National Mathematical Society of Pakistan

Talks Schedule

EMINENT PROFESSORS LECTURES SERIES PROGRAM

By Professor Michel Waldschmidt

From 20-11-15 to 26-11-15

Abstracts:

1. Continued fractions: introduction and applications.

The Euclidean algorithm for computing the Greatest Common Divisor (gcd) of two positive integers is arguably the oldest mathematical algorithms: it goes back to antiquity and was known to \textcolor{macouleur}{Euclid}. A closely related algorithm yields the continued fraction expansion of a real number, which is a very efficient process for finding the best rational approximations of a real number. Continued fractions is a versatile tool for solving problems related with movements involving two different periods. This is how it occurs in number theory, in complex analysis, in dynamical systems, as well as questions related with music, calendars, gears… We will quote some of them

2. An elementary introduction to cryptography

Theoretical research in number theory has a long tradition. Since many centuries, the main goal of these investigations is a better understanding of the abstract theory. Numbers are basic not only for mathematics, but more generally for all sciences; a deeper knowledge of their properties is fundamental for further progress. Remarkable achievements have been obtained, especially recently, as many conjectures have been settled. Yet, a number of old questions still remain open.

3. Some arithmetic problems raised by rabbits, cows and the Da Vinci Code

Mathematic is ubiquitous, it occurs in many unexpected places of our real life. In this lecture we develop one example among many, starting with a sequence of numbers which was introduced around 1200 by an italian mathematician, Leonardo da Pisa, also known as Fibonacci. He proposed a model of growth for a population of rabbits, which yields a sequence of integers starting with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... where each number is the sum of the two preceding ones. These integers occur in Phyllotaxy, the science which studies the position of leaves on a stem and the reason for them. Spiral patterns of pine-cones, pineapples, Romanesco cauliflower, cacti, which permit optimum exposure to sunlight, display numbers of the Fibonacci sequence. The ultimate rate of growth of the population of Fibonacci's rabbits is the golden number, which occurs not only in art, architecture, aesthetic, but also the diffraction of quasi-crystals.

To understand better the world around us requires to understand better numbers in particular and mathematics more generally. Many open problems are challenges for mathematicians.

Date/day	Title/Venue	Time
20-11-15 Friday	Continued Fractions: Introduction and Applications Abdus Salam School of Mathematical Sciences, GC University Lahore	10:00 - 11:30
23-11-15 Monday	An Elementary Introduction to Cryptography Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan	11:00 - 12:30
26-11-15 Thursday	Some Arithmetic Problems Raised by Rabbits, Cows and Da Vinci Code Forman Christian College University Lahore	In after noon(exact time yet to be decide)