Summer school on Finite Geometry 2022
8-12 August 2022, Istanbul
Organisers: Leyla Işık, Fatma Karaoğlu, Michel Lavrauw
8-12 August 2022, Istanbul
Organisers: Leyla Işık, Fatma Karaoğlu, Michel Lavrauw
The summer school is part of SCALE. For practical info click here.
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.
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)
The abstracts and the titles of the lectures, and the abstracts of the contributed talks can be found in the book of abstracts.
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
Check this guide if you need help building your SCALE pencil holder.