tag:blogger.com,1999:blog-20170966985135926032016-03-06T05:23:08.004-08:00Western Undergrad Mathematics Pizza SeminarMasoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.comBlogger24125tag:blogger.com,1999:blog-2017096698513592603.post-23379362614305828592015-07-09T10:58:00.003-07:002015-07-09T10:58:58.488-07:00Regularizing Divergent ProductsHere is a Pdf file of a <a href="http://www-home.math.uwo.ca/~masoud/files/Pizza2015.pdf">Pizza Seminar </a> talk I gave in April 2015.Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-49289679824635714332015-07-09T10:54:00.002-07:002015-07-09T10:54:49.946-07:00First Steps in Quantum Computing-an updateHere is a Pdf file of a <a href="http://www-home.math.uwo.ca/~masoud/files/QuantumComputing2013.pdf">Pizza Seminar</a> talk I gave in April 2013Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-38425089670623075172015-07-09T10:48:00.001-07:002015-07-09T10:48:24.916-07:00Regularizing Divergent Sums-an updateHere is a Pdf file of a <a href="http://www-home.math.uwo.ca/~masoud/files/Pizza2011.pdf">Pizza Seminar</a> Talk I gave in September 2011.Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-61913279141374791422013-04-03T19:10:00.000-07:002013-04-05T21:51:27.210-07:00A Stroll on Strange SpacesPizza Seminar Talk: Tuesday, April 9, 2013, 4:30-5:30 PM, MC 108. <br />Speaker: Mitsuru Wilson<br /> <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-qBEIHdsNvMQ/UVzg17q15pI/AAAAAAAAAM0/4hpHznto7T0/s1600/NitroNCG.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="319" src="http://1.bp.blogspot.com/-qBEIHdsNvMQ/UVzg17q15pI/AAAAAAAAAM0/4hpHznto7T0/s320/NitroNCG.jpg" width="320" /></a></div> Like the above ad promoting a potent Nitro NCG mixture, <a href="http://en.wikipedia.org/wiki/Noncommutative_geometry">Noncommutative Geometry (NCG)</a>, as created by <a href="http://www.alainconnes.org/en/">Alain Connes</a> is a potent<br /> mixture of many ideas in mathematics and physics.<br /><br />One of the grand themes in mathematics, geometry, dates back for thousands of years. Starting from Euclidean spaces, many elementary shapes such as circles, disks, spheres have been studied until today; all spaces we learn in school are still studied today! Riemann then generalized Euclidean spaces with his foundation of manifolds, which was then used by Einstein for his theory which successfully encodes our knowledge of space-time and gravity.<br /><br />With all they knew they did not understand quantum mechanics. The geometry is very fuzzy, uncertain, probabilistic and entangled there. Although it was introduced theoretically, this analogy works to explain noncommutative geometry (NCG) very accurately. The celebrated 1943 paper by Gelfand and Naimark proves the anti-equivalence (functors are contravariant) between (the<br />category of) compact Hausdor spaces X and (the category of) commutative C* algebras A. In the construction, A is nothing but C(X), the algebra of continuous functions . This correspondence gives rise to the viewpoint that any C* algebra is the space of functions C(X) of some noncomutative space X . They are fuzzy spaces in the sense that we cannot directly see them! My goal in this talk is to present the basic idea of Noncommutative Geometry in as elementary terms as possible and discuss a few canonical and fascinating examples in NCG.<br /><br />Absolutely no background is necessary!Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-39427216337471303982013-04-03T19:05:00.000-07:002013-04-14T19:29:34.619-07:00First Steps in Quantum Computing<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;"> Discovery Cafe Talk: Tuesday, April 9, 5:30-6:30 PM, MC 108. Pizza to follow immeditaley after! </div><div class="separator" style="clear: both; text-align: left;">Speaker: Masoud Khalkhali </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-5Gn2Nu68cp4/UV4MbJAvsKI/AAAAAAAAANE/UlaiFWcrVps/s1600/Bit.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://3.bp.blogspot.com/-5Gn2Nu68cp4/UV4MbJAvsKI/AAAAAAAAANE/UlaiFWcrVps/s320/Bit.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Public-key encryption and security of internet communications is based on a certain mathematical hypothesis:factoring a given integer N is a computationally difficult problem. The best current methods take about</div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">$$ O(e^{1.9 (\log N)^{1/3} (\log \log N)^{2/3}})$$</div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">operations. This is almost exponential in log N, the number of digits of N </div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;"></div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">A <a href="http://en.wikipedia.org/wiki/Quantum_computer">quantum computer</a>, running <a href="http://en.wikipedia.org/wiki/Shor%27s_algorithm">Shor's algorithm</a>, can factor N in</div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">$$O((\log N)^3)$$</div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">steps! This is polynomial in log N, or polynomial time, and a huge improvement over current methods.</div><br />This talk will introduce mathematics and physics ideas behind quantum computing and Shor's fast factoring quantum computing algorithm. <br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-27932644973204197762013-01-18T18:02:00.000-08:002013-01-18T18:02:58.570-08:00Random Walks and Brownian Motion<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-XoVRD_a1dng/UPnyb1KEU-I/AAAAAAAAAL8/9GfGRcQr7j4/s1600/random.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-XoVRD_a1dng/UPnyb1KEU-I/AAAAAAAAAL8/9GfGRcQr7j4/s320/random.png" width="272" /></a></div><br /><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;"><br /><br />This Tuesday, Jan 15, we had our first Discovery Cafe meeting in 2013. I had promised before to give some introductory lectures on random walks and Brownian motions. So we started the year with this. Notice that now we meet at 3:30 PM on Tuesdays. Ruth took some notes and I am just using these notes here. I leave out many details. For references you can check out these two books: <a href="http://www.amazon.com/Random-Equation-Student-Mathematical-Library/dp/0821848291">1</a> and <a href="http://www.amazon.com/Random-Electric-Networks-Mathematical-Monographs/dp/0883850249">2</a><br /><br />There are two problems that at first look quite unrelated but our first task is to understand the close relation between the two. <span style="background-color: red;"><b>Random walks</b></span> and the <span style="background-color: red;"><b>Dirichlet problem</b>.</span> In one dimension, on a finite set of points 0, 1, .....up to n, a walker randomly walks around. The chances of going forward or backward is % 50 each . Let P(i) be the probability of the random walker hitting the right end point n before hitting the end point 0, when he starts at i. Say point n is the jail, where the police can keep the drunk overnight, and point 0 can be the drunk's home. So we are looking at the probability that he will end up at the police station, versus ending up at home. This process is potentially infinite - the drunk can take up to an infinite number of steps back and forth without hitting either boundary. But never mind this at the moment and we neglect this possibility. We come back to this in the next week when we carefully define our probability space. </div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;"><br />Finding the probability distribution P(i) is closely related to another problem, the discrete Dirichlet problem. We are looking for a function f(i) for i=1, ....n-1, with given boundary values f(0)=0 and f(n)=1, and such that f satisfies the equation </div>$$f(i) = \frac{f(i + 1) + f(i-1)}{2}$$ <br /><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">We say f has the<b> <span style="background-color: red;">mean value property</span></b>. We shall see that this equation is the discrete analogue of<br />$$ \Delta f=0$$ <br />where \( \Delta f = f'' \) is the second derivative operator, the <span style="background-color: red;"><b>Laplace operator</b></span> in dimension one. Thus f can be regarded as a discrete <span style="background-color: red;"><b>harmonic function</b></span> with given boundary values f(0) and f(n).</div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;"><br /> We checked that this Dirichlet problem has a unique solution. While a proof of this fact in dimension one is very easy and one can write in fact simple formula for f (what is it ?), existence of solutions in higher dimensions even in this discrete case is not totally trivial and in particular no simple formula exist. So the point of this probabilistic approach is that it works in all dimensions with minor modifications. First of all uniqueness follows from the following <span style="background-color: red;"><b>maximum principle:</b></span></div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;"><br />Lemma (maximum principle): Any harmonic function achieves its maximum on the boundary points. </div><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;"><br />This easily follows from the mean value property. Then we used this maximum principle to prove uniqueness:<br />for any two solutions the difference \( f_1-f_2 \) is also a solution with boundary values equal to zero. This will violate the maximum principle unless \( f_1 = f_2 \) at all points.</div><br /><div style="-qt-block-indent: 0; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0px;">Our next goal was to show that the the probability function P(i) has the mean value property. Since it clearly satisfies the required boundary conditions, this will prove that P is the harmonic function we were looking for. This is the first relation between random walks and the Dirichlet problem.<br /><br /><br />This all follows from a cute property of probabilities: Let X be a probability space, L and R be two mutually exclusive events that cover the whole X. Then for any event E we have<br /> $$Prob (E)= Prob (E; L) Prob (L) $$<br />$$+ Prob (E; R)Prob (R)$$<br /><br /> Starting your random walk at i, let L = going left, R = going right, E = the event of getting to the end point n before getting to 0. Now it is not difficult to check, using the above probability identity, that P has the mean value property and since it clearly satisfies the boundary conditions it must be the solution of the Dirichlet problem. <br /><br />Next week we shall carefully defines the probability space X, as the space of all random walks and show how to define a probability measure on it, and will give a general formula for the solution in terms of integration over this path space. We shall then move to higher dimensions where things can get more even interesting! Here is graphs of some random walks all starting at the origin and going up to 100 steps. <br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-FKVjoVwxQfc/UPn5M0Lv9jI/AAAAAAAAAMM/H1dH1zwuF10/s1600/random2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://3.bp.blogspot.com/-FKVjoVwxQfc/UPn5M0Lv9jI/AAAAAAAAAMM/H1dH1zwuF10/s320/random2.png" width="320" /></a></div><br /><br /></div>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com1tag:blogger.com,1999:blog-2017096698513592603.post-62467906424222310032012-11-27T18:29:00.000-08:002012-11-27T18:29:02.007-08:00Discovery Cafe Day 3: guest post by Jonathan Herman<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> <br /><div class="MsoNormal" style="line-height: 200%;"><span lang="EN-CA">In the cafe today, more insight was given as to how mathematics is an ongoing process. In order to discover and learn, a mathematician needs to be constantly asking why, and how? Indeed, we had previously proved that there are infinitely many primes. But the question today was, now what? What other questions can we ask to further understand prime numbers? Who cares that there are infinitely many primes? How is this theorem useful in discovering new concepts and ideas?</span><br /><br /><span lang="EN-CA">Some questions asked by the group included: Is there a pattern among the distribution of prime numbers? Are there infinitely many primes of the form \(n^2 + 1\)? How many prime numbers are there less than n, for any integer n? This last question was actually conjectured by Gauss and Legendre, and a strong approximation is given in the famous Prime Number Theorem. We discussed this interesting result: </span><br /><span lang="EN-CA">$$ \pi(x) \sim \frac{x}{\log x} $$</span><br /><span lang="EN-CA">i.e. \(\lim \frac{\pi(x) }{\log x /x} = 1 \, \, x\to \infty \), where \( \pi (x) \) is the number of primes less than \( x \). The group spent the rest of the time proving Euler’s amazing Product Formula and one of its consequences.<span style="mso-spacerun: yes;"> See the other post about Day 3. </span> </span><span lang="EN-CA" style="font-family: Calibri; font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-CA; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;"> </span><br /><span lang="EN-CA" style="font-family: Calibri; font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-CA; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;">This powerful formula gave a classic example as to why a mathematician, or anyone, should be constantly asking “now what?” Indeed the group then proved an amazing corollary to this formula: that \( \sum \frac{1}{p} = \infty \). This proposition seems very strange indeed. </span></div><div class="MsoNormal" style="line-height: 200%;"><span lang="EN-CA" style="font-family: Calibri; font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-CA; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US;"> </span></div><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-5373023003146514442012-11-25T16:28:00.000-08:002013-01-17T12:41:30.623-08:00Discovery Cafe Day 3: A Prime DayHere is a quick account of what happened in our third Cafe meeting on Nov. 20th. As I was walking down the hall to go to our weekly meeting, I got the idea of expanding a bit on a theme that was touched on last week. As we saw in the last post, Euclid had proved in 300 BC that there are infinitely many prime numbers (I think his actual statement is that there is no biggest prime number). His method of proof was <i>proof by contradiction</i> as students saw last week.<br /><br />Now this result opened the doors to so many questions about the statistics of primes, many of which are still unanswered! I tried to expand on this theme of how many primes are out there.<br /><br />After nearly 2000 years Euler took the next giant step in this question. Around 1737 he not only gave a different proof of Euclid's theorem, but he went way beyond that and showed that the set of primes is not too thin and in a sense there are many primes! See below for more about this. One of his statements is so intriguing when he writes<br />$$ \sum \frac{1}{p} =\log \log \infty $$<br />What he meant by this? Not so clear, but on a heuristic level one can argue that this hints at the famous <a href="http://en.wikipedia.org/wiki/Prime_number_theorem"><i>prime number theorem</i></a> which was precisely conjectured many years later and proved even many more years later down the road in 1890's. But we are of course not yet ready to discuss all this. <br /><br />To prepare for all this we decided to discuss and prove a simpler well formulated statement. More precisely we settled to prove another statement of Euler's that<br />$$ \sum \frac{1}{p} =\infty,$$<br />where the sum is over all primes of course. This statement not only shows that there are infinitely many primes, it shows that in a sense the set of primes cannot be thin! Taking a clue from the fact that the harmonic series <br /> \( \sum \frac{1}{n} =\infty \) is divergent, let us call a subset \( S \) of integers<i> fat </i>if \( \sum_{n\in S} \frac{1}{n} =\infty \) and <i>thin</i> if \( \sum_{n\in S} \frac{1}{n} < \infty.\) For example the set of perfect squares is thin since we know that \( \sum \frac{1}{n^2} <\infty. \) So we set out to show that the set of all primes is a `fat' subset of the set of natural numbers. Students noticed that this is already a huge improvement over Euclid's theorem. <br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-xi7rZOIAxPA/ULK3cOAaB-I/AAAAAAAAALk/h1GbmhT0X_U/s1600/Euler.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-xi7rZOIAxPA/ULK3cOAaB-I/AAAAAAAAALk/h1GbmhT0X_U/s320/Euler.jpg" width="256" /></a></div><br /><br />Euler's main tool in these types of questions related to primes was an amazing formula that he found, now called the <a href="http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function"><i>Euler Product Formula</i></a>: <br />$$\sum_n \frac{1}{n^s}= \prod_p (1-\frac{1}{p^s})^{-1}, \quad s>1.$$<br />(The sum is over all natural numbers and the product is over all primes). We spent some time proving this. To do this we <br />1. Carefully defined infinite products,<br />2. Showed that in an infinite product<br />$$ \prod_n(1+a_n)$$<br />if \( \sum_n |a_n| \) is convergent then the product is convergent and in that case the product is zero iff one of its terms is zero. <br />3. Used unique factorization of integers into primes to prove the identity.<br />4. Take limits as \( s \to 1\) and use the fact that the harmonic series is divergent, to derive the divergence of the inverse primes series \(\sum \frac{1}{p}\), and hence the fact that the set of primes is not thin.<br /><br /><br />Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-84538123600297061762012-11-20T13:24:00.000-08:002012-11-20T17:14:38.295-08:00Discovery Cafe Day 2. Guest Post by Hanbo Xiao<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-16lQlvLB0I0/UKv01dY7BeI/AAAAAAAAALU/_ob49sJuLvU/s1600/euclid.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-16lQlvLB0I0/UKv01dY7BeI/AAAAAAAAALU/_ob49sJuLvU/s320/euclid.jpg" width="213" /></a></div><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> <br /><div class="MsoNormal"><span lang="EN-CA">This week we had another 3 fascinating topics brought up. </span></div><div class="MsoNormal"><span lang="EN-CA">They are: </span></div><div class="ListParagraphCxSpFirst" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><ul><li><span lang="EN-CA" style="mso-bidi-font-family: Calibri; mso-fareast-font-family: Calibri;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN-CA">Euclid’s proof of infinite primes</span></li></ul></div><div class="ListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><ul><li><span lang="EN-CA" style="mso-bidi-font-family: Calibri; mso-fareast-font-family: Calibri;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN-CA">The formulation of the quadratic formula</span></li></ul></div><div class="ListParagraphCxSpLast" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><ul><li><span lang="EN-CA" style="mso-bidi-font-family: Calibri; mso-fareast-font-family: Calibri;"><span style="mso-list: Ignore;"> <span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN-CA">The “Passengers on a Plane” Problem</span></li></ul></div><div class="MsoNormal"><span lang="EN-CA">What we can conclude from our journey into these topics is that problems that are shocking, ie that are very non intuitive, are actually quite simple with the proper point of view. Sometimes when we are stumped, if something seems impossible to prove, a slight out of the box thinking brings the problem to light, which turns into a light bulb moment leaving us with that warm and fuzzy feeling. So, as I have discovered for myself, if there ever is a time where something seems impossible, don’t be afraid to do something random. Perhaps it can open the door to something that no one has thought of before! </span></div><div class="MsoNormal"><br /><br /><u><span lang="EN-CA">Infinite Primes</span></u></div><div class="MsoNormal"><span lang="EN-CA">This proof, brought up by Euclid in 300BC, is one of the most extraordinary examples of ingenuity. His proof that there exist an infinite number of primes, he offered in his book “Elements”. He did so by contradiction:</span></div><div class="MsoNormal"><span lang="EN-CA">First we need to establish a fact: Any integer can be broken down into a product of primes. As we have learned in grade school a number tree not only looks cool, it provides us with apples and those apples are the prime factors </span></div><div class="MsoNormal"><span lang="EN-CA">With that in mind we can continue……….</span></div><div class="MsoNormal"><span lang="EN-CA">Suppose there exist an infinite number of primes and </span><i><span lang="EN" style="mso-ansi-language: EN;">p<sub>n</sub></span></i><span lang="EN-CA"> is the largest amongst them. </span></div><div class="MsoNormal"><span lang="EN" style="mso-ansi-language: EN;">Take the finite list of prime numbers <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, ..., <i>p<sub>n</sub></i>and let <i>P</i> be the product of all the prime numbers in the list: <i>P</i>= <i>p</i><sub>1</sub><i>p</i><sub>2</sub>...<i>p<sub>n</sub></i>. </span></div><div class="MsoNormal"><span lang="EN" style="mso-ansi-language: EN;">Let <i>Q</i>= <i>P</i>+ 1. Then, <i>Q</i> is either prime or not:</span></div><div class="ListParagraphCxSpFirst" style="mso-list: l0 level1 lfo2; text-indent: -.25in;"><span lang="EN-CA" style="mso-bidi-font-family: Calibri; mso-fareast-font-family: Calibri;"><span style="mso-list: Ignore;">1 <span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN-CA">If Q is a prime, then Q would be the largest prime and thus concludes the proof</span></div><div class="ListParagraphCxSpLast" style="mso-list: l0 level1 lfo2; text-indent: -.25in;"><span lang="EN-CA" style="mso-bidi-font-family: Calibri; mso-fareast-font-family: Calibri;"><span style="mso-list: Ignore;">2<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN" style="mso-ansi-language: EN;">If <i>Q</i>is not prime then some prime factor F divides <span style="mso-bidi-font-style: italic;">Q</span>. And the proof is that this F cannot be in the list of primes that we’ve already established. </span></div><div class="MsoNormal"><span lang="EN-CA">See the simplicity in this proof? Neat eh?</span></div><div class="MsoNormal"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> </div><div class="MsoNormal"><span lang="EN-CA"> </span></div><div class="MsoNormal"><br /></div><br /><br /><br /><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><br /><div class="MsoNormal"><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> </div><div class="ListParagraph" style="margin-left: 0in; mso-add-space: auto;"><b style="mso-bidi-font-weight: normal;"><u><span lang="EN-CA">Passengers on a Plane</span></u></b></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-layout-grid-align: none; text-autospace: none;"><span lang="EN-CA">Question: </span><span lang="EN-CA" style="mso-bidi-font-family: CMR12;">An airplane has 100 seats and is fully booked. Every passenger has an assigned seat. Passengers board one by one. Unfortunately, passenger 1 loses his boarding pass and can't remember his seat. So he picks a seat at random (among the unoccupied ones) and sits on it. Other passengers come aboard and if their seat is taken, they choose a vacant seat at random. What is the probability that the last passenger ends up on his own seat?</span></div><div class="ListParagraphCxSpFirst" style="margin-left: 0in; mso-add-space: auto;"><br /></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><span lang="EN-CA">Think about this question for a while before reading forward</span></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><br /></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><span lang="EN-CA">You may think the answer involves a massive beast of a series and you’re right! If you’ve tried to calculate this series……you are a hero. You should have gotten ½, this is the correct answer, which does make some sense. But here’s the funny thing: no matter how many people there are in this question: 100 people or 1000 people or 10000000 people. The last person who steps on the plane will be faced with two possibilities: either the last vacant seat is his own seat, or the vacant seat belongs to the first person who stepped on the plane. No other seat is a possibility. </span></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><br /></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><span lang="EN-CA">My question is: does this make any intuitive sense? I will leave that up to you to think about…….</span></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><br /></div><div class="ListParagraphCxSpMiddle" style="margin-left: 0in; mso-add-space: auto;"><br /></div><div class="ListParagraphCxSpLast" style="margin-left: 0in; mso-add-space: auto;"><span lang="EN-CA">Thus concludes another day in the Discovery Café. Some very interesting things have appeared before our eyes. What I want to end with is this conclusion: Once a problem is solved, the problem becomes simple. But before we can see the path to the solution, even easy problem seems rather hard. This mirrors life. Things only become “easy” when we know the solution, but this rarely happens. My advice is to change your point of view, change your attitude and perhaps you will come up with something ingenious. And when you do share it with the rest of the world, it will make life “easier” </span></div><br /><br /><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><br /><div class="MsoNormal"><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles></xml><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style><![endif]--> </div><div class="ListParagraph" style="margin-left: 0in; mso-add-space: auto;"><span lang="EN-CA">Until next time………………<a href="http://www.blogger.com/blogger.g?blogID=2017096698513592603" name="_GoBack"></a></span></div><br /><br /><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com2tag:blogger.com,1999:blog-2017096698513592603.post-72338912212056734702012-11-08T17:51:00.000-08:002012-11-17T05:55:10.973-08:00Discovery Cafe Opens!On Tuesday, Nov 6, we opened the discovery cafe to all our undergrad students in mathematical sciences here at Western. We plan to have meetings on a regular weekly basis for the whole year.<br />The regular cafe hours are Tuesdays, 4:30-5:30 PM, in Room 108 in Middlesex College building.<br /><br />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!<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-H9_lp5g6jXM/UJxdA9ThjoI/AAAAAAAAALE/UH8BW-C-Ocw/s1600/Discovery.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="194" src="http://3.bp.blogspot.com/-H9_lp5g6jXM/UJxdA9ThjoI/AAAAAAAAALE/UH8BW-C-Ocw/s320/Discovery.jpg" width="320" /></a></div> <br />For the record, I put here the e-mail that was sent out to our undergrad students: <br /><br /><span style="font-family: Times,"Times New Roman",serif;"><i>The Math Department invites all undergrad students in math and applied math to the new "Math Discovery Cafe".</i></span><br /><br /><span style="font-family: Times,"Times New Roman",serif;"><i>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. </i></span><br /><br /><span style="font-family: Times,"Times New Roman",serif;"><i>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). </i></span><br /><br /><span style="font-family: Times,"Times New Roman",serif;"><i>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!</i></span><br /><br />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.<br /><br />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 <a href="http://en.wikipedia.org/wiki/Jordan_curve_theorem">Jordan Curve Theorem</a>. We just tried to prove special cases and surely students proved that a circle divides the plane in two pieces.<br /><br />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:<br /><br /><ul><li><a href="http://en.wikipedia.org/wiki/Space-filling_curve">Peano's space filling curves,</a></li></ul><ul><li><a href="http://en.wikipedia.org/wiki/Weierstrass_function">Weirestrass' nowhere differentiable continuous function,</a></li></ul><ul><li>Cantor's one-to-one correspondence between line and plane,</li></ul><ul><li>Invariance of domain and why the above cannot happen in a continuous world, </li></ul><ul><li><a href="http://en.wikipedia.org/wiki/Smale%27s_paradox">Smales' sphere eversion</a> (this was asked by Johnathan, and we were surely not prepared to go through it!)</li></ul><ul><li> Cantor's theory of cardinal numbers.</li></ul><br />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. <br /><br />If you think that was it, you will surely be surprised to know that at the end we brought up the example of the <a href="http://mathworld.wolfram.com/DirichletFunction.html">Dirichlet function</a> 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. <br /><br />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! <br /><br /><br /> Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com1tag:blogger.com,1999:blog-2017096698513592603.post-58659186827718322962012-10-06T08:39:00.000-07:002012-10-12T05:28:30.195-07:00Fun with Feynman <div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><br /><a href="http://3.bp.blogspot.com/-I-5p6o15WeI/UG8o01B8CzI/AAAAAAAAAKc/bQlwLCoYbPA/s1600/Feynman1.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="196" src="http://3.bp.blogspot.com/-I-5p6o15WeI/UG8o01B8CzI/AAAAAAAAAKc/bQlwLCoYbPA/s400/Feynman1.jpeg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Richard Feynman</td></tr></tbody></table>Have you ever wondered what those diagrams in the above US commemorative postage stamp in honor of <a href="http://en.wikipedia.org/wiki/Richard_Feynman">Richard Feynman</a> mean? Feynman Diagrams are used to compute the amazingly useful, but also notoriously<br />difficult (among mathematicians at least!), Feynman integrals. Notorious because, among other things, the integrals are badly divergent and so far there is no rigorous mathematical definition of them! How is it possible that a mathematically ill defined concept can accurately predict the most precise<a href="http://en.wikipedia.org/wiki/Precision_tests_of_QED"> measurements </a> that are done by mankind? Here we are talking about an agreement to within ten parts in a billion with experiments! This is a very interesting but also very difficult question to which no one has been able to provide a satisfactory answer yet. But we won't deal with these issues in these talks. <br /><br />While Feynman was mostly guided by actual physical processes and saw the edges and vertices of his diagrams as paths of elementary particles smashing with each other creating and annihilating new particles all the time, it is possible to provided a purely mathematical justification for these diagrams, by analogy with some finite dimensional integrals. <br /><br /><br />This year's Pizza Seminar starts with two talks where we probe some of the mathematical underpinnings of Feynman Integrals. This story starts with the well known Gaussian Integral. You can see the mighty Gaussian lurking in the background in this old German 10 D Mark bill honoring Gauss (click on the image for better viewing).<br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-szVq_HjjJqk/UHBNyzSdkgI/AAAAAAAAAKs/s_KgfR7E-sM/s1600/Gauss.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://1.bp.blogspot.com/-szVq_HjjJqk/UHBNyzSdkgI/AAAAAAAAAKs/s_KgfR7E-sM/s320/Gauss.JPG" width="320" /></a></div><br /><br /><a href="http://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin" title="Lord Kelvin, on Wikipedia">Lord Kelvin</a>, an eminent 19th century scientist, is reputed to have once written the Gaussian integral<br />$$ \int_{-\infty}^{\infty}e^{-x^2} dx = \sqrt{\pi}$$ <br />on the blackboard, and then said to his audience <i>"A mathematician is one to whom</i> that<i> is as obvious as that twice two makes four is to you. Liouville was a mathematician</i>".<br /><br />If Kelvin was alive today he would have perhaps used the following extension of the above integral, known as Feynman's Theorem, to drive home his point:<br /><br />$$ \frac{1}{Z_0} \int_{\mathbb{R}^n} e^{h S(x)}dx= \sum_{\gamma \in \Gamma} \frac{h^{\chi (\gamma)}}{|\text{Aut}\,\gamma |} w (\gamma) $$<br />The left hand side is the integral of an exponential type function while on the right we have a sum over isomorphism classes of all finite graphs! Using the coefficients of the polynomial \( S(x) \), to each graph \( \gamma\) a number \(w(\gamma)\), its Feynman Weight, is attached which appears on the right hand side. To get to know the other ingredients of this formula you can attend the lectures! It is remarkable that this amazing result, which has many applications in physics and maths, is within the reach of an undergraduate math student. <br /><br />What is required to understand these two lectures? For the first lecture just calculus, with a touch of class! and for the second: calculus + very basic group theory + a lot of attention! In particular you don't need to know any physics to understand these lecture.<br /><br />Lecture 1: Wick's Theorem, Tuesday, Oct. 16, MC 108, 4:30 PM. (Speaker: Masoud Khalkhali, UWO)<br />Abstract: Wick's theorem allows us to compute any exponential integral by repeatedly differentiating a Gaussian function. It is a bridge that will take us from Gaussian integrals to Feynman integrals and their evaluations in terms of graph sum, and of course to a delicious Pizza! <br /><br />Lecture 2: Feynman's Theorem, Tuesday Oct. 23, MC 108, 4:30 PM.(Speaker: Travis Ens, UWO)<br />Abstract: Feynman's theorem, a fundamental mathematical tool in quantum field theory, provides a way to evaluate complicated integrals by summing over finite graphs. After exploring how to obtain this sum from an integral, we will reverse the correspondence and prove a famous result of Cayley, that the number of labelled trees with \(n \) vertices is \(n^{n-2}\), using known values of integrals.Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-54660175497715570502011-09-30T05:54:00.000-07:002011-10-10T08:16:46.397-07:00Nuit Galois<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-gsSeSj33R4A/TpBcoDtVNqI/AAAAAAAAAJA/qAmlFSMD6dY/s1600/Galois9.gif"><br /></a><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-s_gsmQsGDNQ/ToW-4yNDzeI/AAAAAAAAAI4/JjmKIovzCnI/s1600/Galois.gif"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 224px; height: 250px;" src="http://2.bp.blogspot.com/-s_gsmQsGDNQ/ToW-4yNDzeI/AAAAAAAAAI4/JjmKIovzCnI/s400/Galois.gif" alt="" id="BLOGGER_PHOTO_ID_5658138389619396066" border="0" /></a><br /><br /><br />Program for Galois Night:<br />Tuesday October 11, Place: MC 107, 4 PM,<br />Speaker: Martin Pinsonnault, UWO<br /><br />Title: <span style="color: rgb(255, 0, 0);">Unsolvability by radicals of the quintic.</span><br /><br />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 <span style="font-style: italic;">solvability</span> 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."<br /><br /><br /><br /><br /><br />As usual Pizza will be served after the talk.<br /><br />There will be a second talk, on Tuesday, Oct 25, on Galois' 200th birthday!, at 4 PM:<br />Speaker: Masoud Khalkhali, UWO<br />Title: <span style="color: rgb(255, 0, 0);">A topological proof of the Abel-Ruffini theorem on unsolvability by radicals of the quintic. </span><br />Abstract: TBA<br /><br /><br /><br /><br /><br /><br /><br /><br />This year marks the 200th anniversary of the birth of <a href="http://en.wikipedia.org/wiki/%C3%89variste_Galois">Evariste Galois</a> (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.<br /><br />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:<br /><blockquote style="color: rgb(255, 0, 0);"> <p><i>Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !</i> (Don't cry, Alfred! I need all my courage to die at twenty.)</p> </blockquote><br />But above all it is the power of his ideas, and his vision of mathematics as a conceptual enterprise that interests us here.<br /><br /><br /><br />In France, the birthplace of Galois, they are celebrating his birth by holding a major<br />international conference is his honor. You can learn more about these events<a href="http://www.galois.ihp.fr/manifestations/colloque/"> here</a> and <a href="http://www.galois.ihp.fr/">here.</a><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><a href="http://www.galois.ihp.fr/"></a>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-62827651543289023642011-09-06T18:04:00.000-07:002011-09-26T12:14:16.835-07:00Why 1+2+3+4+.....= -1/12<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-csNJZTngO3k/TmeKTOkWt3I/AAAAAAAAAIw/DmQ-w0YUxno/s1600/Masoud_Khalkhali.JPG"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 400px; height: 300px;" src="http://4.bp.blogspot.com/-csNJZTngO3k/TmeKTOkWt3I/AAAAAAAAAIw/DmQ-w0YUxno/s400/Masoud_Khalkhali.JPG" alt="" id="BLOGGER_PHOTO_ID_5649636320492304242" border="0" /></a>Speaker: Masoud Khalkhali, UWO<br />Time: Tuesday, September 27, 4:30-5:30 PM<br />Place: MC 107<br />As usual pizza and pops will be served after the talk in grad club.<br /><br /><br />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 <span style="font-weight: bold;">divergent</span> and in fact diverges to infinity. So shouldn't we just write 1+2+3+4+....= <span style="font-weight: bold;">infinite</span> ? 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<span style="color: rgb(255, 0, 0);"> </span><a style="color: rgb(255, 0, 0);" href="http://en.wikipedia.org/wiki/Divergent_series"><span style="font-weight: bold;">divergent series</span></a>. 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.<br /><br />One of the first people who realized the importance of divergent series and developed some techniques to sum such divergent series was <a style="color: rgb(255, 0, 0); font-weight: bold;" href="http://en.wikipedia.org/wiki/Leonhard_Euler">Leonhard Euler</a>. 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.<br /><br />This talk is a quick introduction to one such theory of summability: <a style="color: rgb(255, 0, 0);" href="http://en.wikipedia.org/wiki/Zeta_function_regularization"><span style="font-weight: bold;">zeta function regularization</span></a>. I will show how to compute infinite sums like the one on the title, as well as many others like<br />1+1+1+1+.....= -1/2<br />1+4+9+16+25+.....=0<br />etc.<br /><br />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.<br />1.2.3.4.5..... =-1/2 log (2 \pi)<br /><br />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<br /> 1!-2! +3! -4! +5! -.....<br />I shall explain this and end up with Euler's surprising answer!Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com3tag:blogger.com,1999:blog-2017096698513592603.post-17612755193423133232010-10-16T05:52:00.000-07:002010-10-16T06:04:17.372-07:00Achieving the Unachievable<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_T7JQAgSZoaQ/TLmiLSfb1aI/AAAAAAAAAIM/pCxlmkAnBMU/s1600/escher_gallery1.jpg"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 400px; height: 393px;" src="http://4.bp.blogspot.com/_T7JQAgSZoaQ/TLmiLSfb1aI/AAAAAAAAAIM/pCxlmkAnBMU/s400/escher_gallery1.jpg" alt="" id="BLOGGER_PHOTO_ID_5528628332399941026" border="0" /></a><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_T7JQAgSZoaQ/TLmhNjBYajI/AAAAAAAAAIE/mzg-DaTiJDM/s1600/achieving_the_unachievable3.jpg"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 292px; height: 150px;" src="http://2.bp.blogspot.com/_T7JQAgSZoaQ/TLmhNjBYajI/AAAAAAAAAIE/mzg-DaTiJDM/s400/achieving_the_unachievable3.jpg" alt="" id="BLOGGER_PHOTO_ID_5528627271685401138" border="0" /></a><br /><br /><span style="font-family:'PrimaSans BT,Verdana,sans-serif';">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.<br /><br />As usual, pizza will follow in the Grad Club.<br /></span>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-20746007334244142162010-03-23T09:36:00.000-07:002010-03-23T10:37:42.081-07:00An Evening with Leonhard Euler<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S6j6Zu8XRGI/AAAAAAAAAHQ/ope3rclMOhI/s1600-h/euler1.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 300px;" src="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S6j6Zu8XRGI/AAAAAAAAAHQ/ope3rclMOhI/s400/euler1.jpg" alt="" id="BLOGGER_PHOTO_ID_5451882668936873058" border="0" /></a>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: <a href="http://www.philoctetes.org/past_programs/An_Evening_with_Leonard_Euler">An Evening with Leonhard Euler.</a>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com1tag:blogger.com,1999:blog-2017096698513592603.post-40416989568851464722010-03-12T09:19:00.001-08:002010-03-16T08:21:02.195-07:00Pizza Seminar: Remote Coin Tossing<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_T7JQAgSZoaQ/S5-h0-VudsI/AAAAAAAAAHI/G3AR1hdBifs/s1600-h/couple.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 266px;" src="http://3.bp.blogspot.com/_T7JQAgSZoaQ/S5-h0-VudsI/AAAAAAAAAHI/G3AR1hdBifs/s400/couple.jpg" alt="" id="BLOGGER_PHOTO_ID_5449252005600982722" border="0" /></a><br /><div style="text-align: justify;"><br /></div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5mr3OSq3YI/AAAAAAAAAFQ/gDzYs_wL-h4/s1600-h/coin-flip.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 254px; height: 380px;" src="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5mr3OSq3YI/AAAAAAAAAFQ/gDzYs_wL-h4/s400/coin-flip.jpg" alt="" id="BLOGGER_PHOTO_ID_5447574189498883458" border="0" /></a><br />Department of Mathematics Pizza Seminar <p></p><p>Speaker: Ajneet Dhillon (Western) </p><p>Title: Remote Coin Tossing<br /></p><p>Time and Place: Tuesday, March 16, 5 PM, MC 107; All are welcome! </p><p style="color: rgb(255, 0, 0);">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?<a href="http://adhillon.math.uwo.ca/svn/Talks/RemoteCoin/Remote.pdf"> Here</a> you can download a pdf file of this talk.</p><p style="color: rgb(255, 0, 0);"><br /></p><p style="color: rgb(255, 0, 0);"><br /></p><p style="color: rgb(255, 0, 0);"><br /></p>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-23219368313391526802010-03-12T09:17:00.000-08:002010-03-14T20:34:01.475-07:00Pizza Seminar: The mathematics of music from the wave equation to equal temperament<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S52SJEJ8WsI/AAAAAAAAAGw/XIf5ctZH5-k/s1600-h/WaveEq.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 91px; height: 43px;" src="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S52SJEJ8WsI/AAAAAAAAAGw/XIf5ctZH5-k/s400/WaveEq.gif" alt="" id="BLOGGER_PHOTO_ID_5448671808620092098" border="0" /></a><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5mqAdqFWaI/AAAAAAAAAFI/OXkMGiLBUU8/s1600-h/Phys_img020.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 271px; height: 298px;" src="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5mqAdqFWaI/AAAAAAAAAFI/OXkMGiLBUU8/s400/Phys_img020.gif" alt="" id="BLOGGER_PHOTO_ID_5447572149219187106" border="0" /></a><br />Department of Mathematics Pizza Seminar <p></p><p>Speaker: Rasul Shafikov (Western) </p><p>Title: The mathematics of music: from the wave equation to equal temperament.<br /></p><p>Time: 17:30 PM, Tuesday, March 9; Room: MC 107 </p><p style="color: rgb(255, 0, 0);">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.</p><p style="color: rgb(255, 0, 0);"><br /></p> <a href="http://www.math.uwo.ca/calendar/month.php#" class="close"> </a>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-67040637274271869892010-03-12T09:16:00.001-08:002010-03-12T18:51:53.575-08:00Wallpaper groups<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S5mh-ksyKkI/AAAAAAAAAE4/8igerDuJ__g/s1600-h/wall17.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 345px;" src="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S5mh-ksyKkI/AAAAAAAAAE4/8igerDuJ__g/s400/wall17.gif" alt="" id="BLOGGER_PHOTO_ID_5447563320656800322" border="0" /></a> <br /><div style="display: block;" class="content"><div class="ABST" id="info4" style=""><p><b>Pizza Seminar</b> </p><p>Speaker: Zack Wolske (Western) </p><p>Title: Wallpaper groups<br /></p><p>Time: 17:30 PM, Tuesday March 2; Room: MC 107 </p><p> <span style="color: rgb(255, 0, 0);">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.</span></p><p><span style="color: rgb(255, 0, 0);"><br /></span></p><p><span style="color: rgb(255, 0, 0);"><br /></span></p></div></div>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-41634118576311541922010-03-12T09:12:00.000-08:002010-03-12T18:31:16.163-08:00What if we had infinitely many fingers to count on<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5moKHANQKI/AAAAAAAAAFA/uYUh8WD1_ow/s1600-h/Cantor.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 263px; height: 400px;" src="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5moKHANQKI/AAAAAAAAAFA/uYUh8WD1_ow/s400/Cantor.jpg" alt="" id="BLOGGER_PHOTO_ID_5447570115913400482" border="0" /></a><br /><div style="display: block;" class="content"><div class="ABST" id="info12" style=""><p><b>Pizza Seminar</b> </p><p>Speaker: Serge Randriambololona (Western) </p><p>Title: What if we had infinitely many fingers to count on ?</p><p>Time: 17:30 PM, Tuesday February 9; Room: MC 107 </p><p style="color: rgb(255, 0, 0);">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.</p><p style="color: rgb(255, 0, 0);"><br /></p></div></div> <a href="http://www.math.uwo.ca/calendar/newmonth.php?getdate=20100201#" class="close"> </a>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-90681115867608781722010-03-12T09:07:00.000-08:002010-03-12T09:10:22.228-08:00Circular Billiards<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5p1eUM__hI/AAAAAAAAAGI/IrusYcPlbjI/s1600-h/billiard4.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 288px; height: 288px;" src="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5p1eUM__hI/AAAAAAAAAGI/IrusYcPlbjI/s400/billiard4.gif" alt="" id="BLOGGER_PHOTO_ID_5447795862937796114" border="0" /></a><br /><div style="display: block;" class="content"><div class="ABST" id="info20" style=""><p><b>Pizza Seminar</b> </p><p>Speaker: Siyavus Acar (Western) </p><p>Title: Circular Billiards</p><p>Time: 17:00 PM, Tuesday January 26, 2010; Room: MC 107 </p><p style="color: rgb(255, 0, 0);">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.<br /></p><p style="color: rgb(255, 0, 0);"><br /></p></div></div>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-1701467458737667492010-03-12T09:01:00.000-08:002010-03-15T06:10:55.736-07:00What does the spectral theorem say<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S54xj3thJ4I/AAAAAAAAAG4/Yu4IrgHXqPs/s1600-h/ST2.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 104px; height: 59px;" src="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S54xj3thJ4I/AAAAAAAAAG4/Yu4IrgHXqPs/s400/ST2.png" alt="" id="BLOGGER_PHOTO_ID_5448847091484927874" border="0" /></a><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5r28j5-qTI/AAAAAAAAAGo/GnP1xY0WU_A/s1600-h/SpectralT.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 118px; height: 45px;" src="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5r28j5-qTI/AAAAAAAAAGo/GnP1xY0WU_A/s400/SpectralT.png" alt="" id="BLOGGER_PHOTO_ID_5447938219549042994" border="0" /></a><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5r0RHB9MKI/AAAAAAAAAGY/xQ7A3SM9S0Y/s1600-h/Spectra.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 225px;" src="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5r0RHB9MKI/AAAAAAAAAGY/xQ7A3SM9S0Y/s400/Spectra.png" alt="" id="BLOGGER_PHOTO_ID_5447935274040242338" border="0" /></a><br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5p0a0Qa7gI/AAAAAAAAAGA/if0DvulxVYY/s1600-h/solformorkelse.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 340px; height: 272px;" src="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5p0a0Qa7gI/AAAAAAAAAGA/if0DvulxVYY/s400/solformorkelse.jpg" alt="" id="BLOGGER_PHOTO_ID_5447794703310974466" border="0" /></a><br /><div style="display: block;" class="content"><div class="ABST" id="info10" style=""><p><b>Pizza Seminar</b> </p><p>Speaker: Farzad Fathizadeh (Western) </p><p>What does the spectral theorem say?</p><p>Time: 17:00 PM, Tuesday January 19, 2010; Room: MC 107 </p><p style="color: rgb(255, 0, 0);">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.</p><p style="color: rgb(255, 0, 0);"><br /></p><p style="color: rgb(255, 0, 0);"><br /></p><p style="color: rgb(255, 0, 0);"><br /></p><p style="color: rgb(255, 0, 0);"><br /></p><p style="color: rgb(255, 0, 0);"><br /></p></div></div>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-79337889941317875472010-03-12T08:50:00.000-08:002010-03-12T18:31:45.238-08:00An elementary introduction to elliptic curves<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5py8ORAw-I/AAAAAAAAAF4/cIvE1IhN6Hk/s1600-h/ec.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 285px; height: 400px;" src="http://1.bp.blogspot.com/_T7JQAgSZoaQ/S5py8ORAw-I/AAAAAAAAAF4/cIvE1IhN6Hk/s400/ec.jpg" alt="" id="BLOGGER_PHOTO_ID_5447793078205203426" border="0" /></a><br />Pizza Seminar <p></p><p>Speaker: Emre Coskun (Western) </p><p>Title: An elementary introduction to elliptic curves</p><p>Time: 17:00 PM, Tuesday December 8 2009; Room: MC 108 </p><p style="color: rgb(255, 0, 0);">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.<br /></p><p style="color: rgb(255, 0, 0);"><br /></p>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-17022535783033573642010-03-12T08:42:00.000-08:002010-03-12T08:47:14.418-08:00Oppositions and Paradoxes in Mathematics and Philosophy<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5pwBmzH_qI/AAAAAAAAAFg/igovWe9puj0/s1600-h/3paradoxes-4.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 292px;" src="http://2.bp.blogspot.com/_T7JQAgSZoaQ/S5pwBmzH_qI/AAAAAAAAAFg/igovWe9puj0/s400/3paradoxes-4.jpg" alt="" id="BLOGGER_PHOTO_ID_5447789872155197090" border="0" /></a><br /><div style="display: block;" class="content"><div class="ABST" id="info19" style=""><p><b>Pizza Seminar</b> </p><p>Speaker: John Bell (Western) </p><p>Title: Oppositions and Paradoxes in Mathematics and philosophy</p><p>Time: 17:00 PM, Tuesday, October 27, 2009; Room: MC 108 </p><p style="color: rgb(255, 0, 0);">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.<br /></p><p style="color: rgb(255, 0, 0);"><br /></p></div></div> <a href="http://www.math.uwo.ca/calendar/newmonth.php?getdate=20091001#" class="close"> </a>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-2017096698513592603.post-16532173102344325092010-03-12T08:36:00.000-08:002010-03-12T08:39:40.290-08:00Solving Rubik's cube using group theory<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S5puMfiKp1I/AAAAAAAAAFY/dslbhueH_M4/s1600-h/rubiks-cube.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 286px; height: 286px;" src="http://4.bp.blogspot.com/_T7JQAgSZoaQ/S5puMfiKp1I/AAAAAAAAAFY/dslbhueH_M4/s400/rubiks-cube.jpg" alt="" id="BLOGGER_PHOTO_ID_5447787860160325458" border="0" /></a><br /><div style="display: block;" class="content"><div class="ABST" id="info23" style=""><p><b>Pizza Seminar</b> </p><p>Speaker: Sheldon Joyner (Western) </p><p>Title: Solving Rubik's cube using group theory</p><p>Time: 17:00 PM, Tuesday, September 2009, Room: 108 </p><p style="color: rgb(255, 0, 0);">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.<br /></p><p style="color: rgb(255, 0, 0);"><br /></p></div></div> <a href="http://www.math.uwo.ca/calendar/newmonth.php?getdate=20090901#" class="close"> </a>Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0