Friday, September 30, 2011

Nuit Galois

Program for Galois Night:
Tuesday October 11, Place: MC 107, 4 PM,
Speaker: Martin Pinsonnault, UWO

Title: Unsolvability by radicals of the quintic.

Abstract: The aim of the talk is to prove the unsolvability by radicals of the quintic (in fact of the general n-th degree equation for n> 4. That famous theorem was first proved by N. Abel and P. Ruffini around 1821. However, a complete understanding of solvability had to wait Evariste Galois and his introduction of group theory in a 1831 manuscript that was miraculously found by Liouville in 1843. We will present a proof of the Abel-Ruffini theorem, very close to Galois' own exposition, that only uses elementary properties of groups, rings, and fields as they are taught in a first course in abstract algebra."

As usual Pizza will be served after the talk.

There will be a second talk, on Tuesday, Oct 25, on Galois' 200th birthday!, at 4 PM:
Speaker: Masoud Khalkhali, UWO
Title: A topological proof of the Abel-Ruffini theorem on unsolvability by radicals of the quintic.
Abstract: TBA

This year marks the 200th anniversary of the birth of Evariste Galois (October 25, 1811 – May 31, 1832). We are planing to celebrate this very important event in the whole history of mathematics with two Galois Nights! Martin Pinsonnault will deliver a talk on algebraic aspects of Galois's work. Next week, we shall have a second talk on geometric and topological aspects of Galoi's theory by Masoud Khalkhali.

Evariste Galois is undoubtedly the most romantic and most tragic figure among all mathematicians and perhaps all scientists. His tragic death at the age of 20 in a duel, the manuscript he wrote at the eve of his death, his revolutionary republican activities in the aftermath and turmoil of the French revolution, and his almost total rejection by scientific institutions of his time, all add to this image. His last words to his brother Alfred describe the tragedy of his life:

Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans ! (Don't cry, Alfred! I need all my courage to die at twenty.)

But above all it is the power of his ideas, and his vision of mathematics as a conceptual enterprise that interests us here.

In France, the birthplace of Galois, they are celebrating his birth by holding a major
international conference is his honor. You can learn more about these events here and here.

Tuesday, September 6, 2011

Why 1+2+3+4+.....= -1/12

Speaker: Masoud Khalkhali, UWO
Time: Tuesday, September 27, 4:30-5:30 PM
Place: MC 107
As usual pizza and pops will be served after the talk in grad club.

Let us kickoff this year's Mathematics Pizza Seminar Series with some interesting piece of analysis. The title of my talk sounds like an utterly wrong statement! After all, the infinite series divergent and in fact diverges to infinity. So shouldn't we just write 1+2+3+4+....= infinite ? Of course we can. But then with the same, limited and narrow minded, understanding of summation we shall assign the same value, infinite, to a host of other very different types of series like 1+1+1+1+......or 1 +4 +9+16+25+....... etc. The point of my talk is that in doing so we are throwing away a wealth of information hidden in such divergent series. Information that can have practical implications for mathematics and its applications. This situation is a bit similar to set theory and cardinal numbers. Mathematicians used to think that there are only two types of numbers: finite and infinite. Of course, after Cantor, we know that there is a vast hierarchy of infinities and knowing about these different types of infinities is often very useful, though sometimes it creates its own problems. This is an important analogy that we should keep in mind.

One of the first people who realized the importance of divergent series and developed some techniques to sum such divergent series was Leonhard Euler. In fact Euler was of the opinion that any series is summable and one should just find the right method of summing it! In the last 250 years many summation techniques have been designed and there is vast theory of summability: Abel summation, Cezaro summation, Borel summation, zeta summation, etc.

This talk is a quick introduction to one such theory of summability: zeta function regularization. I will show how to compute infinite sums like the one on the title, as well as many others like
1+1+1+1+.....= -1/2

I will also discuss how infinite products like (all positive integers) can be defined and evaluated in some cases, e.g. =-1/2 log (2 \pi)

One of Euler's long standing goals in this area of math was to find (nowadays we say to define!) the alternating sum of factorials
1!-2! +3! -4! +5! -.....
I shall explain this and end up with Euler's surprising answer!