Finite geometry Arcs in finite projective spaces Further applications of finite geometry Finite geometry: Pure mathematics close to practical applications Leo Storme Ghent University Dept. of Mathematics: Analysis, Logic and Discrete Mathematics Krijgslaan 281 - Building S8 9000 Ghent Belgium Francqui Foundation, April 19, 2021 Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry OUTLINE 1 FINITE GEOMETRY 2 ARCS IN FINITE PROJECTIVE SPACES 3 FURTHER APPLICATIONS OF FINITE GEOMETRY Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry OUTLINE 1 FINITE GEOMETRY 2 ARCS IN FINITE PROJECTIVE SPACES 3 FURTHER APPLICATIONS OF FINITE GEOMETRY Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FINITE GEOMETRY What is finite geometry? Work with finite structures. Investigation of geometrical properties. Examples: finite projective spaces, finite generalized quadrangles, partial geometries, semi-partial geometries, :::, Some problems of other research areas can be translated into problems regarding substructures in finite geometries. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FINITE FIELDS q = prime number. Prime fields: Fq = f0;:::; q − 1g (mod q). Calculating (mod q) means calculating with the integers 0;:::; q − 1, but add/subtract multiples of q till the result again is an integer in f0;:::; q − 1g. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FINITE FIELDS Example: q = 7. F7 = f0;:::; 6g (mod 7). 2 + 3 ≡ 5 (mod 7). 5 + 3 ≡ 1 (mod 7). 5 · 3 ≡ 1 (mod 7). Calculating with the days of the week is calculating (mod 7). If today is Thursday, which day of the week are we in five days? Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FINITE FIELDS Finite fields (Galois fields): Fq: exist for every prime power q, q = ph, p prime, h ≥ 1. Évariste Galois (25 oktober 1811, Bourg-la-Reine - 31 mei 1832, Parijs) Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry 3-DIMENSIONAL VECTOR SPACE Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry VECTOR SPACES OVER FINITE FIELDS Vector space V (n; q) of dimension n over the finite field Fq of order q V (n; q) = f(x1;:::; xn): x1;:::; xn 2 Fqg (x1;:::; xn) + (y1;:::; yn) = (x1 + y1;:::; xn + yn). α(x1;:::; xn) = (αx1; : : : ; αxn), for α 2 Fq. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry ADDITION AND SCALAR MULTIPLICATION OF VECTORS Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FROM VECTOR SPACE TO PROJECTIVE SPACE For some problems regarding vector spaces, it is allowed to replace a non-zero vector (x1;:::; xn) by a non-zero scalar multiple α(x1;:::; xn), α 2 Fq n f0g. In finite geometry, a non-zero vector (x1;:::; xn) and its non-zero scalar multiple α(x1;:::; xn), α 2 Fq n f0g, are considered as one object, called a projective point. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FROM VECTOR SPACE TO PROJECTIVE SPACE Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE FANO PLANE PG(2; 2) From V (3; 2) to PG(2; 2) Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE FANO PLANE PG(2; 2) Gino Fano (1871-1952) Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry PROJECTIVE PLANE OF PRIME ORDER Let p be a prime, the projective plane PG(2; p) contains p2 + p + 1 points p2 points (1; y; z), y; z 2 f0; 1;:::; p − 1g (mod p), p points (0; 1; z), z 2 f0; 1;:::; p − 1g (mod p), one point (0; 0; 1). p2 + p + 1 lines p2 lines X + aY + bZ = 0, a; b 2 f0; 1;:::; p − 1g (mod p), p lines Y + bZ = 0, b 2 f0; 1;:::; p − 1g (mod p), one line Z = 0. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry POINT BELONGS TO A LINE Point of PG(2; p) belongs to line of PG(2; p) if and only if coordinates of the point satisfy the linear equation defining the line. Examples (1; y; z) belongs to the line X + aY + bZ = 0 if and only if 1 + ay + bz = 0 (mod p). (0; 0; 1) belongs to the line X + aY + bZ = 0 if and only if 0 + a · 0 + b · 1 = b = 0 (mod p). Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE FANO PLANE PG(2; 2) Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry PROPERTIES OF THE PROJECTIVE PLANE PG(2; p) Properties of the projective plane PG(2; p), p prime: a point belongs to p + 1 lines, and two points belong to one common line. a line contains p + 1 points, and two lines intersect in one common point. there exist four points in PG(2; p), no three on a common line. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE PLANE PG(2; 2) The plane PG(2; 2) contains 7 points and 7 lines. A point belongs to 3 lines and a line contains 3 points. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE PLANE PG(2; 3) The plane PG(2; 3) contains 13 points and 13 lines. A point belongs to 4 lines and a line contains 4 points. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE PLANE PG(2; 5) The plane PG(2; 5) contains 31 points and 31 lines. A point belongs to 6 lines and a line contains 6 points. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE PLANE PG(2; 7) The plane PG(2; 7) contains 57 points and 57 lines. A point belongs to 8 lines and a line contains 8 points. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry PROPERTIES OF THE PROJECTIVE PLANE PG(2; p) Properties of the projective plane PG(2; p), p prime: a point belongs to p + 1 lines, and two points belong to one common line. a line contains p + 1 points, and two lines intersect in one common point. there exist four points in PG(2; p), no three on a common line. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry PROPERTIES OF THE PROJECTIVE PLANE PG(2; p) Every two lines intersect in one common point. No parallel lines in PG(2; p), p prime. Idea: parallel lines share a point at infinity. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry PARALLEL LINES SHARE A POINT AT INFINITY Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry THE PLANE PG(2; 3) The plane PG(2; 3) contains 13 points and 13 lines. A point belongs to 4 lines and a line contains 4 points. Points 10, 11, 12 and 13 are points at infinity. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry FROM V (n + 1; q) TO PG(n; q) 1 From V (1; q) to PG(0; q) (projective point), 2 From V (2; q) to PG(1; q) (projective line), 3 ··· 4 From V (i + 1; q) to PG(i; q) (i-dimensional projective subspace), 5 ··· 6 From V (n; q) to PG(n − 1; q) ((n − 1)-dimensional subspace = hyperplane), 7 From V (n + 1; q) to PG(n; q) (n-dimensional space). Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry PG(3; 2) From V (4; 2) to PG(3; 2) Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry OUTLINE 1 FINITE GEOMETRY 2 ARCS IN FINITE PROJECTIVE SPACES 3 FURTHER APPLICATIONS OF FINITE GEOMETRY Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry BENIAMINO SEGRE (February 16, 1903, Turin - October 2, 1977, Frascati) Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry BENIAMINO SEGRE Beniamino Segre: one of the founding fathers of finite geometry. Stimulated research on finite geometry: Finite geometry is an interesting research area. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry DEFINITION DEFINITION (SEGRE) n-Arc in PG(k − 1; q): set of n points, every k linearly independent. Example: n-arc in PG(2; q): n points, no three collinear. Leo Storme Finite geometry Finite geometry Arcs in finite projective spaces Further applications of finite geometry NORMAL RATIONAL CURVE Normal rational curve (NRC) = classical example of (q + 1)-arc in PG(k − 1; q): k−1 f(1; t;:::; t )jjt 2 Fqg [ f(0;:::; 0; 1)g: For k − 1 = 2, NRC = conic 2 f(1; t; t )jjt 2 Fqg [ f(0; 0; 1)g 2 is set of points of conic X1 = X0X2.