UWO Mathematics Pizza Seminar

Nuit Galois

2011-09-30T05:54:00.000-07:00

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.

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

2011-09-06T18:04:00.000-07:00

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 1+2+3+4+.....is 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
1+4+9+16+25+.....=0
etc.

I will also discuss how infinite products like 1.2.3.4.5..... (all positive integers) can be defined and evaluated in some cases, e.g.
1.2.3.4.5..... =-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!

Achieving the Unachievable

2010-10-16T05:52:00.000-07:00

The 1st Pizza seminar in our 2010-2011 series will be held Tuesday October 19, from 4:30 to 5:30 in room 105B. Instead of a talk, we will present the documentary "Achieving the Unachievable". This documentary explores one of the most fascinating enigmas of modern art – the empty circle at the centre of Print Gallery by Dutch artist M. C. Escher. In 1956, Escher challenged the laws of perspective with Print Gallery and found himself trapped behind an impossible barrier... This uncompleted masterpiece quickly became the most puzzling enigma of Modern Art, for both artists and scientists. Half a century later, mathematician Hendrik Lenstra took everyone by surprise by drawing a fantastic bridge between the intuition of the artist and his own, shattering the Infinity Barrier.

As usual, pizza will follow in the Grad Club.

An Evening with Leonhard Euler

2010-03-23T09:36:00.000-07:00

We have no live presentation in the Pizza Seminar this week, but you can watch and enjoy a nice lecture on several aspects of Euler's work here: An Evening with Leonhard Euler.

Pizza Seminar: Remote Coin Tossing

2010-03-12T09:19:00.001-08:00

Department of Mathematics Pizza Seminar

Speaker: Ajneet Dhillon (Western)

Title: Remote Coin Tossing

Time and Place: Tuesday, March 16, 5 PM, MC 107; All are welcome!

Abstract: Judy and Andrew are going through a bitter divorce. They live thousands of miles apart. They wish to toss a coin over the phone to see who will keep the car. How can they do this without anyone cheating? Here you can download a pdf file of this talk.

Pizza Seminar: The mathematics of music from the wave equation to equal temperament

2010-03-12T09:17:00.000-08:00

Department of Mathematics Pizza Seminar

Speaker: Rasul Shafikov (Western)

Title: The mathematics of music: from the wave equation to equal temperament.

Time: 17:30 PM, Tuesday, March 9; Room: MC 107

In this talk I will explain how the solution of the wave equation can be used to explain music scales, temperament (i.e., music tuning) and harmony. We will also do a few experiments on a guitar.

Wallpaper groups

2010-03-12T09:16:00.001-08:00

Pizza Seminar

Speaker: Zack Wolske (Western)

Title: Wallpaper groups

Time: 17:30 PM, Tuesday March 2; Room: MC 107

A planar tiling is a repeating symmetric pattern in the plane. Because of their common everyday appearances such patterns are called "wallpaper groups." We follow Conway's orbifold notation, which describes the 17 wallpaper groups as certain topological spaces: quotients of the plane by some finite group. Completeness is given by computing the Euler characteristic of such spaces. No knowledge of groups, topology, orbifolds, or how to hang wallpaper required.

What if we had infinitely many fingers to count on

2010-03-12T09:12:00.000-08:00

Pizza Seminar

Speaker: Serge Randriambololona (Western)

Title: What if we had infinitely many fingers to count on ?

Time: 17:30 PM, Tuesday February 9; Room: MC 107

Natural numbers encompasses at least two way of counting. The first one tells how many objects a collection has: there are 84 students in the class, 4 apples in my lunch box or 223,647,852 inhabitants in Indonesia. In the second way of counting, we care for the position of an event in a sequence of events: the final exam will be the 106th day of the academic year, "trois" is the name of the numeral that comes after "deux" in French and the 8,000,000,000th human birth has already happened. As far as we only consider finite collections, these two notions of counting lead to the same arithmetic. But when we try to generalize them to infinite collections, surprising phenomena appear.

Circular Billiards

2010-03-12T09:07:00.000-08:00

Pizza Seminar

Speaker: Siyavus Acar (Western)

Title: Circular Billiards

Time: 17:00 PM, Tuesday January 26, 2010; Room: MC 107

There is an old question in optics that has been called Alhazen's Problem. The name Alhazen honours an Arab scholar Ibn-al-Haytham who flourished 1000 years ago. The problem itself can be traced further back, at least to Ptolemy's Optics written around AD 150. The problem - while considered one of the 100 great problems of elementary mathematics - is very easy to state: Given two arbitrary balls on a circular billiard table, how does one aim the object ball so that it hits the target ball after one bounce off the rim. In this talk we introduce various methods of approach that has been studied, but mainly focus on the number of solutions and their distribution on the table.

What does the spectral theorem say

2010-03-12T09:01:00.000-08:00

Pizza Seminar

Speaker: Farzad Fathizadeh (Western)

What does the spectral theorem say?

Time: 17:00 PM, Tuesday January 19, 2010; Room: MC 107

The Spectral Theorem, and the closely related Spectral Multiplicity Theory is a gem of modern mathematics. It is about the structure, and complete classification, up to unitary equivalence, of normal operators on a Hilbert space. This theorem is the generalization of the theorem in linear algebra which says that every normal, in particular selfadjoint, matrix is unitarily equivalent to a diagonal matrix; or, in simple terms, is diagonalizable in an orthonormal basis. The extension of this result to infinite dimensions is by no means obvious and involves many new subtle phenomena that have no analogue in finite dimensions. The final result has many applications to pure and applied mathematics, mathematical physics, and quantum mechanics. In this talk, a proof of the spectral theorem for Hermitian operators on a Hilbert space will be outlined and some applications will be discussed. This talk should be accessible to undergraduate students.

An elementary introduction to elliptic curves

2010-03-12T08:50:00.000-08:00

Pizza Seminar

Speaker: Emre Coskun (Western)

Title: An elementary introduction to elliptic curves

Time: 17:00 PM, Tuesday December 8 2009; Room: MC 108

The theory of elliptic curves is a fascinating field with many connections to algebraic geometry, number theory, complex analysis and even computational problems. In this talk, we introduce these objects in a very elementary manner, describe some of their properties and as an application, we show how they can be used to prove special cases of Fermat's Last Theorem.

Oppositions and Paradoxes in Mathematics and Philosophy

2010-03-12T08:42:00.000-08:00

Pizza Seminar

Speaker: John Bell (Western)

Title: Oppositions and Paradoxes in Mathematics and philosophy

Time: 17:00 PM, Tuesday, October 27, 2009; Room: MC 108

From antiquity mathematics and philosophy has been beset by a number of oppositions, such as the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable. These oppositions have on occasion crystallized into paradox and they continue to haunt fundamental thinking to this day. In my talk I'll analyze some of these and describe their impact on the development of mathematics and philosophy.

Solving Rubik's cube using group theory

2010-03-12T08:36:00.000-08:00

Pizza Seminar

Speaker: Sheldon Joyner (Western)

Title: Solving Rubik's cube using group theory

Time: 17:00 PM, Tuesday, September 2009, Room: 108

Group theory is the mathematical language of symmetry, and as such has many real world applications, ranging from the study of crystals to fundamental ideas about the workings of the universe. In this talk, we will introduce group theory and see how it is used to create a wonderful algorithm to solve Rubik's cube.