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).

Computer Science and Math

At the end of last semester, I decided to double major in computer science and math, rather than just in math. This decision was based on several reasons:

  • Practicality. As much as I love theoretical math, most of it is totally irrelevant to the real world. CS is closely related but far more useful.
  • Opportunity. Cornell has a top-rate CS department, and it would be a shame for me to not take advantage of it. I am virtually done with my math major as well, so it does not cut into that.
  • Expanding my skill set. I think CS is a strong backup in case I didn’t get anywhere with math.

The catch is, being a junior already, I need to rush the major in my remaining 3 semesters (including this one). This will require quite a bit of work, but due to my math major, I have much of the foundation done. I also have every liberal arts distribution requirement out of the way. In addition, during my sophomore year I took CS 2110, so that is another requirement done. My schedule for this semester is below.

2013 Spring Schedule

The most interesting class will be Math 7370, which is Algebraic Number Theory at the graduate level. I have some background in analytic number theory but not in algebraic number theory, so it will be interesting to see the differences. Also, the same professor is teaching 4340 and 7370, so I should have ample opportunity to ask any questions about algebra.

As for CS classes, I hear CS 3410 has a lot of work, but I am prepared. Also, given that I did decently in Combinatorics (Math 4410) last semester, CS 4820 should not be too hard. And given the knowledge from a math major, I doubt 4850 will be that difficult.

In addition, I have planned a more consistent posting schedule for this blog. There won’t necessarily be more posts, but they should be spaced more evenly. Also keep an eye on my math blog epicmath.org—I will continue to update it this year.

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.

Off to Ithaca

As I am leaving for college, I originally intended this post to be a reminiscence of and a goodbye to Austin, but when I tried to write it, the words would not easily come out. Therefore, this is not solely a farewell to Austin, but an anticipation of where I’m heading next.

Austin has been my home for 10 years, almost exactly. We moved here from Greenville, South Carolina in September 2000, and it is now late August 2010.

Even though I wasn’t born in Austin, I consider Austin to be my primary hometown. Besides relatives, nearly all the people I know I have met in Austin. I was too young to remember people from Greenville or earlier places.

I also received most of my primary and secondary education here in Austin, from third grade straight through twelfth. I attended Forest North Elementary School, Laurel Mountain Elementary School, Canyon Vista Middle School, and Westwood High School (’10).

And I almost continued this with University of Texas (’14):

UT TowerYet, I chose Cornell University instead. I have nothing against UT. In fact, logically I should have gone with UT, for it was a little bit cheaper, and I was in two honors programs (Dean’s Scholars and Plan II) to boot.

But for me, it lacked only one thing: fresh air. Don’t get me wrong—Austin is one of the cleanest cities in the nation, and is reputed one of the best places to live. What I mean by “fresh air” is, I have lived in Austin for 10 years, and it was time for a change. Between familiarity and uncertainty, I had to choose the latter:

Cornell TowerIt was certainly not an easy choice. Since most of the people I know are in Austin, I would miss them all. Furthermore, no one else from my high school class is going to Cornell (though I do know a couple of people in a different grade). And from my high school, a million (okay, more like at least 50) people in my class are going to UT. So those are the numbers. And, oh yeah: Winter? Snow? What’s that? 😀

We’re flying tomorrow morning. Today is my last day in Austin. I feel like I have a lot more things to say, but I don’t know what order, so I’ll just make a list:

  • I’m going to miss all those “Keep Austin Weird” bumper stickers.
  • An online shoutout to the Westwood Class of ’10!
  • And to the Cornell Class of ’14!
  • I am moving from a liberal city to an apparently even more liberal town.
  • Ithaca, at 7.92%, has the highest percentage of residents holding Ph.D.s in America. [Source: Forbes on MSNBC]
  • Perhaps Austin and Ithaca won’t be too different. Who knows?
  • I still have some final packing to do, and my room is nowhere near clean.
  • We’ve said so many goodbyes in the last few days as we’re going off to college. I would love nothing better than to say goodbye to all my friends in person, but that is obviously impossible. To all those whom I didn’t catch in person: Goodbye! And to those whom I did: Goodbye again!
  • Even this morning, the UT McCombs School of Business entrepreneur-in-residence Gary Hoover commented on this blog out of the blue, and gave me a goodbye present. That was very pleasant, thank you.
  • This blog WILL continue to be updated as I become a college student.
  • Cleaning out my room and finding old things is creating a lot of nostalgia. Right now there are about 253 things on the floor, so I’d best get back to cleaning. My next post shall come from Ithaca. 🙂

College Decisions

Decision

I had four college choices:

  • University of Texas at Austin
  • University of Chicago
  • Carnegie Mellon University
  • Cornell University

California Institute of Technology (Caltech) was a fifth possibility as it offered me a spot on its waiting list; however, I decided to reject the offer and instead choose among certain acceptances.

My decision was mostly based on location and cost. After living in Austin for nine years, I felt it was a time for change. It’s not that I have anything against the city—Austin is a wonderful heck of a place. But I wanted to get out of Texas, and have that idyllic new experience. When I began college applications in the fall of 2009, I only had one choice in state: UT Austin—I didn’t apply to Texas A&M University or Rice University, as many of my schoolmates did. (I noticed an interesting pattern in my acceptances/waitlists/rejections.)

That said, money was a also pretty important issue, if not the most. UT’s list price was approximately $24,000 per year, while other schools I applied to had list prices of over $50,000 per year, up to $57,790 per year (UChicago). This is of course changed by scholarships and financial aid, but this was a striking difference. Would it really be worth an extra $25,000 to $30,000 per year for a “better” education? And would it actually be “better” at all?

Regarding my interests, I would probably say I’m an intellectual. And UT does have great academics. My primary interests right now are in math and science (I listed mathematics as my first-choice major), but I do think I would enjoy a liberal education. UT’s undergraduate studies are split into multiple programs; I chose the College of Natural Sciences and the College of Liberal Arts. This actually a reflects a sort of liberal arts change in myself just the past year: Had I applied for college one year earlier, I think I would have picked engineering instead. This would have been a great fit at UT as the Cockrell School of Engineering is very strong and ranks in the top ten engineering programs in the nation. I’m not saying the other programs are not good—the Dean’s Scholars and Plan II honors programs in the College of Natural Sciences and College of Liberal Arts respectively were both very attractive, especially Dean’s Scholars. I attended an invitational for this and was really pleased with the atmosphere, the openness, the excitement.

The other three schools I hadn’t visited, but I found plenty of helpful information on their websites (*edit: later on, I did visit UChicago). I researched a lot into UChicago, and also spend much time looking into CMU, though not as much. For both I also had a helpful alumni interview with an interviewer who was very excited about the school. I thank both of you! For Cornell, however, I barely reviewed its website, and was not contacted for any interview.

UChicago was the first to give me a real acceptance. (UT has this 10% auto-admission rule, and being admitted to somewhere to which you know you are going to be admitted is nothing like being admitted the normal way.) I was very happy, and one main thing it had going for me was its strong math department. Of course, it’s also renowned in economics, and I saw possibilities in a math/economics mix. As an Early Action admit, I also received various gifts from UChicago, including a calendar, various post cards and letters, a course catalog, and a scarf. Plus, other publications it sent to me struck out as a very intellectual school—the one I remember in particular is “The Power of Ideas.” Admittedly, that title is pretty cheesy, and the letter accompanying it even acknowledged that, and that it should be a much, much, longer name. Basically, the pamphlet was filled to the brim with intellectual curiosity, and that got a two-thumbs up from me.

The next acceptance letter came from CMU, about three months later under the Regular Decision cycle. I was somewhat disappointed as I was accepted to the Mellon College of Science but waitlisted to the School of Computer Science. In my interview, my interviewer and I discussed the Computer Science program a lot, and that was the main reason I applied. So, at this point, it was UT versus UChicago versus CMU. We decided to check off CMU. (When I say “we,” I mean to include my parents.)

Soon came Cornell. UChicago versus Cornell was a very hard choice. I had still not yet received the financial aid decision from Cornell, so I was considering UT with slightly reduced cost (small scholarship from Dean’s Scholars honors program), UChicago with a massively reduced cost, but still significantly more expensive than UT, and Cornell with unknown cost. But I had expected Cornell’s financial aid to be about the same as that of UChicago, so the choice was basically between UT and some out-of-state school. (By the way, UChicago did offer more aid than CMU, so that was another reason for rejecting CMU.)

Then arrived Cornell’s financial aid decision. This was the most shocking document I received in my entire college application run. Cornell basically offered far, far more than UChicago, bringing the cost of UT and Cornell to the same price range, with Cornell still slightly more expensive. (I guess one thing the Ivies have more than other schools is money.) And because Cornell’s academics are comparable to UChicago’s, we took UChicago off the list. Finally, we were left with two choices.

UT versus Cornell. Money was mostly out of the question. Cornell would probably come out ahead in academics, that is, even when compared against two honors programs. Since Cornell still costs slightly more money, we concluded that the academics and financial concerns cancel each other out.

So now it was a question of style and location. I chose Cornell.

Cornell

Cornell Class of 2014!

College Interviews Part 2

College Interviews Part 1 contained interviews with MIT, University of Chicago, Yale, and Harvard. They are described with much more detail than the ones in this post.

This post contains interviews with Carnegie Mellon, University of Texas at Austin (twice, for Dean’s Scholars scholarship interviews), and Princeton.

Carnegie Mellon

Interviewer: Eric Stuckey
Setting: Saturday, 1/16/10, 1 pm – 2:15 pm, Genuine Joe’s (West Anderson Lane)

The interview can be summarized in two words: Computer science.

As of my writing this, a week has passed since this interview. I actually cannot remember some of the specific questions that he asked me; perhaps this is because it was for the most part a fascinating conversation. I got to the coffeehouse about 10 minutes early and bought two coffees; he arrived about 5 minutes late, and declined the coffee because it contained caffeine, and went for hot chocolate instead. So, I had two cups of coffee for myself.

He first asked me why I was interested in CMU. I mentioned how I knew a few Westwood students who went there, and how its high rankings in the computer science departments caught my eye. It seemed like an interview for only five or so minutes. The remaining hour and some was just a nice and mostly intellectual discusison.

We talked a lot about computer science, and CMU’s involvement in it. One point of discussion was the self-driving car, or rather, CMU’s pioneering of it. In the mid-1990s CMU designed, built, and tested such a thing. That’s right, in the 1990s! I guess I didn’t do my research; I thought engineers were just starting to work on that.

In 2005-6, there were two competitions for a self-driving car. CMU won the first, and placed second the next year. However, the first-place team, Stanford, included someone who had just been on the CMU team the year before. In other words, CMU is a leader in the advancement of computer technology.

We went on to other topics in computer science, and he really emphasized CMU’s strengths in the field. I asked about the rankings in particular—why is CMU ranked so high in computer science, i.e. what exactly does it have? The answer was simply two things: good students and good faculty.

For a while we discussed the intellectual realm in general. Computer science itself, even though a relatively new field, has expanded considerably and is now a fairly broad field.

UT

These two are scholarship interviews, not admissions interviews. They are both pretty short, so I will not elaborate further.

Interviewer: Alan Cline
Setting: Friday, 1/22/10, 2 pm – 2:15 pm, On campus

He first mentioned that my school, Westwood HS, had more students in Dean’s Scholars than any other school. We discussed my IB Extended Essay on a modification of the Riemann zeta function.

Interviewer: Jim Vick
Setting: Friday, 1/22/10, 2:15 pm – 2:30 pm, On campus

Similar to the previous one. The main point of discussion was the Riemann zeta function.

Princeton

Interviewer: Amy Mitchell (’77)
Setting: Saturday, 1/23/10, 3 pm – 4:15 pm, Starbucks (Research and Anderson Mill)

This was a pretty amazing interview. We covered an extensive range of topics with a lot of depth. It was very conversation-like. It felt similar to the Yale interview.