Autumn 2013

Congratulations

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Many congratulations to you all on your exam results — they were tremendous, and thoroughly deserved!

I would like to thank you all for your participation in the course. The spirit throughout the course was fantastic — it was a pleasure to teach you all! I am very happy to see that you have learnt so much!

Frode T. Børseth and Ulrik B. R. Enstad represented you. Two meetings took place during the term. Thank you very much to Frode and Ulrik for an excellent job!

In the mathematics courses you have taken so far, everything has been expressed in the language of sets.

But what is a set? Is there a rigorous definition? Is there more than one possibility? Do we need a rigorous definition at all?

How can we formulate the idea that mathematics is built upon sets? Do we have to build mathematics upon sets, or are there other possibilities?

Can we use mathematics to study mathematics itself? What does this have to do with the liar paradox?

Can every mathematical statement either be proven or disproven? How can we prove that something cannot be proven?!

What do we mean by a mathematical statement? We could agree that 2+2=4 and 2+2=5 are mathematical statements — one is true and the other is false. However, is 2+2=ℝ, where ℝ is the set of real numbers, a meaningful mathematical statement? Probably you would say no, and I would agree — yet most mathematicians today accept as a foundation for mathematics a theory in which 2+2=ℝ is as meaningful a statement as 2+2=4 or 2+2=5!

If any of these questions intrigue you, this course is for you!

In a nutshell, we will explore many examples of what are known as type theories — they will be drawn from different kinds of mathematics, some with a geometric flavour. We will look into tools and ideas from an area of mathematics known as category theory.

There are lots of directions in which the course can go, depending upon the interests and background of the participants. Well-known topics that can be looked at include Gödel's incompleteness theorems and the continuum hypothesis.

Homotopy type theory may be introduced at the end of the course. This is a new, very exciting topic, of high current research interest. Its key idea is that there is deep connection between topology and certain kinds of type theories.

Upon successful completion of the course you will have developed a new understanding of what it means to do mathematics! You will have acquired a deep knowledge of what it means to give a rigorous argument — in an indirect way this will, I believe, be valuable to you in your future courses, and in your future career.

The lectures will be given in English.

The official title of the course is Mastergradsseminar i Matematikk. Its description can be found here.

A few people have experienced difficulties in registering for the course. If this is the case for you, don't worry! The administrative staff can register you for the course directly — the best thing to do is to send me an email. I will be happy for you to take the course whichever your degree programme.

If you have previously taken a course with code MA3001, it is still possible to take the exam this time around, but you must register for TMA4310 instead.

There are no formal pre-requisite courses. No knowledge of type theory, category theory, set theory, or logic will be assumed. A little topology — for example having taken MA3002 Generell Topologi — would be nice, but is not required.

You would definitely have sufficient background to follow the course if you have previously taken one or more courses of about the level of MA3002 Generell Topologi or MA2201 Algebra. Feel free to get in touch if you are unsure.

You are welcome to contact me or to come to my office at any time.

My contact details can be found here.

Last updated at 16:57 (GMT+2), 29/12/2013.