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

 

Hmm

I guess I should then also disavow concern with my image, serious or not….

 

As motivation,

consider that if you showed serious initiative in working out a scholarly theory of the sort mentioned in the previous post, I would probably write you a reference letter that might well render poor performance on the exam completely irrelevant

 

In fact

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.

 

Logic question

Dear Prof. Kim,

Sorry to disturb your working.I am one of your students. Since we do not have tutorial during this reading week,I have got a problem of understanding one of the question from the homework. Not only me, but also other students have the same problem.The question is followed.
Q: Show that every truth table in 2 variables is the truth table of some formula of Propositonal Calculus.(Hint: 8=16/2)

I will be really appreciated if you can explain it for me.

Thanks,
Austin

Reply:

Call the variables A and B. Then given any assignment of truth values for A and B, you should be able to cook up a formula that’s true for that assignment and none else. This is very simple. For A=F and B=T, for example, you can use

(not A) and B

Make sure you understand this elementary construction. Now start experimenting with some truth tables:

A B P

—————-

T T

T F

F T

F F

That is, start with some sample fill-ins for the third column, and see if you can combine the elementary formulas above to come up with a P that works. If you experiment with a few, you may see a pattern. Alternatively, since there are only sixteen truth tables possible, you could just do them all separately in an ad hoc way