A lecture, a conjecture and a problem...

Here I am, back after a not so hectic,but a  busy week. Well good news first, my guides back from Canada, so that means  a bonus to my reading.
Yesterday I attended a lecture on a the 'Fuglede's conjecture'. The statement of the conjecture by itself is pretty messy, but the underlying idea is beautiful.Actually I reached late,and missing the first few minutes was too dear for me. The reason was simple, every mathematical talk will begin with a definition, and if you have skipped that, you could as well skip the entire lecture. The speaker was Prof. Shobha Madan from IIT Kanpur.

Before saying anything more about the lecture let me mention something about the speaker. For one, she's a woman, and women in mathematics are very few- in fact those that exist are super-smart, talking from my experience. There was a rumour floating around that Shoba Madan had withdrawn her membership from the American Mathematical Society, because America attacked Iraq, in a lopsided battle. It is no longer a rumour though, as I found her letter to AMS regarding her resignation on the net,you can find it here- http://www.thehindu.com/fline/fl2009/stories/20030509006913000.htm
It is not that I support such boycott, yet her move is exemplary.

Getting to the lecture, my unpunctuality got me just a peripheral understanding of the topic. A lot of it was mathematical jargon, hower some ideas were good. There was mix of different branches of mathematics in her proofs, especially Functional Analysis, Algebra and good amount of Combinatorics.

Let me tell you the problem. You are given a bounded region D, and a certain shape T.You have to tell me if it is possible to tile (in the usual sense of tiling the floor or the roof) D using tiles of shape T.
Sounds pretty banal right. But thats not the case. I will not elaborate on this problem anymore however will give instances where one can apply these ideas. The Fuglede's Conjecture asserts that such a tiling always exists when the domain D is nice(in a certain mathematical sense- which you need not bother).
For example imagine a hexagonal room,can one tile this room with triangles? well seems obvious right. Cutting the floor along the diameter of the hexagon you see that just six equi-triangular tiles are sufficient. Note that here we made a choice in the type of triangle, so in a sense we have not answered the question in full generality. The heart of the problem is to check for the possibility of a tiling starting with an arbitrary triangle, which again is not true(relying on intuition). Then we need to classify all the possible triangles that fit the bill, and this step will put the problem to rest.
This problem has been the impetus for many  masons of the past,,and a matter of secrecy, about their successful tiling assignments. With ever changing designs it is imperative that one answers a question of this type, as it will go a long way in efficient resource management.
In mathematics such problems are also called 'Packing problems' , and solutions to these are of utmost importance, in theory and and practice.

 Let me get a little rigorous, one can easily picture that it is not possible to tile a region using circles, because there will always exist a non-trivial gap between the tiles. Extending this idea, one can ask ' what kind of regular polygons will result in possible tiling' It seems intuitive(but need not necessarily be true) that there should be a bound on the number of sides of the polygon-this is because as the number of sides increases the boundary of the polygon attains the shape of a circle(convince yourself,its easy!), and we know that a circle cannot tile ( this idea is due to Sandeep Bhupati ,my senior at IISc), its cool right. One often uses such methods to prove stuff in mathematics.

As the talk progressed I grappled with the proofs, finally decided to lay back and relax. It is not worth breaking ones head on technical jargon.
The talk was nice, actually I liked the speaker, she had this healthy confidence and composure throughout. I like to attend talks by experts, their understanding of the subject galvanizes me with a positive attitude to research in particular and life in general!!!

Here is a small problem if you would like to work out-
You are given a 8x8 chess board with two diagonally opposite corners deleted(note that we now have only 62 squares). The question is that if you have dominoes which cover exactly two squares of the chess board at a time,then is it possible to cover the 62 squares using exactly 31 pieces of dominoes?Whatever the answer, you got to tell me why!!!