Category Archives: Personal Experiences

My mathematics related experiences

Intra-mathematical Dependencies

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Recently I completed all of my undergraduate level maths courses, so wanted to sum up my understanding of mathematics in the following dependency diagram:

mat-dependency (1)

I imagine this like a wall, where each topic is a brick. You can bake different bricks at different times (i.e. follow your curriculum to learn these topics), but finally, this is how they should be arranged (in my opinion) to get the best possible understanding of mathematics.

As of now, I have an “elementary” knowledge of Set Theory, Algebra, Analysis, Topology, Geometry, Probability Theory, Combinatorics and Arithmetic. Unfortunately, in India, there are no undergraduate level courses in Mathematical Logic and Category Theory.

This post can be seen as a sequel of my “Mathematical Relations” post.

Teaching Mathematics

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One of the most challenging and rewarding thing associated with being a math enthusiast (a.k.a. mathematician) is an opportunity to share your knowledge about the not so obvious truths of mathematics. A couple of years ago, I tried to communicate that feeling through an article for high school students.

When I joined college, I tried to teach mathematics to some kids from financially not-so strong family. Since they had no exposure to mathematics, I had to start with  concepts like addition and multiplication of numbers. My experience can be summarized as the following stand-up comedy performance by Naveen Richard:

After trying for about a couple of months to teach elementary mathematics, I gave up and now I discuss mathematics only above the high school level. Last week I delivered a lecture discussing the proof of Poncelet’s Closure Theorem:

Whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

I had spent sufficient time preparing the lecture, and believed that I was aware of all possible consequences of this theorem. But, almost half way through the lecture one person (Haresh) from the audience of 10 people, pointed out following fascinating consequence of the theorem:

If an n-sided polygon is inscribed in one conic section and circumscribed by the other one, then it must be a convex polygon and no other m-sided polygon (with m≠n) can be inscribed and circumscribed by this pair of conic sections.

This kind of insights by audience motivates me to discuss mathematics with others!

Division algorithm for reals

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You must have seen long-division method to compute decimal representation for fractions. Astonishingly, I never pondered about how one would divide an irrational number to get decimal representation. Firstly, this representation will be approximate. Secondly, we have been doing this in name of “rationalizing the denominator” stating the reason that division by irrationals is not allowed. But, in fact, this is the same problem as faced while analysing division algorithm for Gaussian integers.

Bottom line: Numbers are just symbols. We tend to assign meaning to them as we grow up. Since the set of real numbers, rational numbers and integers  form an Euclidean domain, we can write a division algorithm for them. For example, we don’t have special set of symbols for 3 divided by π, but 3 divided by 2 is denoted by 1.5 in decimals.

Borsuk-Ulam Theorem

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Yesterday, I was fortunate enough to attend a lecture delivered by Dr. Ritwik Mukherjee, one of my professors, to motivate the study of algebraic topology. Instead of using the “soft targets” like Möbius strip etc. he used the following profound theorem for motivation:

If f: S^n \to \mathbb{R}^n is continuous then there exists an x\in S^n such that:  f(-x)=f(x).

This is known as Borsuk-Ulam Theorem. To appreciate this theorem, one need to know a fundamental theorem about continuous functions known as Intermediate Value Theorem:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

Here is a video by James Grime illustrating Borsuk-Ulam Theorem in 3D:

Though the implications of the theorem itself are beautiful, following corollary known as Ham sandwich theorem is even more interesting. Here is a video by Marc Chamberland explaining this theorem:

Also, yesterday Grant Sanderson uploaded a video exploring the relation of Borsuk-Ulam Theorem with a fair division problem known as Necklace splitting problem:

But, to my amazement, this theorem is related to one of the other most astonishing theorem of algebraic topology called Brouwer fixed-point theorem:

Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

Here is a video by Michael Stevens illustrating Brouwer fixed-point theorem in some interesting cases:

 

Now the applications of this theorem are numerous, and there is a book dedicated to this theorem: “Fixed Points” by Yu. A. Shashkin. But my favourite application of this fixed point theorem is to the board game called Hex, explained by Marc Chamberland here:

If you come across some other video/article discussing the coolness of “Borsuk-Ulam Theorem” please let me know.

Geometry of Virus

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This semester I am taking a course about protein structures. Here is a quick intro to proteins:

Though I have taken some other biology courses in past years, I found this course very much relatable to mathematics. Proteins are made up of “amino acids”. Though, chemistry allows large number of possible structures for amino acids (considering steric hindrance etc.), nature uses only 20 unique amino acids to make billions of different proteins. In my opinion, these 20 amino acids are “axioms” of protein building just like the 5 axioms of euclidean geometry.

Using just 20 amino acids we can get a large variety of protein structures, just like creating any kind of shape in euclidean space using just 5 axioms. Even more fascinating is the existence of “Quasisymmetry in Icosahedral Viruses”. An awesome article explaining this is available here. Note that, the term “triangulation number” stated in that article was not borrowed from mathematics. It’s a term used to study symmetries in icosahedral viruses and refers to “the square of the distance between 2 adjacent 5-fold vertices.”

rainbow

200 Icosahedral Viruses from the PDB (source: http://pdb101.rcsb.org/learn/resource/200-icosahedral-viruses-poster)

Moreover, the structures which don’t conform to classic quasisymmetry are similar to Escher print and Penrose tiling, as visible in following picture:

If you are interested in doing a fun activity, you may refer to: http://pdb101.rcsb.org/learn/resource/quasisymmetry-in-icosahedral-viruses-activity-page

Hello 2017

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2017 is a prime number!! In fact this is 306th prime year A. D. The previous prime year was 2011 and next one will be 2027.

Moreover, 2017 leaves a remainder of 1 when divided by 4. So, this can be represented as sum of two squares. How do we know?

Fermat’s two-square theorem: An odd prime p can be written as sum of two squares if and only if it is can be written as 4n+1 for some integer n. Moreover, this representation is unique.

Unlike the three-square theorem discussed in previous post, this is not that difficult to prove (the only challenging part is to show that if the prime leaves a remainder 1 when divided by 4 then it “can” be written as sum of two squares). There are many ways to prove this, and there is a Wikipedia page dedicated to the popular proofs of it.  But my favorite proof is the one by Richard Dedekind, it requires knowledge of Gaussian integers and some College algebra (properties of unique factorization domain). You can find “existence” proof here and “uniqueness” proof here.

I will end this post with writing step-by-step procedure of writing 2017 as sum of two squares by following the algorithm explained here:

Step 1: Find z such that z^2+1 is divisible by 2017.

Choose any quadratic non-residue a modulo 2017 because then a^{1008} +1 is divisible by 2017. Since half of the residues modulo 2017 are quadratic non-residue, it’s easy to check our guess using a^{1008} +1 divisibility. Easy to observe that a=5 is smallest solution (in fact here is the list of quadratic non-residues modulo 2017 generated by WolframAlpha). Hence z\equiv 5^{504} \pmod{2017}, we get z=229.

Step 2: Compute the greatest common divisor of 2017 and 229+i using the Euclidean algorithm for the Gaussian integers. The answer will be x+yi where x^2+y^2=2017.

Note that norm of any Gaussian integer r+si is r^2+s^2. Hence, the norm of 229+i, N(229+i) = 52442. For euclidean algorithm I will use long division/calculator as:

For Gaussian integers we first multiply the denominator by its conjugate and then use calculator to compute real and imaginary parts of the quotient separately.

2017 = (229+i)(8) +(185-8i) ;   N(185-8i) = 34289

229 + i = (185-8i)(1) + (44+9i);   N(44+9i) = 2017

185-8i = (44+9i)(4-i) + 0

Hence, the gcd is 44+9i.

Finally, we get: 44^2 + 9^2 = 2017.

You may find this property of 2017 not so special, since there are infinitely many primes of form 4n+1. Please let me know more interesting properties of this number…

What is Topology?

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A couple of years ago, I was introduced to topology via proof of Euler’s Polyhedron formula given in the book “What is Mathematics?” by Richard Courant and Herbert Robbins. Then I got attracted towards topology by reading the book “Euler’s gem – the polyhedron formula and the birth of topology” by David S. Richeson. But now after doing a semester course on “introduction to topology” I have realized that all this was a lie. These books were not presenting the real picture of subject, they were presenting just the motivational pictures. For example, this is my favourite video about introduction to topology by Tadashi Tokieda (though it doesn’t give the true picture):

Few months ago I read the book “The Poincaré Conjecture” by Donal O’Shea and it gave an honest picture of algebraic topology. But, then I realized that half of my textbook on topology is about point-set topology (while other half was about algebraic topology). This part of topology has no torus or Möbius strip (checkout this photo) but rather dry set theoretic arguments. So I decided to dig deeper into what really Topology is all about? Is is just a fancy graph theory (in 1736, both Topology and graph theory started from Euler’s Polyhedron formula) or it’s a new form of Geometry which we study using set theory, algebra and analysis.

The subject of topology itself consists of several different branches, such as:

  • Point-Set topology
  • Algebraic topology
  • Differential topology
  • Geometric topology

Point-set topology grew out of analysis, following Cauchy’s contribution to the foundations of analysis and in particular trigonometric representation of a function (Fourier series). In 1872, Georg Cantor desired a more solid foundation for standard operations (addition, etc.) performed on the real numbers. To this end, he defined a Cauchy sequence of rational numbers. He creates a bijection between the number line and the possible limits of sequence of rational numbers. He took the converse, that “the geometry of the straight line is complete,” as an axiom (note that thinking of points on the real line as limits of sequence of rational numbers is “for clarity” and not essential to what he is doing). Then Cantor proved following theorem:

If there is an equation of form \displaystyle{0=C_0+C_1+\ldots +C_n+\ldots} where C_0 = \frac{d_0}{2} and C_n = c_n\sin{(nx)} +d_n\cos{(nx)} for all values of x except those which correspond to points in the interval (0,2\pi) give a point set P of the \nuth kind, where \nu signifies any large number, then d_0=1, c_n=d_n=0

This theorem lead to definition of point set to be a finite or infinite set of points. This in turn lead to definition of cluster point, derived set, …. and whole of introductory course in topology. Modern mathematics tends to view the term “point-set” as synonymous with “open set.” Here I would like to quote James Munkres (from point-set topology part of my textbook):

A problem of fundamental importance in topology is to find conditions on a topological space that will guarantee that it is metrizable…. Although the metrization problem is an important problem in topology, the study of metric spaces as such does not properly belong to topology as much as it does to analysis.

Now, what is generally publicised to be “the topology” is actually the algebraic topology. This aspect of topology is indeed beautiful. It lead to concepts like fundamental groups which are inseparable from modern topology. In 1895, Henri Poincaré topologized Euler’s proof of Polyhedron formula leading to what we call today Euler’s Characteristic. This marked the beginning of what we today call algebraic topology.

For long time, differential geometry and algebraic topology remained the centre of attraction for geometers.But, in 1956, John Milnor discovered that there were distinct different differentiable structures (even I don’t know what it actually means!) on seven sphere. His arguments brought together topology and analysis in an unexpected way, and doing so initiated the field of differential topology.

Geometric topology has borrowed enormously from the rest of algebraic topology it has returned very scant interest on this “borrowed” capital. It is however full of problems with some of the simplest, in formulation, as yet unsolved. Knot Theory (or in general low-dimensional topology) is one of the most active area of research of this branch of topology. Here I would like to quote R.J. Daverman and R.B. Sher:

Geometric Topology focuses on matters arising in special spaces such as manifolds, simplicial complexes, and absolute neighbourhood retracts. Fundamental issues driving the subject involve the search for topological characterizations of the more important objects and for topological classification within key classes.
Some key contributions to this branch of topology came from Stephen Smale (1960s), William Thurston (1970s), Michael Freedman (1982), Simon Donaldson (1983), Lowell Edwin Jones (1993), F. Thomas Farrel (1993), … and the story continues.

References:

[1] Nicholas Scoville (Ursinus College), “Georg Cantor at the Dawn of Point-Set Topology,” Convergence (May 2012), doi:10.4169/loci003861

[2] André Weil, “Riemann, Betti and the Birth of Topology.” Archive for History of Exact Sciences 20, no. 2 (1979): 91–96. doi:10.1007/bf00327626.

[3] Johnson, Dale M. “The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part I.” Archive for History of Exact Sciences 20, no. 2 (1979): 97–188. doi:10.1007/bf00327627.

[4] Johnson, Dale M. “The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part II.” Archive for History of Exact Sciences 25, no. 2–3 (December 1981): 85–266. doi:10.1007/bf02116242.

[5] Lefschetz, Solomon. “The Early Development of Algebraic Topology.” Boletim Da Sociedade Brasileira de Matemática 1, no. 1 (January 1970): 1–48. doi:10.1007/bf02628194.