On Senior Year

Cornell Tower

I’m a couple of months into my final year of school. This post is a reflection on my senior year so far and the Cornell experience in general.


Senior year has been quite different from any other year. This is largely due to a more carefree attitude resulting from having post-college employment already lined up. In addition, this is the first semester in which I don’t have to take any distribution requirements, so I get to take whatever I want.

At first glance I seem less incentivized to do work. But in fact, it has made me more productive than ever before. Not having to research companies/grad schools, fill out applications, prepare for interviews, etc. frees up a lot of time. I feel much less stressed than in earlier years, and I feel happier in general. I now have the time for introspection, to put aside the act and think about what I truly care about.



Even though I am majoring in math, most of the greatest classes I’ve taken were not in the math department. Intro classes in astronomy by Steve Squyres and sociology by Ben Cornwell were very eye-opening. Computing in the Arts (by resident genius Graeme Bailey) was a refreshing multidisciplinary class that truly combined everything together. And in math, the honors intro sequence (2230 & 2240, by Ravi Ramakrishna and John Hubbard respectively) shattered and rebuilt what I thought math was.

But the learning extended far beyond classes. I’ve met some really amazing people here from all over the country (I wanted to say world, but that would be a lie). And of course, the Cornell experience wouldn’t be complete without seeing famous people, whether through lectures, connections, alumni status (Bill Nye), or even pure coincidence (*cough* Bill Murray).

The Future

It feels strange knowing this will be my last year of school. If you count kindergarten as a grade, that’s school for 17 years consecutively, and that could have been more if I were going to grad school. I’ve lived the vast majority of my life in the academic life, and it feels like almost a relief to be headed next year into the real world.

Thoughts on Classes, Spring 2013

In a previous article, I posted my schedule and about my decision to double major in mathematics and computer science. The computer science department seems to be quite backed up at the moment, so I have not received any official response yet.

I can see why the CS department is really backed up. In most of my experience at Cornell, I had class sizes of 10-30, with larger class sizes (150-250) only at introductory level courses such as Sociology 1101 or Astronomy 1102. It would be quite rare to have an advanced level course with that many people in them.

goldwin smith 132

But, CS easily has 150-250 people in each class. In the first few days of class, even in large lecture halls, there were no seats left and the late-arrivers had to sit in the aisles. I think students here see CS as too lucrative of a skill to pass up. Some difficult or otherwise time-consuming homework assignments have caused class sizes to drop significantly, but there are around 150-250 people in the CS classes. On the other hand, my math classes have 14 and 6 people respectively.

Math 4340 – Abstract Algebra

Professor: Shankar Sen


This is a fairly trivial class so far. We are covering basic group theory and it is quite a relief compared to some of the more intense math I did last semester (*cough* topology). The course is supposed to move on to rings and modules later; however, in linear algebra we actually covered much of the foundations of ring theory and modules.

However, given the lack of difficulty of the topic so far, the homework grading has been quite harsh. I usually skip writing down every rigorous step if I think some part is obvious. Learning the material is more important than writing down every detail of the proof, in my opinion.

Math 7370 – Algebraic Number Theory

Professor: Shankar Sen


There are no exams, no prelims, and no homework. However, it is a graduate level seminar-type class and it is pretty insane. I have put up my lecture notes on Scribd, and even if you know nothing about college math, if you click that link, you can probably see how much more difficult 7370 is than 4340.

It is a really good thing I had a basic introduction to ring theory and modules before taking this course. Knowing what PID (principal ideal domain) and UFD (unique factorization domain) mean, knowing the difference between prime and irreducible, etc., was extremely helpful.

This class is even more difficult than the graduate Complex Analysis course that I took last year. Before I took complex analysis, I actually knew quite a bit about complex variables, complex functions, and contour integrals. I had even studied the Riemann zeta function in high school. And on top of that, I was not the only undergraduate in that class—there were at least 3 others.

But for algebraic number theory, this is really new material, most of which I haven’t seen or even heard of, and moreover, I am the only undergrad in the class. However, I talk with the professor outside of class and I am confident that I can learn the material if I really try.

Math 4900 – Independent Research/Reading – Elliptic Curves


Since I felt that I was doing too much CS and not enough math, I decided to add on an independent reading class. The book is The Arithmetic of Elliptic Curves by Joseph Silverman.

I have seen elliptic curves in complex analysis in the form of the Weierstrass P-function and equating points in the complex plane by a lattice. To see the algebraic side of it will be interesting though, especially because I am interested in number theory for possible research.

In addition to this official reading, I am also reading and doing problems from Tom Apostol’s Introduction to Analytic Number Theory, so that I can get both the algebraic and analytic sides to it.

CS 4820 – Introduction to Algorithms

Professor: Dexter Kozen


This is a really fun theoretical and mathematically oriented class. After all, Kozen is practically a mathematician.

Given my mathematical background, especially the combinatorics class I took last semester, this algorithms course is not too difficult and in fact fairly trivial so far. But, I expect it to get more sophisticated once we get over the introductory stuff. For instance, on our discussion board on Piazza, one student asked how to use a contradiction proof. In just topology alone, I probably used about a hundred.

In addition, Kozen shares some very interesting stories during lecture. Just last Friday, he was talking about dynamic programming and discussed a project using body scan data to analyze the number of dimensions it took to store the size information of a human body. “Are women 2-dimensional? I don’t think so,” said Kozen. In fact, he recalled from the study that women were around 5-dimensional and men were fewer.

Also, when he was explaining the growth of the Ackermann function A(n), he noted that even A(4) was an extraordinarily large number, and in fact that it was “even higher than Hopcroft’s IQ.”

CS 4850 – Mathematical Foundations for the Information Age

Professor: John Hopcroft


From the title of this course, one might think it is really easy, but even as a math major, I find it nontrivial (that means hard, in math terms). In fact, I’d say at least 30-40% of the class has dropped since the first day. The fact that Hopcroft won a Turing award makes the class no easier.

It is essentially a mathematical and statistics course with applications. We proved the Central Limit Theorem on the first day of the class and then looked at spheres in high dimensions, with the intent of generating random directional vectors in high dimensions. As it turns out, most of the volume of a high-dimensional sphere is on a narrow annulus or shell, and when a given point is taken to be the north or south pole, the rest of the volume is located at the equator.

Currently we are studying properties of large random graphs, in particular, properties that appear suddenly when the edge saturation of the graph passes a certain threshold. For instance, below a certain number the components of the graph are all small, but above that number, a giant component arises. For an assignment I showed how this giant component phenomenon arises in connections of the Reddit community.

CS 3410 – Computer System Organization and Programming

Professor: Hakim Weatherspoon


In contrast to the high-level programming I have done in the past, this course is about low-level programming and the hardware-software boundary. The programming language for this course is C.

We are building up a processor from the ground up, one could say, with basic logic gates to begin with. The first project was to design a 32-bit arithmetic logic unit (ALU) using Logisim, a circuit simulation program. For instance, for a subcircuit we needed to create a 32-bit adder with overflow detection.

The above picture is actually a screenshot of the overall ALU that I designed for the class. The subcircuits are not shown (this project is not due yet, so it would break academic integrity to show a more coherent solution).

The Math You Weren’t Allowed to Do

You probably learned a bunch of things in school math about what you can and can’t do. When you were a first grader, perhaps you learned that that you can’t subtract 5 from 2, but later on, you learned about negatives: 2 – 5 = -3.

You also might have been told that you can’t divide 2 by 6, but then you learn about fractions. And by now, you are no doubt an expert at splitting 2 pizzas among 6 people.

Even so, there is much more that you may not have known…

1. You Can’t Count to Infinity

Actually, you can. It can be done via ordinal numbers.

You start out counting by

1, 2, 3, 4, 5, 6, 7,… 700,… 30,000,000, etc.

When you played a “what’s the highest number?” game with someone, every time you said a number, they countered by saying your number plus one, that is, unless you said infinity. Because infinity plus one is still infinity, right?

Here is where ordinals come into play. The ordinal number \omega (ω, omega) is defined as the first number after ALL of the positive integers. No matter what normal number they might say, whether it’s ten billion or a googol, the ordinal number \omega is far, far larger. It is practically infinity.

But then you can add one to it, and it becomes an even bigger number. Add two, and it becomes even bigger.

\omega, \omega+1, \omega+2,..., 2\omega,...,3\omega,...,n\omega,....,\omega^2,...

What the heck is going on? If you count an infinite number of numbers after omega, you get two omega? Is this two times infinity? And then three omega? And then omega squared?

It turns out to keep on going. Eventually you will get \omega^\omega, and then \omega^{\omega^\omega}, etc. And then you reach \Omega (big omega), which is larger than all things that can be written in terms of little omegas. And then you can make bigger things than that, with no end.

So the next time someone claims infinity is the largest number, you can confidently reply, “infinity plus one.”

2. You Can’t Divide by Zero

Actually, under certain conditions, you can.

The field of complex analysis is largely based around taking contour integrals around poles. Another word for pole is singularity. And another word for singularity is something you get when you divide by zero.

Consider the function y = 1/x. When x is 1, y is 1, and when x is 5, y is 1/5. But what if x is 0? What happens? Well, 1/0 is undefined. However, if you look at a graph, you see that the function spikes up to infinity at x = 0.

What you do in complex analysis is integrate in a circle around that place where it spikes to infinity. The result in this case, if done properly, is 2\pi i. It’s quite bizarre.

3. You Can Only Understand Smooth Things

Actually, there is much theory on crazy, “pathological” functions, some of which are discontinuous at every point!

The image above is kind of misleading, as it is a graph of the Cantor function, which is actually continuous everywhere (!), but nonetheless manages to rise despite having zero derivative almost everywhere.

There is another function with the following properties: it is 1 whenever is x is rational and 0 whenever x is irrational. Yet this function is well understood and is even integrable. (The integral is 0.)

Then you have things that are truly crazy:

The boundary of that thing is nowhere smooth, and is one of the most amazing things that have ever been discovered. Yet it is generated by the extraordinarily simple function z^2 + c, which most people have seen and even studied in school.

4. You Must “Do the Math” and Not Draw Pictures

Actually, math people use pictures all the time. The Mandelbrot set (the previous picture) was not well understood until computer images were generated. There is no such thing as doing the math in a “correct” way. Some fields are quite based on pictures and visualizations.

How else would anyone have thought, for example, that the Mandelbrot set would be so complex? Without seeing that in pictures, how would we have realized the fundamental structure behind the self-similarity of nature?

Yeah, that’s a picture of broccoli. Not a mathematical function. Broccoli.

5. If It Doesn’t Make Sense, It’s Not True

Actually, many absurd things in math can be perfectly reasonable.

What’s the next number after 7?

8, you say. But why 8? What’s wrong with saying the next number after 7 is 0? In fact, I can define a “number” to only include 0, 1, 2, 3, 4, 5, 6, and 7. Basic operations such as addition and multiplication can be well defined. For example, addition is just counting forward that many numbers. So 6 + 3 = 1, because if you start at 6 and go forward 3, you loop back around and end up at 1.

Even weirder is the Banach-Tarski Paradox, which states a solid sphere can be broken up into a finite number of pieces, and the pieces can be reassembled to form TWO spheres of the exact same size as the original!

I hope this was understandable for everyone. May the reader live for ω+1 years!

Cornell Fall 2011

Cornell Tower

It’s been a nice semester so far. The weather is so much better than in Austin, where temperatures are still surpassing 100°, and it feels great not being a freshman—I actually know where things are!  Classes I’m taking:

  • Math 4130 (Honors Introduction to Analysis I): Mostly review so far. It’s only been a week, I’m pretty sure it will get harder as the semester goes on. But I definitely think Math 2230/2240 are great preparation.
  • Phys 1116 (Mechanics and Special Relativity): Also mostly review.
  • Gerst 1210 (Exploring German Contexts I): Intense immersion learning. I feel like I’ve learned more German in the past week than I learned in my first month of Spanish in middle school. This makes perfect sense, since college is quite a step up from middle school.
  • Econ 3010 (Microeconomics): We’re covering some of the underlying theory of economics, which is intrinsically heavy on math. Particularly, a lot of what I learned in Math 2230 last year is coming into use. Having independently studied some game theory over the summer, I found this introduction very interesting; things are clicking already.
  • CS 2110 (Data Structures and Object-Oriented Programming): Almost all review. In yesterday’s section we covered linked lists, which seemed to baffle at least half the group. The 25-hour assignment due next week does not look fun though.

Of course, I am also in the band, which has started up. Once again I am impressed by the speed at which it learns the show. Everyone seems ready for performance after a mere one rehearsal.

My residence for this year is 14 South Avenue, which is on the southwest corner of campus. This is a total change from last year, when I was on the northeast side of campus. The change of perspective is certainly nice.

Here is my schedule from Schedulizer:

Reflections on High School

Today I took my final final exam in high school, and in a couple of days is graduation.

In my mind, however, school ended a long time ago (in a galaxy far, far—nevermind). I was mentally finished by the end of the first semester, when I began receiving college acceptances. I still remember December 15, 2009, the day I was accepted to the University of Chicago—it was my first real acceptance, as UT Austin accepted me earlier but I knew I was already guaranteed admission from the 10% rule. And though I’m not going to UChicago—I’m going to Cornell—that acceptance letter in my mind sealed away high school.

And when I say “sealed away,” I do not mean I stopped caring. I merely started to view everything from a philosophical, artistic point of view, by questioning things and by not being so rational, for no purpose other than being creative and trying new ideas. Most of my examples of this would come from second-semester during English class, in which I made liberal use of puns and tried to incorporate wit in many other ways. My particular memories include a discussion on Kafka’s Metamorphosis; a short play based on Shakespeare’s Othello and Stoppard’s Rosencrantz and Guildenstern Are Dead as a creative group project after study of the former; and the independent study project, for which my topic was the Physics of the Falcon Punch. My teacher, Ms. Gaetjens, was quite understanding—or perhaps forgiving—of my creative side.

I would elaborate very much on each of these, but the title of this post is “Reflections on High School” and not “Reflections on One Class in One Semester.” It is true that I already touched on many things relating to high school in my similar post at the end of the previous semester, in my Reflections on 2009, so I’d do best to cover the opposite of what I covered in that post.

To be continued… EDIT: Or not.

IB Theory of Knowledge

IB Logo
International Baccalaureate

One interesting component of the International Baccalaureate (IB) is the Theory of Knowledge (TOK) class, a one-year course that, at my school, is taken in the second semester of junior year and the first semester of senior year. The reason I would describe it as an interesting component is that the class is so different, so bizarre in comparison to the other IB classes we take. Instead of teaching a set curriculum about a particular subject and then preparing for an end-of-year examination, TOK emphasizes thinking, or at least, the way we think, or the “ways of knowing.” It has some elements of a philosophy course, and though it does not completely qualify as one, our teacher Dr. Schaack would categorize it under “Applied Philosophy.”

Regarding the purpose of TOK, our teacher described it as in part to find out whether students can think. Thinking is quite a different activity from test-taking. The IB wants to make the most out of an individual, and one part of this is to tweak the way we think, or at least make us aware of different theories of knowledge. As such, it is an interdisciplinary course, where matters from all other subjects are discussed.

What does one do in TOK? This is a frequently asked question, and I had asked this myself to IB seniors several times last year. If I had to describe the class in three words, I would say, “discussion,” “thought,” and “application.” Discussion of what? Of almost any topic you can imagine. In just my class, I have heard and participated in discussions about current events, the subjectivity of knowledge, quantum mechanics and its relation to reality, the Iraq War, dreams, the three-second present moment, Nobel prizes and laureates, the fourth dimension, essay writing, college application, the authority of science, and much more. Our discussions have mainly been centered about two texts: Sophie’s World (1991) by Jostein Gaarder in junior year, and Zen and the Art of Motorcycle Maintenence: An Inquiry into Values (1974) by Robert Pirsig in senior year. These are only two of the many examined works (a list not limited to only books), and especially with the latter, we have had some extraordinarily thought-provoking discussions.

Other works studied in my class include: “Allegory of the Cave” by Plato, Waking Life (2001) by Richard Linklater, “The Dimension of the Present Moment” (1990) by Miroslav Holub, Flatland: A Romance of Many Dimensions (1884) by Edwin Abbott, Nobelity (2006) by Turk Pipkin, and “The Fourth Dimension,” a chapter of The New Ambidextrous Universe (1991) by Martin Gardner.

The second word, “thought,” necessarily accompanies the first. After all, one cannot participate in a high-level discussion without thinking about what to say. Thus, to participate, one must review what has been discussed already and then continue on or take a different path. It is a class where any relevant, insightful thought is welcomed.

Finally, on to “application.” What use are thoughts that do not apply to the world? In discussions, even of abstract concepts, we often cite concrete examples to demonstrate the implications of our ideas. Even if a topic does not directly affect our daily lives, for example the existence of black holes, a discussion of such in a talk about the advancement of science is more grounded than one that only refers to science in general.

In addition, two essays of prime importance are written in this class. The first is the Theory of Knowledge essay, an essay that I actually have added to this site (under Essays). It is an essay that allows the writer to select from ten possible choices and write more or less freely about it for around 1500-1600 words. It is also a mostly free-form essay, written in a highly personal manner; the teacher recommends up to three sources in the bibliography.

The Extended Essay, on the other hand, is a wholly different matter. It is essentially a research paper, on the topic of the student’s own choosing, and should contain at the minimum 10 sources (except for exceptional cases such as essays on mathematics and experimental sciences). The length is up to 4000 words. Mine is currently not finished yet. As of today, half the essay is due next class.

It can be said that TOK is an incredibly unique class. Simply, I have never before had such a thought-provoking and thought-changing experience.