Summer school on Finite Geometry 2022

About

The summer school is part of SCALE. For practical info click here.

Finite Geometry

Finite geometry is a branch of mathematics situated within discrete mathematics and combinatorics, which is concerned with the study of discrete objects using geometric and combinatorial techniques. Typical examples are Galois geometry (the study of objects embedded in projective spaces over finite fields), and finite incidence geometry (axiomatic study of geometries). Foundational work and major contributions can be credited to Beniamino Segre (Galois geometry) and Jacques Tits (Buildings). Finite geometry has strong connections with algebraic coding theory, graph theory, finite group theory, algebraic geometry over finite fields, design theory, finite fields, algebraic combinatorics, and more. Applications are typically found in the theory of error-correcting codes (e.g. MDS codes, MRD codes), and cryptography. With the rapid development of computer algebra systems, research problems in finite geometry are often a combination of a theoretical and a computational approach, and theories/conjectures are usually motivated/supported by computational evidence. The aim of this school is to introduce some of the main concepts in finite geometry, to increase the audience's knowledge of the many results and conjectures, to introduce some techniques available in computer algebra systems to approach problems, to provide an opportunity for PhD students to discuss and share their own projects, and to enable mathematicians in this area to meet and plan scientific collaboration.

Lecturers

Bence Csajbók (Politecnico di Bari) Polynomials over finite fields and applications (notes)

Jan De Beule (Vrije Universiteit Brussel) Geometries over finite fields (notes)

Maria Montanucci (Technical University of Denmark) Algebraic geometry codes (notes)

Morgan Rodgers (Istinye University) Matrix techniques in finite geometry (notes)

Abstracts

The abstracts and the titles of the lectures, and the abstracts of the contributed talks can be found in the book of abstracts.

Schedule

Monday

Jan 1: Finite projective spaces

Morgan 1: Graphs and incidence structures

Bence 1: Functions over finite fields

Maria 1: Algebraic curves and their function field (basic notions and definitions)

Jan 2: Spreads, arcs and caps (the basics)

Tuesday

Morgan 2: Finite geometries (including GQs and polar spaces)

Bence 2: Linearised Polynomials

Maria 2 More on algebraic curves

Jan 3: Generalized polygons

Morgan 3: Distance Regular graph point of view, eigenvalue techniques

Wednesday

Contributed talks

Thursday

Bence 3: Applications and constructions

Maria 3: AG codes part 1

Jan 4: Polar spaces

Morgan 4: Cameron-Liebler and related problems (tight sets, m-ovoids, spreads, m-covers)

Bence 4: Polynomials associated with point sets

Friday

Maria 4: AG codes part 2

Jan 5: Polar spaces

Morgan 5: Integer Programming techniques for search, some Kramer-Mesner type techniques for assuming a prescribed stabilizer group

Bence 5: Resultant method

Maria 5: Decoding AG codes

Daily schedule

09:30-10:20 Lecture 1

10:20-10:35 Break

10:35-11:25 Lecture 2

11:25-11:55 Coffee break

11:55-12:45 Lecture 3

12:45-14:00 Lunch break

14:00-15:30 Discussion and study session

15:30-16:00 Coffee break

16:00-16:50 Lecture 4

16:50-17:05 Break

17:05-17:55 Lecture 5

Kalemlık

Check this guide if you need help building your SCALE pencil holder.