My College Experience

Cornell

Yesterday, I took my final final exam. Now, short of receiving a piece of paper, I am done with college and also with the formal education system (for at least the time being).

I’m not a sentimental person, but I am a reflective person, so I feel compelled to write about my experience.

Several other posts already covered various aspects of college and also of Cornell specifically:

There are in total 35 blog posts (as of writing this) under the College category, including the ones listed above. But the most important post comes from before any of these, before even stepping onto the Cornell campus, and it is related not directly to Cornell, but to the University of Chicago, a post on Andrew Abbott’s “The Aims of Education” speech.

Abbott’s main argument is that education is not a means to an end, but the end in itself. He goes through why education is not best viewed as a way to improve financial status, a way to learn a specific skill, a way to improve general life skills, or a way to survive in a changing world. Instead, “The reason for getting an education here—or anywhere else—is that it is better to be educated than not to be. It is better in and of itself.

This philosophical point I carried throughout my college experience. It is why I find it absurd to worry about the GPA of oneself and others so much: you’re here not to beat other people, but to be educated.

There is a lot of interest in the relation between academic study and the real-world job market. One hears jokes about English or psychology majors working in jobs having nothing to gain from an English or psych degree. But my situation is actually similar. As a math major pursuing a theoretical track (originally thinking about academia), I’ve encountered concepts that, at least currently, have no practical application. That’s a blessing and a curse. In the post I wrote about why I chose math, one of the pro points was precisely the abstraction of it. So, even though I will be working in a math-related area, it is almost certain that knowing that normal spaces are regular, or that the alternating group on 5 elements is simple, is useless.

Of course, it does help to know calculus and to have a good understanding of probability. But at least over the summer, we rarely ever used concepts that were outside my high-school understanding of probability or calculus. In other words, I could have majored in English and have been just as qualified.*

*(Perhaps taking many math classes trains you with a certain type of thinking, but this is hard to specify. I haven’t thought too much on this so if anyone has other ideas, please share them.)

Another thing I haven’t really talked about in other posts is socializing. I’m an introvert (INTP), and I could easily spend all day reading thought-provoking books or watching good movies without the slightest urge to unnecessarily talk to another person. I used to ponder this, but after reading Susan Cain’s wonderful book Quiet, I’ve decided to not worry.

Academically, I’ve expanded my horizons a lot since coming to Cornell, though not from math courses. While academia in general can be thought of as an ivory tower of sorts, math (and/or philosophy) is the ivory tower of ivory towers, so it is sometimes refreshing to take a class in a different subject that is only one step removed from reality.

In addition, I managed to keep this blog alive through college, though there was a period of time in late freshman/early sophomore year where there were few posts. By junior year, I was back in a weekly posting routine. And a couple of months ago, I started doing 2 posts per week, and that has been consistent so far.

Finally, I also subscribe to a quote allegedly by Mark Twain: “I have never let my schooling interfere with my education.” Even after college, I will always find opportunities to learn.

Overall, Cornell has been a great experience, and I would definitely recommend it, even if not for the reasons you were looking for. Enjoy, and keep learning!

When Principles Collide

One of the things about growing up with a sheltered life is that you rarely ever have to stand up for your principles. This could be due to several reasons: maybe they’re not really your principles, but your parents’; maybe you’re just not placed into situations where conflicts occur; maybe your principles themselves seek to avoid confrontation. I recall so many times when I was younger that I had some well thought-out idea for something but then instead went along with someone else’s idea without question, in the interest of avoiding conflict. I’m not saying that you should always insist what you’re doing is correct, but I think on the spectrum I was too far on the side of passivity.

Throughout college (and perhaps starting senior year of high school), I found myself more often at points where I needed to disagree. It wasn’t really conflict for the sake of conflict, but rather to get to the truth or to make a situation better, by challenging faulty ideas or plans. I think this change is evident on my blog: in the past, most of the topics I wrote about were very non-controversial, but recently, they have been more questioning of commonly held ideas. Granted, my online persona (including on Facebook) and my real life character are still quite different—in real life I don’t go around seeking to criticize peoples’ religious beliefs, an activity that is reserved for the internet. That’s another topic.

Contradictory Principles

For a really simple example, consider the principles “be honest” and “don’t be a jerk.” Everyone follows these principles, and most of the time they support each other. You’d be quite a jerk if you lied to your friends about so many things to the point where nothing you say has any credibility. However, when you find minor fault in something someone did, you could be honest and tell them, but most of the time it’s better to be silent about it. Of course, the best choice depends all on the situation.

contradiction-signs
I respect both ownership rights and aesthetic cleanliness—do I pollute whitespace by citing the image source, especially if the image isn’t all that special?

Perhaps a more pertinent contradiction is that between tolerance of others and… tolerance of others. For example, most of my audience probably tolerates the LGBT community. Yet, there are many people in America who do not. This leads to a tolerance paradox (that I think many of us don’t think about): Is is possible to simultaneously be tolerant of LGBT individuals and tolerant of people who are intolerant of them? Is a hypothetical all-tolerant person also tolerant of intolerance?

This depends somewhat on how you define tolerance, but it points to a deeper issue, that simply using the principle “tolerate others” is insufficient in these fringe cases. There must be some overriding principle that tells you what to be tolerant of and what not to be tolerant of. I think that being intolerant of intolerance is still tolerant.

In chess, one of the most important principles, among the first to be taught to new players, is to never lose their queen unless they can get the opponent’s queen as well. While this is a great principle 99.9% of the time, there are cases where losing your queen (for no pieces or just a pawn in return) is the best move, and there are even cases where it is the only non-losing move. It’s because the principle of “win the game” overrides the principle of “don’t lose your queen.”

Interestingly enough, even meta-principles can contradict one another. For me, “stand up for your principles” is a good principle, and so is “be open-minded about your principles.” Often blindly standing up for principles is a very bad idea (in the typical novel/movie, the antagonist may have good intentions but focuses on one idea or principle to the exclusion of all others, thus causing more overall harm than good; on the other hand, this principle seems required to become a politician).

Throughout my first two years of college, I wanted to go into academia, and I naively shunned finance because I thought people went into it just for money. Of course, once you start thinking about what to do after college and the need for money comes closer, you realize that you need money to live(!) and that despite the negative outside perception, the industry is not all evil people trying to figure out how to suck away all your money. Of course, on the “stand up for your principles” front, this change fails pretty hard, but it follows “being open-minded about your principles,” which I consider to override the first in this case. After all (to add one more layer of contradiction), it is standing up for the principle of being open-minded.

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.

Differences

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.

Cornell

Cornell

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.

My Spring 2013 Semester (Part 2)

Part 1 can be found here.

I’m finally done with the semester. As I wrote in part 1, this has been my busiest semester at college. Most of the time was spent on one class: CS 3410, or Computer System Organization and Programming. I’ve probably spent twice as much on this class than all of my other classes combined.

On the other hand, I did learn a lot from this course. While I do not regret taking it, this kind of workload does call into question the decision to go for the CS degree. As I wrote before, going for the CS degree will negatively affect my ability to take more advanced math courses. Even this semester, I felt I had almost no time to study math on my own. In my math classes I was pretty much doing the bare minimum so that I would have time to work on CS. Next semester I will most likely be going pure math.

Relatively-Prime-Grid-Points

Anyways, for the summer I have an internship in New York City, with one week in London.

Stress and GPA-centrism in College

studying

Every time papers, projects, and prelims come around, the campus stress level rises dramatically. Sleep is lost (or outright skipped), meals are avoided, and all activities other than studying are brushed off. This happens not once a semester but throughout, corresponding to large assignments for every class.

And every time this happens, it seems that many students are focused not on actually learning the content, but on scoring higher grades than others. Of course, this phenomenon occurs in certain majors (engineering) much more than others. And I would guess that it happens at Cornell at an above average rate compared to that of the typical university. But it raises some questions that I want to explore.

Just a couple of notes. First, this article will mainly focus on the math/science/engineering side. And second, I do not think I need to mention why Cornell should be concerned about student stress.

Should Competition Be GPA-Focused?

Competition to a certain degree is beneficial, and I think no one would argue with that. As a math major I know very well that competition leads to efficiency. But there is a line where the marginal benefit in efficiency is not worth the huge increase in stress levels, and in this respect I think Cornell has crossed the line for good.

In addition to the GPA competition, there is the additional factor that students are competing for jobs, internships, and research positions. I think the competition here is mostly fine (except regarding salaries vs societal contribution; this topic deserves its own post). Combined with academic competition, however, this induces a vast amount of anxiety and stress in the students.

Without mentioning names, I will list some of the extreme behaviors I have observed of people I know:

  • A student pulled multiple all-nighters in a row to finish a project. While this might be plausible in real life for a rare occasion such as a doctoral thesis or a billion dollar merger, the student did this regularly for his classes.
  • A student has problem sets due on four out of five of the weekdays, and spends literally all his time outside of class eating, sleeping, or doing problem sets. In one particular class, the problem sets he hands in are 10-20 pages per week.
  • A student took 50+ credits in one semester, though he claims to know of someone who took 61.
  • A student brought a sleeping bag, refrigerator, and energy drinks to one of the school computer labs, and pretty much lives there, returning to his living place once every few days to shower.

Interestingly enough, I think these particular students will do fine, as they seem to know their own abilities and limits, and more importantly, they are all aware of what they are doing. They are also all very smart people who can actually learn the material. This “top tier” of students is not really adversely affected by competition, since they are smart enough to excel regardless of whether competition exists. Moreover, these students don’t seem to be grade-focused: they learn the material, and the grade comes as a byproduct of learning.

The group I am actually concerned about is the second tier. (Note: I just realized after typing this how judgmental that statement sounds, but hey, from statistics, unless you define the first tier to include everyone, there must be a second tier.) This group I would define to be the smart people who don’t seem to understand how the first tier operates. They see the students in the first tier getting high grades and know those students are smart, so they think that if they prepare the tests well and get high grades, they will become as smart.

What they don’t realize is the difference in cause and effect. The first tier prioritizes understanding first and the grades come as a byproduct, whereas the second tier prioritizes grades first and hope to gain some understanding as a byproduct.

Again, just as a disclaimer, these are just arbitrary definitions for first and second tier I made up for this particular observation. I am not saying that this criterion is the final say, and of course, there are numerous other factors regarding how well one does in college.

But from my experience, it is precisely the students in this second tier who are stressed. They are the ones trying so desperately to beat the test instead of to learn the material. And they are the ones who make college seem so competitive, as you can always hear them talking about tests and what their friends got on the tests and how they are being graded and what the format of the test will be.

On the other hand, the student I mentioned above who lives de facto in one of Cornell’s computer labs—I have not once heard him talk about anything specifically regarding a test.  The closest was talking about the material that was to be covered, but he was talking about stuff that was beyond what the class taught for the topic, stuff that he knew was not going to be on the test.

Some of you might be thinking, “That’s great, but what kind of job is he going to get if he is not grade-focused?” Good question. After working there for a summer internship, this same friend rejected a return offer from Goldman Sachs.

How to Break from GPA-centrism

I am not worried about this person’s career at all. I am worried about the second tier, the smart people doing mundane tasks, wasting a lot of potential creative brainpower that the world needs more than ever. Renewable energy, bioengineering, artificial intelligence, space exploration and colonization, nanotechnology—there are so many people here who would be excellent for these fields, yet many of them seem too bogged down by current competition-induced stress to even think into the future.

Indeed, this GPA competition is a force to be reckoned with, as it really is self perpetuating. Those students in other groups or who are apathetic to grades will tend to become more grade focused just from sitting in lecture, as there are always people who ask for as much details of the test as possible. I feel that this defeats the purpose of a test, which is to measure how well you know the material or how well you can apply a certain skill, not how much of the test structure you can memorize or how much content can you cram the night before.

Overall though, there are some measures that can be taken to reduce this kind of stress.

  • Reduce the importance of the GPA. I do not know if I would go so far as to remove it, though. For example, at Brown University, the GPA is not calculated. Somewhere in the middle ground should be best.
  • Stop showing score distributions, or show them only for major tests like a midterm/final. In many engineering classes at Cornell, the first thing that is requested after a test is graded is to see the score distribution (often in graph form), along with the mean, median, standard deviation, etc. In fact, this has become so common that it is now the first thing the professors put on their lecture slides. Moreover, the computer science department uses an online course management system which automatically tells students the mean, median, standard deviation, etc. for every single assignment, not just tests. Being a math major who would normally love extra statistics, I thought this was cool at first. But now I despise it—it is just too much information that I don’t need in order to learn the material, and it only detracts from my learning experience. And the way the page is setup, it is not something you can just ignore.
    • A side note: One of my classes in the CS department actually does not list grades, and I definitely feel more pressured to actually learn in that class, not more pressured to beat other people at grades like in other CS classes. Props to Professor John Hopcroft.
  • Teach better math/science/engineering/CS much earlier in the education system. A friend showed me this article today, a comparison between the CS education systems of the US and Vietnam, a comparison that is horrific for the US. If students already knew the foundations, then college would be what it was supposed to be: going really deep into a topic in a learning atmosphere, not treating us like elementary school children because, well, frankly that’s the level of engineering/CS of many college entrants. For instance, I think it’s great that students are trying to take Calculus 1 freshman year and then do engineering. Hard work is certainly a virtue. But wouldn’t it be much better for both the student and the college if they mastered calculus in high school? Imagine how much better our engineers would be.

What I envision is a class where students are trying to learn, not to beat each other on a test. I hope this vision is not too far-fetched.

Math or Computer Science?

Well this is an interesting situation. Just a month ago I announced that I was adding a computer science degree, so that I am now double majoring in math and computer science. The title of the post, after all, is “Computer Science AND Math.” Given the circumstances at that time, I think it was a good decision. My work experience had been mostly in software, and a CS degree from Cornell should look pretty good. In addition, I was wanting a more practical skillset.

decisions-2

In the past week, however, things have changed. I received and accepted an internship offer from my dream workplace, based on my background in mathematics and not in CS (though my prior CS experience was a plus). Based on this new situation, I have considered dropping the CS major (next semester) and taking more advanced math:

  • The CS degree has some strict course requirements, and I am afraid that if I go for the degree, I may be forced to skip certain math classes that I really want to take. For instance, I may have to take a required CS class next semester that has a time conflict with graduate Dynamical Systems, or with Combinatorics II. And given that I am currently a second-semester junior, I don’t have that much time left at college.
  • Even this semester, I am taking Algorithms, which meets at the same time as graduate Algebraic Topology. While Algorithms is pretty interesting and the professor is excellent, I am already very familiar with many if not most of the algorithms, extremely familiar with the methods of proof, and I feel that the experience is not as rewarding as possibly taking Algebraic Topology with Allen Hatcher, who wrote the textbook on the subject. I feel that I could learn algorithms at virtually any time I want. But learning algebraic topology with Allen Hatcher is a once-in-a-lifetime opportunity that I am afraid I am missing just because I want to get a CS degree to look good.
  • Even not being a CS major, I will still be taking some CS classes out of curiosity. However, these classes will no longer feel forced, and will not restrict me from taking the higher level math courses that I want to take.
  • My risk strategy for grad school is different now because of the internship. In the past, I would have been willing to take a decent grad school in math or really good job. (I would prefer grad school over getting a job, but of course, a good job is better than a mediocre grad school.) However, now that I have my dream internship, I am willing to play the grad school game with more risk.
  • But whether for grad school, trading, or just for curiosity, I would prefer taking advanced (graduate) math classes over undergraduate CS classes. In a sense, my taking of the CS degree was a hedge bet, as I wanted to reduce the possible cost of the worst case scenario. I knew that it would directly inhibit my ability to take advanced math classes via class time conflicts, but the thought was that if I couldn’t get into a good math grad school or get a good job using math, at least I would have a CS degree from Cornell. But, in this new situation, I think the risk is significantly reduced and the hedge is no longer necessary.

Interestingly enough, the primary motivation for dropping CS wouldn’t be to slack off, but to be able to explore more advanced math. (At least, that’s what I tell myself.)

I think this might be the second time in my life where I have had to make an important decision. (The first time was deciding where to go to college, and I certainly think I made the right choice there.) Unfortunately, I really can’t be both taking as many interesting math courses as I can, and at the same time be pursuing a CS degree. As much overlap as there is, I can’t do both. In an ideal world this might be possible, but not currently at Cornell.

So instead of the idea of having math and computer science, I am now having to think in terms of math or computer science. I am currently in favor of going with math, but I am not completely sure.

Edit: Thanks for the discussion on Facebook.

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

Cayley-Graph

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

Relatively-Prime-Grid-Points

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

Elliptic-Curve

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

Minimum-Spanning-Tree

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

Large-Graph

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

32-bit-ALU

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).