giant mind map that is the gist of your thesis.
  • Have a “research summary” every month to consolidate all the things you’ve come across in that month. Well, I actually did this in my summer project—I wrote 3 reflections in those 3 months. But I have to admit those reflections were more on feelings than on science.
  • It is the act of sorting and organizing ideas and references that is the most important. It is a mental exercise that helps you have a “helicopter” view of your work. The actual end results of the sorting is less important.
  • Sometimes it is more important to let people experience things and come to the conclusion on their own than telling them everything.

Now, those are practical and specific tips, but here’s the coolest one: in your PhD, it is super important to work with other students. SL has seen very smart PhD students working alone and ending up nowhere while less smart PhD students succeeding by drawing supports from a group.

I have to say there are many cool people around here. It’s great!

I’m happy

I am doing a PhD, which I enjoy, on water resources management, which I care about.

I am taught, trained and mentored by brilliant nice people (not only now, in my past works too).

I’m surrounded by highly intellectual people. The amount of knowledge I learn every day is amazing. Back in my working days, I often felt so bored that I had to look out for podcasts to enrich my mind. Now, my brain is never hungry, and podcasts become virtually unnecessary (except for an occasional dose of Freakonomics).

I have a wonderful wife, more than any guy could ever ask for. And she just baked me a jar of chocolate cookie to munch in the office.

I have a happy and healthy baby boy who is super active, constantly moving about experiencing the world.

I have a strong family support system.

I’m living very comfortably.

I have no reason to be unhappy. I can be worried and upset, but I should always remember that I have all the reasons to be happy.


Little one is sick. He has an eye infection and a fever. On Friday night his temperature peaked at 39.3 °C. But he’s recovering well, and today, he’s active and happy as if he’s fully recovered. He’s amazing. And these few days when he’s sick, he’s clinging on me more.

His waking up time has been oddly early for about a week, which makes my wake-up-early and-do-work plan no longer feasible. So I switched back to staying up late at night for my lucubration. I still wish that I could resume the early waking up soon—I think I work better in the early morning.