Monthly Archives: December 2016

Happy Birthday Ramanujam

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Today is 129th birthday of Srinivasa Ramanujam Iyengar. Let’s see some properties of the number 129:

♦  It is sum of first ten prime numbers. [Tanya Khovanova’s “Number Gossip”]

\displaystyle{2+3+5+7+11+13+17+19+23+29 = 129}.

♦  It is the smallest number that can be written as the sum of 3 squares in 4 ways. [Erich Friedman’s “What’s Special About This Number?”]

\displaystyle{11^2+2^2+2^2 = 10^2+5^2+2^2 = 8^2+8^2+1^2 = 8^2+7^2+4^2 = 129}.

Though the first property appears like a coincidence (to me!), but the second fact is connected to a well-known theorem in number theory.

Legendre’s three-square theorem: A non-negative integer n can be represented as sum of three squares of integers if and only if n is NOT of the form 4^a (8b+7) for some integers a and b.

Therefore, to make sure that a number can be written as sum of three squares or not, we just need to check its divisibility with 4 and 8. Since 129 is a small number (and a multiple of 3) we can easily factorize it as 129 = 3 \times 43.  Now, since 4 is not a factor, we get a=0 and just need to check 8b+7= 129 = 8\times 16 + 1, but no integer b can satisfy this condition. Completing the verification of our example.

It’s not difficult to prove that no integer n=4^a(8b+7) can be sum of three squares. But, it’s difficult to prove the converse of this statement. Till now I didn’t know the complete proof of this theorem. Interestingly, the two available proofs use some deep results like:

◊  quadratic reciprocity law + Dirichlet’s theorem on arithmetic progressions + equivalence class of the trivial ternary quadratic form.

◊  quadratic reciprocity law + Minkowski’s theorem on lattice points contained within convex symmetric bodies + Fermat’s theorem on sums of two squares

I would prefer proving this result using second approach, since proving Dirichlet’s theorem is very difficult (as compared to the main theorem we wish to prove). The proof using second approach was given by Nesmith . C. Ankeny, in  “Sums of Three Squares,” Proceedings of the American Mathematical Society, vol. 8, no. 2, p. 316, Apr. 1957. Also, note that Ankeny prove the three-square theorem only when n is square-free because it’s easy to prove that

Lemma: If an integer is a sum of squares of three positive integers, so is its square.

For proof see this article by Alexander Bogomolny, “Sum of Three Squares” from Interactive Mathematics Miscellany and Puzzles

Now, what remains to determine is the number of ways we can write a non-negative integer which satisfies Legendre’s three-square theorem as sum of three squares. Interestingly, this is a research level problem if we are talking about a formula to calculate the number of presentations of a given number as sum of three squares. Indeed, there is a (very long) paper by Paul T. Bateman titled “On the Representations of a Number as the Sum of Three Squares”, Transactions of the American Mathematical Society, Vol. 71, No. 1 (Jul., 1951), pp. 70-101. Though I didn’t have patience to read this paper, this MathOverflow discussion gives an overview of the Sum of Squares Function. The output of this function is available as A005875: Number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed) but it is useless since it counts the “tuples”. For example, 129 can be represented by 144 =  (3+6+6+3)8 tuples because we are allowed to replace positive by negative integers before squaring them (hence multiplied by 8).

So, we will have to “manually” check the number of ways we can write the non-negative integers less than 129 as sum of three squares (A000408: Numbers that are the sum of three nonzero squares). As of now, I don’t know how to find the distinct representations. Will try to answer this question in future.

 

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Hilbert Effect

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Human beings tend to measure the influence of a person(s) on a particular field of study by associating their name to cornerstones. For example: Urysohn lemma, Tychonoff theorem, Gauss Lemma, Eisenstein Criterion, Chinese Remainder Theorem, Hensel Lemma, Langlands program, Diophantine Analysis, Clifford algebra, Lie Algebra, Riemann Surface, Zariski Topology, Banach–Tarski paradox, Russell paradox, Bernstein polynomial, Bernoulli Number ……

In mathematics their have been some fights about naming the cornerstones, which ended up creating a compund-name. For example, Lobachevsky-Bolyai-Gauss geometry (in textbooks it is generally referred as hyperbolic geometry), Bolzano–Weierstrass theorem (Bolzano prove it in 1817, later Wierstrass proved it again rigorously and popularized it), Schönemann–Eisenstein theorem (in textbooks it is generally referred as Eisenstein Criterion), ……

But, David Hilbert influenced mathematics at a whole new level. Apart from terms like Hilbert Cube (and many more..) named after him, he introduced exotic words in mathematics which are very popular in (research-level) mathematics. Following are some of the terms:

  • Eigen: This word troubled me a lot when I came across the term “eigen-vector” and “eigen-values” a couple of years ago. Hilbert used the German word “eigen”, which means “own”, to denote eigenvalues and eigenvectors of integral operators by viewing the operators as infinite matrices. You can find more information about the history of introduction of this term in mathematics in this web-page by Jeff Miller.
  • Entscheidungsproblem: It is german word for “decision problem”, but still mathematicians tend to use this particular term. For example, the famous paper by Alan Turing titled “On computable numbers, with an application to the Entscheidungsproblem“.
  • Syzygy: Interestingly, “syzygy” is greek word used in astronomy to refer to the nearly straight-line configuration of three celestial bodies in a gravitational system. In Hilbert’s terminology,  “syzygies” are the relations between the generators of an ideal, or, more generally, a module. For more details refer to this article by Roger Wiegand titled “WHAT IS…a Syzygy?“.
  • Nullstellensatz: It is german for “Set of zeros” (according to google translate). But today, just like syzygy, it has whole new meaning in mathematics. For more details, refer to this MathOverflow discussion: What makes a theorem *a* “nullstellensatz.”

Apart from the terms used in mathematics, Hilbert popularized the term “ignorabimus” in philosophy during his famous radio address. For more details read this short Wikipedia article.

It appears that mathematicians (sometimes) tend to use their creativity in naming theorems like Snake Lemma

 

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.