general


As another academic year begins, I urge students in my LInear Algebra class and my Algebraic Number Theory class to make full use of this blog. You might start by going to the very first posts that were around late October of the previous year. As you move forward, you will get a sense of the kind of things that were discussed here, and how you might yourself best take advantage of this resource. Also useful might be to look at the posts that go from March to May of 2008. There, you will find material that deals with many different aspects of the examinations. You might find that even vague questions you have are already answered there. Looking at them at the very beginning could be a substantial aid to you as you study through the year.

The search function can be used for specific topics. For example, typing ‘Jordan’ will bring up all kinds of articles on the Jordan canonical form. My hope is that browsing through the previous discourse will help you focus your attention on the important issues, as well provide specific help with mathematical questions.

Three students are in the midst of summer reading projects under my supervision, Acyr Locatelli, Alex Tao, and Nikhil Mehrotra.

Acyr is reading Ideals, Varieties, Algorithms by Cox, Little, and O’Shea, a very nice introduction to computational algebraic geometry.

Alex is reading Rational Points on Elliptic Curves by Silverman and Tate. This book deals with integral and rational solutions to cubic equations like

y^2=x^3+x-1000.

This class of problems, seemingly limited in scope, turns out to have many remarkable attendant structures, making it one of the most important directions of contemporary number theory.

Nikhil is reading Goedel, Escher, Bach by Douglas Hofstadter, a great classic on consciousness, mathematics, and the possibility of artificial intelligence.

I will be posting here their weekly reports, in case someone else wants to follow along. For now, here are

Alex’s report 1

Acyr’s report 1

Dear Prof. Kim,

with regard to the Non-Commutative Constructions in Arithmetic and Geometry workshop, please confirm the date and time since the e-mail says:

time: 7-8 June.

Also how formal is it.

secondly, please recommend me some books for MATH 2201, something that is more elementary or explicitly explains whats in the course. i have consulted the books you had recommended for the course but did not find the very helpful for me. i want something that gives me the roots of the topics studied.

Reply:

Yes, 7-8 June for the workshop is correct. You can see a precise schedule on the webpage. The level of formality is hard to describe precisely. There is a sense in which mathematics meetings are all rather informal, so it’s certainly no cause for concern. However, I should warn you that the lectures will be at a very high level. For serious students, I still think it’s good to come into contact with presentations by world-class researchers (this description obviously doesn’t include me) as early as possible. That’s why I issued the general invitations to students.

As far as linear algebra is concerned, there are two recommendations I can make:

1. Finite-dimensional vector spaces by Paul Halmos

This is a classic text that deals primarily with the *concepts* of linear algebra abstractly, and at a rather deep level.

2. Linear algebra in action by Harry Dym

This book is heavily computational and provides a very solid understanding of the important techniques in matrix algebra. It also look toward quite advanced work in analysis.

It could be better to move on to other things at this point to see really how linear algebra functions in higher level mathematics. That can often help consolidate your understanding of the basic material. The textbook `Algebra’ by Michael Artin is not about linear algebra, but contains a quick summary of the basics at the beginning. This is because he emphasizes throughout the text the examples from linear algebra, even when discussing groups, rings, fields, etc.

To my tedious post on `Practical responsibility,’ I’ve added a few words just in case someone misunderstands the the intent of my anecdote on the manager at Eli Lilly. Furthermore, I hope no one interprets my remarks as a discouragement of input from students.

To be a bit theoretical (or mystical), there is a rather age-old tension in education between

1. the intellectual gains of dialectic (e.g., argumentation); and
2. the spiritual gains of acceptance.

This is well-known, but I wished to relate this tension somewhat to mundane matters, perhaps from the perspective of the side that gets less explicit attention (I think). Of course there is no way to decide on one or the other once and for all.

Let me use this occasion to pose a question that might even be serious research material. Can one come up with a good measure of similarity between successive exams? That is to say, suppose you wanted to show me with suitably convincing rigor that I was really deviating from previous years. How would you do it? Of course, this would require a good theory combining formal logic, linguistics, epistemology, and of course, mathematics, both in the form of the theory and in the specific construction of the models that the theory tries to work with. I’m sure that a truly satisfactory theory would be close to impossible at this point. But it still might be fun to start thinking about it.

Very roughly speaking, there should be

a space of linear algebra exams

possibly of very high dimension, endowed with a natural inner product, using which we can measure the distance between exams. In this space, you might attempt to show that the exams of previous years form a pretty tight cluster, while my exam is convincingly distant from that cluster.

If you’re interested in thinking about a problem of that sort, let me know. I’m not an applied mathematician, but I think we might be able to do something.

I wish to thank again everyone who wrote me very kind words and also the ones who sent constructive critiques. Perhaps it’s not out of place to post just one of the messages, since it came from the student representative. Thanks again.

——————————————————————————————————–

Dear Professor Kim,

As the second year Mathematics representative I would like to apologize for some of the remarks fellow students have been making and have been sending you. I strongly condemn what has happened. The comments thats have been made are despicable, this is no way to go around such a situation, especially of people of such an age and at such a good institution.

I would just like you to know that not everyone feels the same way as the person who sent the messages I have seen on the course blog.

Everyone has their own opinion in life about different matters but one thing that can not be contested is the effort you have put in to help us with your course blog, which no other lecturer has set up and you have been open to consultations at any time and hardly turned anyone away as far as I am aware. The enthusiasm you brought to lectures was refreshing and it was easy to see you really put in a lot of effort to help us understand.

Personally I found the paper hard-ish too but looking at the questions now, it is a good paper, even though I made some silly little errors on my quest to solve them but the processes you were looking for us to execute/apply were easy to see.

Once again I would just like to apologize on behalf of the second year. You should not feel discouraged by what has happened. Keep up the good work and do NOT leave UCL!

Best wishes,
Ali H. Khalid

Upon re-examining the previous post, it occurs to me that it dwells excessively on platitudes that most people are undoubtedly aware of already. But I’ll let it stand anyways, in case someone does find it actually useful.

I’ve received now quite a few comments in the course of the day. I plan to share them with you eventually once I’ve judged that the danger of misinterpretation has passed.

Of course essentially all remarks, even the complaints, were of a thoughtful nature and I’m very grateful to have received them. Needless to say, I’ll do my best to take them reasonably into account. In any case, I hope my own handling of obligations, faulty as it may be, appears sincere enough to allow insertion of a few words about a constructive approach to a personal sense of responsibility, hoping all the while that they don’t appear unbearably preachy. As I do so, I wish to make clear my intention in doing so:

My hope is for *every one* of the students in my class to be successful in the path of life he or she chooses.

This hope is of course rather grand, and you may be skeptical about any possibility of its fulfillment or even about the sentiment itself. For example, an overly cynical person might ask why I care at all. Since I don’t believe such cynics to exist among the students here, we could move on to a somewhat more serious doubt a skeptic might address to me, personally: Why these students, of all people in the world? After all you were born and raised elsewhere, made a career somewhere else again, and then seem to move around the world all the time. Why should you care in particular for the students in your class? To such a question, there is no easy answer. Certainly, any sense of obligation to one group needs to tempered by a universalist consciousness that benefit to one could be a loss to another. Nevertheless, perhaps because of the truth inside the cliche `think globally, work locally’ my honest commitment at the moment is primarily to the students I meet here and now.

With that preliminary over with, let me go on to a moralistic injunction:

In taking an exam, be in the habit of taking your own portion of the responsibility very seriously.

Even though I’ve described it as moralistic, and the moral dimension is an important one, I wish here to just discuss the practical aspect of this advice. That is to say, as I speak of `success,’ I’m sure we all know that the moral aspect of success is a crucial one. But today, I’ll focus purely on the practical side.

Now, one can go on endlessly about the fairness of the exam content, the exam schedule, similarity to previous exams and so forth. When I write a sentence of that sort, my intention is not to downplay your sense of frustration (and suffering, as alluded to in an earlier post). Indeed, as I hope I’ve expressed clearly enough, I do value your perspective and input. However, in this discussion, I am going to entirely sweep the details of the argument under the rug for the sake of making one important practical point.

The context is this: As I understand it, the majority of you are not bound for an academic career and will seek employment as soon as you leave the university. Then you must know that a crucial ingredient in an application packet is a set of references. According to monster.co.uk, 60 percent of employers claim they have had to turn down otherwise successful applicants because of inadequate references. I strongly suspect that this percentage increases substantially when one considers only jobs of the most desirable sort. Now I’m sure many of you have had work experience of a temporary nature. But I gather from conversations with several of you that for the most part, the job after you graduate, at least the one desired, will be your first serious one, that is, the first one likely to lead to a career. If so, I suspect that references from your professors will be regarded as quite valuable to a prospective employer. This is certainly true in the US (I’ve written many myself and have spoken to many personnel committees on the phone), but given the transnational character of business in our times, it doesn’t seem likely to be much different here. Let us ask then why it should be important. It certainly seems to give the professors enormous power, possibly arbitrary. (In fact, I hope you don’t interpret this whole post as some kind of a threat :) .) But we should first admit that the truly capricious cases are likely to be rare in human life, just as entirely thoughtless comments from students about professors are in the definite minority. Through an evolution of social know-how, institutions have a way of understanding this statistical rarity, and hence, taking the effects of caprice in due strides. But still, why ask the professors about prospective employees, whose expertise may often be related to the actual work only in marginal ways? There is one `obvious’ reason, that I will put in emphases:

A professor’s perspective is likely to be one well-approximating a managerial perspective.

That is, in addition to specific skills, a professor who knows you somewhat will have experienced your work in a way analogous to a manager in a company, and will have tried to evaluate it as well in similar ways. Here, we need to note that work refers not just to the part that reflects technical skills, but, very importantly, `work-ethics,’ to use a term that’s unfortunate but unavoidable. That is why I spoke of sweeping the detailed arguments about specific fairness under the rug for the moment. It’s not hard to imagine the challenges of industry coming at you in often unpredictable packets of pressure that many buckle under and but which the (potentially) successful people meet with flexible skills and mental resources. You might find it convincing or not to compare practical challenges with an examination, but what’s uncontroversial is that a manager in a company has typically much less patience for complaints of `unfairness.’ Of course he/she will be looking for someone who is able take personal responsibility and initiative seriously, and `get the job done’ in a varying range of challenging situations. If anything, the sentiment I’ve encountered most often from industry leaders I’ve met (at least in the US) is that university professors, similar in their role to managers as they may be, are far too lenient to train the right sort of personality. Obviously, it’s much easier to be expelled for poor performance from a company than from a university.

From a student’s perspective, the university provides the service of an advanced degree and, hopefully, the means of acquiring useful skills (as I’ve said, I’m leaving aside for now the loftier points). From an industrial perspective we provide a means not just to train the workforce, but also to discriminate within it, and to sort out the potential leaders. From the view of societal economics, this `filtering’ demand on the university is a perfectly reasonable one since the portion of the budget coming from governmental or industrial sources outweighs the tuition income in any sizable university. (Once again, I know this for a fact in the US. Perhaps it’s even more true here.) For the reasons outlined above, in this process, a professor’s opinion can end up playing an even a more important role than transcripts. (Once again, this is not a threat.) Of course, you can choose to avoid altogether using a professor as a reference. But then, you’ll be ceding at least one advantage to someone who is able to get a good reference from a reputable `surrogate manager’. (Incidentally, I’ve read in several independent sources that `personal’ references are practically worthless, even for people new to the workforce.) In any case, I hope it’s obvious that the importance I’m attaching here to a professor’s opinion is not because of direct benefits or losses that may result from it. The point, which I’ll repeat for clarity, is rather that whether or not you ask him/her for a reference,

a professor’s view of you is likely to be a good indicator for the view of an employer.

If you would like to examine this issue from yet another angle, consider going into business with a friend, seriously. When the wishful thinking is done away with, you might see the difference between someone to have a good time with and a reliable business partner. (I’m not denying the common enough confluence of the two.) In fact, if there’s a professor both of you know well enough, you might do well to seek his/her advice about the friend before taking out that big loan! (Yet again, I remind you I’m ignoring moral issues.)

The key theme here then is that in all these processes of evaluation (taken literally, measuring the value, unfortunately monetary, of someone), the ability to be responsible to one’s work plays an enormous role. I’ll leave it to you now to convince yourself further that this is the case. I did once write a letter for a lady who was in my linear algebra course at Purdue University. She was quite weak in the prerequisites, and received a `C’ even after hard work. (In the US system, this mean roughly, `below average.’) However, it was clear through the term that her basic diligence was far above average, in fact, extraordinary. When she came by during office hours, always with great punctuality, she typically had reams of paper where she had worked out exercises for herself (with considerable difficulty, I might add) far beyond what was required, and the textbook looked worn out by repeated readings. Nor did she complain at all about receiving the C. Of course, I discussed all this at length in the letter I wrote for her. Frankly, I never learned how her case worked out. But it is one situation I recall where the work ethic should have far outweighed in importance the mark received. By the way, if I briefly come back to situations that I’m somewhat more familiar with, that is, applications to post-graduate programs, a reputable professor’s good opinion of you can easily overrule poor transcripts.

Even if it refers a bit to regional stereotypes, I’ll recount a rather flippant opinion I heard from a manager at Eli Lilly, the transnational pharmaceutical company. He preferred Indian employees to North American ones: `The Indians in my division, when they make some mistakes, I admonish them, and then they shape up. The American employees are so difficult and sensitive about criticism of their work and always have some excuse. After a while, a manager gets tired of dealing with all that.’ These are just one person’s words (which I certainly don’t endorse), not expressive of any general reality, so don’t take them at all seriously. But do remember that it is the kind of (harsh) thought a manager you actually encounter may well have.

To mention a personal perspective closer to education, I can tell you honestly that I never allow my own children to complain about their teachers. I don’t expect ever to have an occasion, but if there were something to complain about, I’d do it myself secretly, without letting my children know. This is not primarily because of my regard for the teachers’ sensibilities, although such considerations are part of any civilized human interaction. It’s simply that it would be very damaging to the children’s outlook on life to acquire the habit of avoiding responsibility for their own learning. Since it’s probably easier to believe in a commitment to the welfare of my own children, I don’t mind stating even more emphatically that a complaining mind-set would be absolutely disastrous for them.

In this regard, even if I’ve discussed the issue of personal responsibility at length in terms of a professor’s opinion of you, obviously the more important point is the intrinsic habit you either have or can acquire, which will then eventually be recognized by the relevant people. The whole point of having this discussion is that all of you are still young (although older than my children!), and able to change modes of thought and action that contribute negatively to your future.

In closing, I should admit the obvious fact that not all successful people are endowed with an expansive sense of personal responsibility. But many are. Furthermore, and this is where the spiritual dimension comes back in, the ones who manage to succeed without compromising their true inner self definitely fit that mold. Sufficient practical pressure can induce many people to act in a responsible way for a period of time. For this action to come with enduring ease requires something deeper.

In case you’ve already seen the earlier version, I thought I’d explain that the previous post was modified by deleting the last copied message from the student. It seemed my point was sufficiently clear without publicizing the unfortunate language, which, after all, couldn’t have been sincere.

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