Category Archives: general




Text of comments on the sculpture ‘Axiom’ by Mat Chivers at the unveiling ceremony on 28 October, 2014 at the Andrew Wiles building in Oxford:

Mathematics in Society

After several conversations recently about the social status of mathematics, I thought I’d put here links to two short essays I wrote on this.

1. An exchange on Mathoverflow

2. Mathematical Vistas

Mathematical problems

This week, I was given a very short deadline to write one of those introductory blurbs to the Korean edition of Ian Stewart’s book, ‘The Great Mathematical Problems.’ I could only skim through it, but I managed to get enough of a sense to write something. Here it is, in case some students find it interesting.

New Year

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.

Student reports

Acyr’s report 2

Alex’s report 2

Summer projects for students

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


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

Workshop and 2201

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.


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.

World history

In my previous post, I made a reference to the `global community.’ This is a concept I think about frequently and, given the composition of the student body at UCL, I can hardly be alone in my preoccupation with its complexities. Of course I haven’t the competence to discuss such issues in any depth. But I thought to share with students a clumsy essay written a few years ago with the title `Is there a world history?’, perhaps to provide some comic relief during this stressful period. I was taking a German class in Bonn, and the teacher proposed homework dealing with the question of Turkish membership in the European Union, a topic of some immediacy at the time (and maybe even now). The original German I wrote in is unreadable even to me, but I also prepared an English version. As usual with essays that are dated, the ideas expressed look quite half-baked at this point, if not entirely wrong-headed, even in the span of two short pages. Nevertheless.

Another remark on travel

Because I’ve recently begun posting links to conferences or seminars I go to, I thought I’d add a clarifying comment. The purpose of it, at least consciously, was not to show off that I’m a busy guy or something of that sort. It’s rather that for any serious mathematician, a complex web of obligations makes demands that oftentimes take time away from efforts aimed towards the direct benefit of students at his or her home institution. For example, I was absent for several days of this week because my two post-graduate students at Purdue were having their thesis defense, the last step in the process of obtaining their doctoral degrees. But then, by making the nature of such activities transparent, I hoped that my immediate absence might be rendered more understandable. In fact, I thought it might even create a better sense of global connectedness for the students at UCL in that teachers, after all, are shared the world over, hopefully to the benefit of the whole global community. (Probably I flatter myself.) This is also why I encourage students at UCL to come to lectures delivered by visiting academics or to try in general to take part in academic activities that go beyond the classroom.