Discretization and tropicalization: How are they related? Christer O. Kiselman Member of the Reference Group for Mathematics of ISP First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm, Sida Headquarters 2017 March 07 15:00–15:45
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important. Discrete objects, like carpets and mosaics, have been around for thousands of years, but the advent of computers and digital cameras has made them ubiquitous.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important. Discrete objects, like carpets and mosaics, have been around for thousands of years, but the advent of computers and digital cameras has made them ubiquitous. Tropical mathematics is by comparison a relatively new branch of mathematics.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important. Discrete objects, like carpets and mosaics, have been around for thousands of years, but the advent of computers and digital cameras has made them ubiquitous. Tropical mathematics is by comparison a relatively new branch of mathematics. I shall compare the two and also pose a philosophical/mathematical problem in relation to them. 6
Contents 1. Introduction 2. What is a discrete set? 3. To discretize a set or a function 4. Are discrete objects easier than non-discrete? 5. What is tropicalization? 6. Comparing discretization and tropicalization 7. Is there a Rosetta Stone? 7
Introduction Carpets, embroidery work, and mosaics are examples of artifacts that are discrete.
Introduction Carpets, embroidery work, and mosaics are examples of artifacts that are discrete. A carpet can consist of thousands of knots, but they are only finite in number. 9
Persian carpet.
A mosaic is made up of finitely many little stones, tessellas. 11
Ravenna mosaic. 12
Embroidery ... 13
All three can present pictures quite well.
All three can present pictures quite well. These objects have been around for a long time, but now, with computers and digital cameras, they are everywhere. A photo consists of many pixels (picture elements) but only finitely many. 15
So all this makes digital geometry into a kind of geometry of growing interest. Discretization of sets and function is studied from many points of view.
So all this makes digital geometry into a kind of geometry of growing interest. Discretization of sets and function is studied from many points of view. Tropicalization is somehow similar to discretization ...
So all this makes digital geometry into a kind of geometry of growing interest. Discretization of sets and function is studied from many points of view. Tropicalization is somehow similar to discretization ... Here I will talk about both procedures. 18
Discretization What is a discrete set? Intuitively, a discrete set is a set where every point is a bit away from all the other points.
Discretization What is a discrete set? Intuitively, a discrete set is a set where every point is a bit away from all the other points. To make this precise, we need some kind of distance. So denote by d ( x , y ) the distance between two points x and y in some kind of space, like the plane or three-space.
Discretization What is a discrete set? Intuitively, a discrete set is a set where every point is a bit away from all the other points. To make this precise, we need some kind of distance. So denote by d ( x , y ) the distance between two points x and y in some kind of space, like the plane or three-space. Then a set A is said to be discrete if for every element a of A , there is a positive number r such that d ( a , b ) > r for all elements b 6 = a of A . 21
Let us take R as an example, with d ( x , y ) = | x � y | . Then for every number a there are other numbers b that are as close as we like to a . So R with this metric is not discrete.
Let us take R as an example, with d ( x , y ) = | x � y | . Then for every number a there are other numbers b that are as close as we like to a . So R with this metric is not discrete. The subset Z of integers is discrete, because now d ( a , b ) > 1 for all points b 6 = a .
Let us take R as an example, with d ( x , y ) = | x � y | . Then for every number a there are other numbers b that are as close as we like to a . So R with this metric is not discrete. The subset Z of integers is discrete, because now d ( a , b ) > 1 for all points b 6 = a . But beware! Every set is discrete if we define the distance as d ( x , y ) = 1 when x 6 = y and d ( x , x ) = 0 . So also R is discrete with this kind of distance.
Let us take R as an example, with d ( x , y ) = | x � y | . Then for every number a there are other numbers b that are as close as we like to a . So R with this metric is not discrete. The subset Z of integers is discrete, because now d ( a , b ) > 1 for all points b 6 = a . But beware! Every set is discrete if we define the distance as d ( x , y ) = 1 when x 6 = y and d ( x , x ) = 0 . So also R is discrete with this kind of distance. This means that the distance between π and 3.141592653589793238462643383279502884197 is 1 with this metric, whereas it is rather small in the unusal metric. 25
Mathematical models based on real or complex numbers have been very successful—think of celestial mechanics and the theory of electric circuits.
Mathematical models based on real or complex numbers have been very successful—think of celestial mechanics and the theory of electric circuits. However, we should not think of real numbers as being more real than complex numbers or than discrete models. The attribute real is misleading. 27
That time is discrete was a theory developed in Spain in the Middle Ages:
That time is discrete was a theory developed in Spain in the Middle Ages: “As regards the theoretical and philosophical analysis of time, the most important and original contribution of medieval Islamic thinkers was their theory of discontinuous, or atomistic, time.” (Whitrow 1990:79) 29
Moshe ben Maimun (born in 1135 or 1138 in C´ ordoba and deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma ¨ imwn–dhc , wrote:
Moshe ben Maimun (born in 1135 or 1138 in C´ ordoba and deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma ¨ imwn–dhc , wrote: “Time is composed of time-atoms, i.e. of many parts, which in account of their short durations cannot be divided. ... An hour is, e.g. divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subject to division, and in fact are indivisible.” (Whitrow 1990:79)
Moshe ben Maimun (born in 1135 or 1138 in C´ ordoba and deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma ¨ imwn–dhc , wrote: “Time is composed of time-atoms, i.e. of many parts, which in account of their short durations cannot be divided. ... An hour is, e.g. divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subject to division, and in fact are indivisible.” (Whitrow 1990:79) Often we do not go from models based on real numbers to discrete models, but from one very fine discrete model to another model, also discrete, but less fine.
Moshe ben Maimun (born in 1135 or 1138 in C´ ordoba and deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma ¨ imwn–dhc , wrote: “Time is composed of time-atoms, i.e. of many parts, which in account of their short durations cannot be divided. ... An hour is, e.g. divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subject to division, and in fact are indivisible.” (Whitrow 1990:79) Often we do not go from models based on real numbers to discrete models, but from one very fine discrete model to another model, also discrete, but less fine. Physicists study discrete models of spacetime ... and not just recently—but much later than Maimonides ... 33
To discretize To discretize a set means to map it in some way to a discrete set.
To discretize To discretize a set means to map it in some way to a discrete set. A simple example of a discretization is the mapping P ( R n ) 3 A 7! A \ Z n 2 P ( Z n ) , mapping any subset of R n to the set of its points with integer coordinates. 35
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