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!

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.

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).

Why Math?

As a math major, I am often asked the question, “Why math?” In particular, why theoretical math, when it doesn’t seem to be related to anything?

I often have trouble coming up with a full and satisfying answer on the spot. Math is one of the subjects whose material and categorization can be confusing. It spans several fields that many have not even heard of. When I say “topology” or “analytic number theory,” it often draws blank stares; in fact, topology is often misunderstood as “topography,” the study of terrain.

Well, here is my more thought-out answer to why I study math.

Is It Relevant?

Take a good look at the following equation:

\displaystyle \sum_{n} \frac{1}{n^s} = \prod_{p} {\frac{1}{1 - \frac{1}{p^s}}}

Through the cryptic jumble of symbols, you might ask yourself, how is this useful?

An even more relevant question for most readers is, what does it even mean?

As it turns out, this particular equation has very little practical value. Yet it is one of the most fundamental equations in the field of analytic number theory, and it is a remarkable statement about the prime numbers.

It basically links the infinite set of natural numbers (1, 2, 3, 4, 5,…) with the infinite set of prime numbers (2, 3, 5, 7, 11,…). The proof is relatively simple, but I am not going to give it here. It is known as the Euler product formula.

There is virtually no “useful” information given by this highly abstract formula. It doesn’t help with daily finance. It doesn’t solve traffic congestion. It doesn’t even help in landing a rover on Mars. But it does is provide us with an insight into the fundamental truth of nature. In a way, this equation exceeds the known universe, as according to current theory, the universe is finite. The equation, by contrast, deals with the infinite.

In fact, modern mathematics is full of statements and theorems that have currently no practical use. There are entire branches and fields of study that are, in essence, useless. Sometimes, useless things have applications in the far future. Complex numbers, for example, were invented centuries ago, but didn’t really find any use until modern electronics and physics were developed.

Maybe everything we know today in math will be applied somehow. But this cannot happen forever. The universe is finite, after all, and knowledge is infinite. Sooner or later, or perhaps even now, we will have found knowledge that serves no use in our universe. This leads to the next question.

Is Knowledge Worth Seeking?

Should we seek knowledge for the sake of knowledge?

Is a culture with more knowledge inherently richer than one without?

Historically, knowledge in the form of technology had the power to save oneself, one’s family, and even one’s country. Entire civilizations were wiped out due to the technological superiority of the invaders. Knowledge has for a long time acted as a defense tool.

So perhaps we should embrace new knowledge for the sake of defending against a future alien force. But what about afterwards?

Assuming humans survive long enough to establish a galactic presence, and have enough technology to be virtually indestructible as a species, so that survivability is no longer an issue, what will be the point of further knowledge? What will be the point of knowledge for the sake of knowledge?

That picture above is the Mandelbrot set, a fractal generated by the fairly simple quadratic function

z \mapsto z^2 + c

where c is a complex number.

There could easily be no purpose to this fractal, yet it certainly holds some value. It is aesthetically pleasing, and the ability to zoom in on the image forever raises some old philosophical questions. In this sense, it is almost like art, only the rules are completely different.

In essence, knowledge for the sake of knowledge is what math is all about. There is no intrinsic need for math to apply to the real world, nor does any topic in mathematics need an analogy in real life. Math is knowledge at the abstract level.

Recently someone asked me what classes I was taking, and when I mentioned topology, he asked if that was a map making course. Topology and topography sound quite similar, I suppose.

In any case, topology is a great example of what pure math is about. It is the underlying foundation behind geometry. Geometry is highly applicable in real life, because shapes, sizes, and angles of things all affect the way they work. But in topology, sizes and angles do not matter. A line is the same thing as a curve, a square is really the same thing as a triangle or a hexagon, and a sphere is really the same thing as a cube or amoeba.

And a donut is really the same thing as a coffee mug.

These fields of math are totally alien to the math taught at the pre-college level. Geometry, basic algebra, and calculus are about sizes of things and comparing objects to determine their shapes, lengths, volumes, etc.

But when you get to the higher fields, such as analysis, number theory, abstract algebra, and topology, everything completely changes. They feel like entirely different subjects than the math taught in middle school and high school.

Previously, you were told that dividing by zero is impossible and that it is pointless to think of infinity. But in complex analysis, you can actually “cancel out” zeroes and infinities provided certain properties are counted, and you actually care about where functions hit infinity and how often they do so. And in set theory, you discover that there are actually different sizes of infinity. These facts are much more interesting than, say, the quadratic equation, which is taught in every high school algebra course.

The fact that zeros can actually cancel out infinities, or that there are different sizes of infinities, is much more interesting than such a formula.

This graph, showing a region of the gamma function, generalizes the notion of factorial (i.e., 5! = 5 * 4 * 3 * 2 * 1) to complex numbers.

The gamma function is also closely related to the equation at the very top of the page, with the natural numbers on one side and the prime numbers on the other. Those two expressions also define the Riemann zeta function.

You might be able to see some relation between the two images. It turns out that the trivial zeroes of the zeta function, which can be seen as the strange color mismatches on a line going from the center to the left, are the result of the poles of the gamma function, which are the vertical spikes in the other picture.

Basically, that is why I study math. The point is not to memorize formulas or to calculate quickly. It is to discover fundamental truths out of ridiculous-sounding things, and to make sense out of them. In a way, this is what people do in other academic fields as well. Sometimes math goes over the top and seems completely useless. This is bound to happen. But some things, like art and mathematics, don’t need a practical purpose to exist. Such things are valuable in their own right.

Year 3 and Blogging

My third year of Cornell starts in a just few days!

I’ll try to blog a bit more than before, but that’s not a guarantee. Although, between this and my other blog, I’ve written more in the last 2 months (52 posts) than in the previous 1.5 years combined (41 posts). With luck, that momentum will remain strong and I might catch up in the next few months to my 2010 posting frequency (235 posts in 1 year).

A glimpse at some of the classes I’ll be taking:

  • German 2000 – Intercultural Context: Continuation of my German learning experience. Guten Tag!
  • Math 4330 – Honors Linear Algebra: According to my peers, this is supposed to be a difficult class. I think I’m prepared.
  • Math 4530 – Intro to Topology: This will be interesting. I have always wanted to learn topology. It just seems like one of the strangest things math has created.
  • Math 4810 – Mathematical Logic: Given all the logic posts I’ve written, I think it will be worthwhile to take a class in formal logic. Gödel’s Incompleteness Theorem, here I come!
  • ???: I haven’t decided on a fifth class yet. I was originally planning to take Math 6170 – Dynamical Systems, but it turns out to be the same time as Intro to Topology, and I don’t happen to have one of those magical time turners. With Cornell’s add/drop system, I can wait a while to finalize this. If any Cornellian happens to be reading this and wishes to suggest a good class in any subject, please let me know!

Well, my next post will probably come from Ithaca. Stay tuned!

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:

Life at Cornell

Because of the experiment I mentioned last post, I haven’t been posting much, so with this post I’d like to return to my normal posting schedule. Well, a “schedule” never really existed, so what I mean, then, is a more frequent schedule. Until my next experiment…

Question Mark
Still looking for ideas for my next experiment...

Anyway, on to life outside of WoW in the last 20 days. I’ve been doing okay in my classes overall. Here are my courses my order of easiest to most difficult:

  • CS 1610 (Computing in the Arts): We still have not had a prelim or received any grades yet. The content is pretty straightforward.
  • SOC 1101 (Intro to Sociology): I’m at an A- right now, but we just had the second prelim yesterday. I felt I didn’t do as well on it as on the first prelim, but that seems to be the general consensus, so with the curve, it may be similar.
  • HIST 2500 (Technology in Society): We don’t have prelims, but instead, essays. We have three such essays that each count for 25%, and the other 25% is participation. I received an A on the first essay, but admittedly, I pulled an all-nighter for it, and the grade was very hard earned. In contrast, I do barely any work or studying for Sociology.
  • ENGL 1170 (Short Stories): This class has a lot of reading and a lot of writing. By the end of this semester I’ll probably have written more in this class than in all my other classes combined, then doubled. Plus, all the writing is in the form of literary analysis, which is not exactly my favorite style. I think I have a B in it right now, and I doubt I will be able to raise it by very much.
  • MATH 2230 (Theoretical Linear Algebra and Multivariable Calculus): This is by far my hardest class. The class median score on the first prelim was a 47, which I happened to get. It curved up to a B. Not bad, but it is so different from high school, where I was used to A+’s in math without doing any work. Plus, I used to be able to understand the concepts without doing the homework, and now, in college, I am starting to not understand the concepts even though I am doing the homework. My old theory: Math is easy. New theory: Math is tough.

I should probably mention some other aspects of Cornell as well. The weather has recently turned cold. For example, it is, at the time of this post, 40° F, and according to the Weather Channel, this will drop to 33° F later tonight.

Snowman
Just one degree lower...

I hear that in Austin, the daytime temperatures are still reaching the 80s. Lucky! 😛

Moving on… One thing I love about Cornell are the libraries. My favorite ones so far are the Uris Library and the music library (in Lincoln Hall). Uris has the appearance of being old-fashioned, and for some reason, that makes my productivity increase dramatically (though the most important aspect is likely the quietness). On the other hand, the PCL at the University of Texas looks new and modern, and for some reason, I never had much productivity in it.

The music library at Cornell is quite modern as well (and despite the name, it is actually more quiet than say the Olin library). What makes it modern is, well, one day, I heard this mechanical sound, and saw, with my own eyes, one of the bookshelves moving! It was like a scene from a Harry Potter movie…

Andrew Dickson White Library
The Andrew Dickson White Library within Uris Library. It's not the one with moving bookshelves, but still...

I’ve probably spent more time in libraries in this semester so far at Cornell than during all of high school combined. I also find them very good for creative work.

Moving on again… Band! I will just have to say here once again that the BRMB (Big Red Marching Band) is amazing! It’s so much better than high school marching band. On October 8/9 (which was during the middle of my experiment), we traveled to Boston for the Cornell–Harvard game! Neither team was that great (I’m from Austin, so I am qualified to judge football competency), and we somehow managed to let Harvard catch two of their own punts. Seriously? (Harvard won 31–17.)

There are many things I would say about the trip, which was very interesting and eventful, but I am forbidden from saying anything about the bus ride. (What happens on Bus 5 stays on Bus 5.) I stayed, as did the majority of the trumpet section, with a couple (both in number and in marital relation) of Cornell band alumni on Friday night before the game. It was a fun night.

Wow, I’ve written nearly 800 words so far. It’s about time I get to the second, and what I originally intended as the main, subject of this post:

The Principles of Scientific Management

The what of what? Actually, most people whom I know in my audience have heard of this work before, as they have likely taken AP US History or a related history course at some point. When the course gets to economic progress the early twentieth century, the textbook mentions: Henry Ford and Frederick Winslow Taylor, the latter for whom the concept of “Taylorism” is named.

A refresher: Taylorism, or scientific management, is an economic theory that focuses above all on efficiency. It is concerned with maximizing productivity. That’s about all that’s mentioned in APUSH. (Here are Wiki links for Frederick Taylor and scientific management if you are interested.)

Frederick Taylor
Frederick Taylor (1856-1915)

In our HIST 2500 class, “Technology in Society,” we just read Taylor’s work that founded this theory: a treatise called The Principles of Scientific Management (1911). Near that period of time, labor and employers were generally not on friendly terms with each other. Remember all those labor strikes and unions you had to memorize for APUSH? Yeah…

Taylor was an engineer who proposed a solution, scientific management, to deal with this social issue. His goal was to resolve the management–labor conflict with a system that would be beneficial to both employers and workers. Scientific management, he argued, would enable workers to be much more efficient, and thereby more productive. This would allow a smaller number of specialized workers to produce much more than a larger number of normal workers, which would in turn allow the employer to raise wages and still increase profit.

We are not talking about minor improvements here. Taylor didn’t argue that 10-20% increases in productivity would solve the labor issue. His analysis in the book shows that in many industries the daily productivity of one worker could be doubled, and in some cases, tripled or even more. It means that not only were the employers gaining more revenue, but the workers were also earning higher wages. And, as Taylor implies, this increase in production would also lower the prices of manufactured goods, which helps the common people: they have more money and can buy cheaper goods. It’s a win-win-win situation.

So how exactly does this increase in productivity occur? The idea is to make every part of every task as efficient as possible. For a shoveler, a group of scientists carefully analyzed which type of person was most suited for shoveling. They also figured out the optimum load on the shovel (21 pounds—any more or less in one scoop would reduce the overall efficiency), which type of shovel should be used for different materials, and even what material the bottom of the container that is being shoveled from should be. They figured out how many rest breaks the workers should have, and for how long they should last, and when they are scheduled. And they analyzed each motion in shoveling as to figure out which ones are necessary and which ones are useless, which movements are faster and which are slower, and how to shovel as to move the greatest amount of material in the least amount of time.

Stopwatch
The single most important tool in scientific management.

My crazy idea is to apply the theory of scientific management to other things. Oh wait, that’s already been done. Often with unremarkable consequences.

What I really should do is to have some degree of scientific management in my life, that is, have a schedule. At college I am going pretty much without a schedule. Then again, NOT playing WoW is probably much more significant in productivity-increasing than whatever I could I apply from scientific management. Plus, the application of scientific management requires at least two people, so if I were to try to apply this, someone would need to be my “manager.” Interesting, but no thanks.