Posts from the ‘Mathematics’ Category

Seven Wonders of Mathematical World

In the words of Alex Bellos the world has no order and math is a way of seeing it in an order.  According to Charles Caleb Colton the study of mathematics, like the Nile, begins in minuteness but ends in magnificence.  Mathematics is the most beautiful and most powerful creation of the human spirit as viewed by Stefan Banach. In Leo Tolstoy‘s experience some mathematician has said that true pleasure lies not in the discovery of truth, but in the search for it.   While in the search of the beauty of mathematics I came across with seven wonderful happenings(recent) of the mathematical world.  Lets adore its magnificence, enjoy its beauty and keep searching.

  1. A mathematical proof of size as big as 200 terabytes for Boolean Pythagorean Triples problem

  2. The implausible existence of the sphere packing problem in dimension 8 and dimension 24 and its staggering solution

  3. Beauty of Mathematics Yee Power I Pie Equal to Zero

  4. Homework theorem of Tamar Barbi

  5. Calculus, long-long ago

  6. A Nobel Prize for a spare time Mathematician

  7. Andrew Wiles’ proof for Fermats Last Theorem after 3 centuries


Andrew Wiles’ proof for Fermats Last Theorem after 3 centuries.

T4.JPGIn 1963, when he was a ten-year-old, Andrew Wiles found a copy of a book on Fermat’s Last Theorem in his local library. He became captivated by it which was easy to understand but which had remained unsolved for 300 years. “I knew from that moment that I would never let it go,” he said. “I had to solve it.”  Fermat’s Last Theorem, first formulated by Pierre de Fermat in the 17th century, is the assertion that the equation x^n + y^n = z^n has no solutions in positive integers for n>2.  Fermat proved his claim for n=4, Leonhard Euler found a proof for n=3, and Sophie Germain proved the first general result that applies to infinitely many prime exponents.  The complete proof found by Andrew Wiles relies on three further concepts in number theory, namely elliptic curves, modular forms, and Galois representations.  Elliptic curves are defined by cubic equations in two variables. They are the natural domains of definition of the elliptic functions introduced by Niels Henrik Abel.  Modular forms are highly symmetric analytic functions defined on the upper half of the complex plane, and naturally factor through shapes known as modular curves. An elliptic curve is said to be modular if it can be parameterized by a map from one of these modular curves.  Andrew Wiles, in a breakthrough paper published in 1995, introduced his modularity lifting technique and proved the semistable case of the modularity conjecture and also Fermat’s Last Theorem..  Few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem.  Wiles’ proof was an epochal moment for mathematics and also the culmination of a remarkable personal journey that began three decades before.

A Nobel Prize for a spare time Mathematician, Angus Deaton

T7.JPGYes in “Puzzles and paradoxes” Angus Deaton reported to have said he was a mathematician of sorts in his spare time and that he had studied economics only to escape from mathematics.  Of course it is his economics, not mathematics, earned him a Nobel Prize.  As we all know, today there is no mathematics prize by Nobel Foundation.  Alfred Nobel in his will dictates that his entire remaining estate should be used to endow “prizes to those who, during the preceding year, shall have conferred the greatest benefit on mankind.” There is a Nobel Prize for Physics, Chemistry, Medicine, Literature, Peace and for Economics then, why don’t add Mathematics which among all conferred the greatest benefit on mankind. Nobel Foundation has to consider this. In 1968, a new prize called the Sveriges Riksbanks Prize for Economic Science in the Memory of Alfred Nobel was created, but since then no requests for new prizes has been granted.  However, when Angus Deaton confessed he was helped by the little mathematics that he had learned, it was a proud moment for every mathematician.

His love for math is comprehended when he say one of his child was a math major and the breadth and depth of experience of his child was much superior to what he had.  Let’s be optimistic and hope that Nobel Foundation be persuaded to include Mathematics as a category in Nobel Prize.  With lots and lots of question branching in my mind for ‘why not a Nobel for math?’ I am looking forward to witness awarding a Nobel prize to a full time Mathematician (like John Forbes Nash, Jr. with both an Abel in math and Nobel in another discipline) too in near future.

Calculus, long-long ago

T1.JPGA concept essential to modern calculus was understood 1500 years earlier than historians have ever seen.  Jupiter’s erratic pace across the sky (appearing to slow down and speed up from day to day based on the combination of its orbit and Earth’s) must have perplexed ancient astronomers and tested their best computational techniques. A surprisingly modern technique was used to calculate how far the bright dot traveled through the sky over the course of months. Their process requires a leap in understanding in how position and speed relate to time, one that wouldn’t appear again until 1350 and that was a precursor to modern calculus.  The connections between speed, position and time are known to most modern travelers (people easily understand speed as a measure of miles or kilometers per hour). Locations are often described in terms of time (it’s only an hour away) rather than distance. The insight that led to calculus demonstrated the connection between a graph of the traveler’s changing speed and the total distance traveled.  Examining the tablets at the British Museum, Mathieu Ossendrijver figured out that the trapezoid calculations were a tool for calculating Jupiter’s displacement each day along the ecliptic, the path that the sun appears to trace through the stars.  The computations recorded on the tablets covered a period of 60 days, beginning on a day when the giant planet first appeared in the night sky just before dawn.  The distance travelled by Jupiter after 60 days, 10º45′, is computed as the area of the trapezoid whose top left corner is Jupiter’s velocity over the course of the first day, in distance per day, and its top right corner is Jupiter’s velocity on the 60th day. In a second calculation, the trapezoid is divided into two smaller ones with equal area to find the time in which Jupiter covers half this distance.  It’s like a precursor if you like of what we know today as integral calculus, which allows us to calculate the movements of decelerating or accelerating objects.  Further, the little tablet Text A(officially named BH40054) had markings that served as a kind of abbreviation of a longer calculation that looked familiar to him. By comparing Text A to the four previously mysterious tablets, he was able to decode what was going on: This was all about Jupiter. The five tablets computed the predictable motion of Jupiter relative to the other planets and the distant stars.

Homework theorem of Tamar Barabi

T2.JPGTamar Barabi spends most her time after school taking ballet lessons.  While doing her geometry homework she discovered that the theorem she was using to solve one of the problems on her homework didn’t actually exist.  Like most discoveries, the eureka moment happened by accident.  Yes, the teacher said the theory she used to solve the homework problem didn’t actually exist. Further, she was told if she could prove it, it could be her theory. So that’s what happened.  ‘Three Radii Theorem’ or ‘Tamar’s Theory’ came to existence, though there are some disputes comparing it to Euler’s work (for a point in a circle that is not the center at most two points lie on the circumference at any distance from that point).  With help from her dad, who is also a math teacher, they sent the theorem to experts around the world.  According to Tamar’s Theory if three or more equal lines leave a single point and reach the boundary of a circle, the point is the center of the circle and the lines are its radii.  As said by Professor Ron Livne, Einstein School of Mathematics at Hebrew University, Tamar deserves praise for finding a new twist of stating that a circle has only one center and only one radius.  The teacher-student team sent the theorem along with proof to a mathematician from the Massachusetts Institute of Technology.  Tamar’s theorem can give elegant proofs for other important theorems as recognized by Ofer Grossman from MIT.

Beauty of Mathematics Yee Power I Pie Plus One Equal to Zero

T6.JPG Many have written of the experience of mathematical beauty as being comparable to that derived from the greatest art. This makes it interesting to learn whether the experience of beauty derived from such a highly intellectual and abstract source as mathematics correlates with activity in the same part of the emotional brain as that derived from more sensory, perceptually based, sources. To determine this, the researchers (Semir Zeki, John Paul Romaya, Dionigi M. T. Benincasa and Michael F. Atiyah) used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.  The formula most consistently rated as beautiful (average rating of 0.8667), both before and during the scans, was Leonhard Euler’s identity.  The formula links 5 fundamental mathematical constants with three basic arithmetic operations, each occurring once as e^(iπ) + 1 = 0.  As precisely said by Bertrand Russell, Mathematics, rightly viewed, possesses not only truth, but supreme beauty.   The beauty of mathematical formulations lies in abstracting, in simple equations, truths that have universal validity.  If the experience of mathematical beauty is not strictly related to understanding (of the equations), what can the source of mathematical beauty be? That is perhaps more difficult to account for in mathematics than in visual art or music. Whereas the source for the latter can be accounted for, at least theoretically, by preferred harmonies in nature or preferred distribution of forms or colors, it is more difficult to make such a correspondence in mathematics. The Platonic tradition would emphasize that mathematical formulations are experienced as beautiful because they give insights into the fundamental structure of the universe.

The implausible existence of the sphere packing problem in dimension 8 and dimension 24 and its staggering solution.

T5.JPGThe sphere packing problem asks how to arrange congruent balls as densely as possible without overlap. The density is the fraction of space covered by the balls, and the problem is to find the maximal possible density. This problem plays an important role in geometry, number theory, and information theory.  Aside from the trivial case of one dimension, in 1892 the optimal density was previously known in two dimensions.  Then in 2005 for three dimensions by Thomas C. Hales and then in 2015 by the team consisting of Thomas Hales(Thomas Hales, Mark Adams, Gertrud Bauer, Dat Tat Dang, John Harrison, Truong Le Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Thang Tat Nguyen, Truong Quang Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, An Hoai Thi Ta, Trung Nam Tran, Diep Thi Trieu, Josef Urban, Ky Khac Vu, Roland Zumkeller).  Lattices are far more algebraically constrained, and it is widely believed that they do not achieve the optimal density in most dimensions.  By contrast, periodic packings at least come arbitrarily close to the optimal sphere packing density.  A lattice is a discrete subgroup of Rn of rank n, and a lattice packing uses spheres centered at the points of a lattice, while periodic packings are unions of finitely many translates of lattices.  In this connection, Henry Cohn and Abhinav Kumar in 2009 recognized Optimality and uniqueness of the Leech lattice among lattices.  Now, Maryna Viazovska proved, based on linear programming bounds, that no packing of unit balls in Euclidean space R8(8-dimension) has density greater than that of the E8-lattice packing.  Subsequently a team consisting of Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska proved that the Leech lattice is the densest packing of congruent spheres in 24 dimensions, and that it is the unique optimal periodic packing.