The regular cafe hours are Tuesdays, 4:30-5:30 PM, in Room 108 in Middlesex College building.
For a long time I thought we can enhance the training and education of our undergraduate students, beyond regular classes, tests, and exams etc. For several years we had round table discussions on an irregular basis. Now I think there is enough energy in the department to test derive this idea on a regular weekly basis. And we have got a new name for this event too: The Discovery Cafe!
For the record, I put here the e-mail that was sent out to our undergrad students:
The Math Department invites all undergrad students in math and applied math to the new "Math Discovery Cafe".
The mathematics you see in books and lectures tends to be very polished. But it has not always been like this! At the beginning, new ideas are rough and sometimes fuzzy. Only after decades or even centuries of research and teaching does one arrive at the polished form you are used to. This might help to understand the theories, but it does not teach how to get to new ideas yourself.
The goal of the Math Discovery Cafe is to give an idea of how research is actually done in mathematics. Through informal discussions, we will see how problems can be approached, and we will look at how mathematical ideas evolved over time. The Math Discovery Cafe is aimed at interested undergrad students in math and applied math. Its core group will consist of the Math Scholars group in our department, but it is open to all and we certainly encourage all our undergraduates to take part and become active in Cafe. It opens every Tuesday 4:30-5:30PM in room MC 107 (except when there is a conflict with the Pizza Seminar).
What you can expect to happen during each math discovery cafe? A free, informal, impromptu, and in depth discussion of any math issues that may be raised or asked by participants. We won't leave any stone unturned! Behind the counter you will meet, alternately, the Cafe owners Prof. Franz and Prof. Khalkhali! We look forward to meet you there!
Okay, now for the record I want to tell you what happened during our first Discovery Cafe meeting. I am planing to post later events as well on a regular basis so that hopefully it can be used as a resource by our students. I was surely amazed by the number of questions and topics that we touched! Obviously we did not have time to go through any of them in any details, but surely planing to revisit all these questions in due time and with due respect! So this first meeting was a kind of brainstorming session and getting to know who is interested in what at the moment.
The first question was asked by Hanbo. He was wondering why it is hard to prove that a closed curve in the plane divides it into two pieces. Rightly formulated, which we carefully did in class, this is the famous Jordan Curve Theorem. I was not prepared to give an elementary proof on the spot! I later checked that Wiki has a very good introduction to this and so I cite it here Jordan Curve Theorem. We just tried to prove special cases and surely students proved that a circle divides the plane in two pieces.
Then somehow our discussions turned into major shocking counterexamples (or examples!) in analysis that shaped the modern form of the subject from late 19th century onward. They included:
- Cantor's one-to-one correspondence between line and plane,
- Invariance of domain and why the above cannot happen in a continuous world,
- Smales' sphere eversion (this was asked by Johnathan, and we were surely not prepared to go through it!)
- Cantor's theory of cardinal numbers.
Again we were either not prepared or did not have time to discuss these in any details, but surely can go back to them in time.
If you think that was it, you will surely be surprised to know that at the end we brought up the example of the Dirichlet function and, for easy landing!, we carefully proved that it is continuous at irrationals and discontinuous at rational and is nowhere differentiable. It was asked how this functions looks like? I should say this question had never occurred to me before, but Johnathan could find a graph in his textbook.We also very briefly touched the issue of how well an irrational number can be approximated by a rational number and if there is a hierarchy of irrationality in this way. That is, can we somehow measure/decide if one irrational number is more irrational than the other? Is there a most irrational number out there? Again no time to discuss this, but I promised we shall look at some of the amazing rational approximations to the number pi that were discovered by Archimedes and go from there.
So that was it for the first meeting, but of course we shall go in much slower pace next time and spend more time at each question or topics that we touch!