__Finite Geometry Seminar__

# Program of 2020 Spring

## On some combinatorially defined point sets of \(\mathrm{PG}(2,q)\) + virtual pizza

In [1] we investigated point sets \(S\) of \(\mathrm{PG}(2,q)\), where \(q\) is a power of the prime \(p\), with the property that for each point \(P\) of \(S\) the lines incident with \(P\) are as follows:

- a unique line meeting \(S\) in \(t \mod p\) points and

- \(q\) lines meeting \(S\) in \(m \mod p\) points.

When \(t\) and \(m\) give different residues modulo \(p\) then we proved the following "lemma of renitent lines": the \(t\)-secants incident with points of \(S\) on a fixed \(m\)-secant of \(S\) are concurrent.

In this talk I will speak about point sets \(S\) of \(\mathrm{PG}(2,q)\) with the property that for each point \(P\) of \(S\) the lines incident with \(P\) are as follows:

- there are \(r\) lines meeting \(S\) in not \(m \mod p\) points and

- \(q+1-r\) lines meeting \(S\) in \(m \mod p\) points.

In particular, I will show different generalizations of the lemma of renitent lines in this setting, and some further results when \(r=2\).

This is an ongoing research together with Péter Sziklai and Zsuzsa Weiner, and it is part of my UNKP-19-4 project.

[1] B. Csajbók and Zs. Weiner: Generalizing Korchmáros-Mazzocca arcs, submitted in January 2020.

## No seminar due to educational break

## No seminar due to public holiday

## The necklace folding problem

We give counterexamples for some conjectures about the necklace folding problem, also known as the separated matching problem. The problem has two variants
which we call the heterogeneous and the homogeneous model. Consider a necklace which consists of $n$ red and $n$ blue beads. In both models, the aim is to find a specific folding of the necklace, that is a subsequence of the beads of the specific form. If \(X\) is a subsequence of the beads then let \(-X\) denote the sequence in which the color of every bead is changed. One would like to find a subsequence of the form (\(X\), \(-X\) reversed) and (\(X\), \(X\) reversed) in the heterogeneous and homogeneous models, respectively. Lyngso and Pedersen conjectured (and later independently Kyncl, Pach and Tóth and Giberti, Preissmann and Sebő formulated the same conjecture), that there always exist an appropriate subsequence of the form (\(X\), \(-X\) reversed) and (\(X\), \(X\) reversed) in the heterogeneous and homogeneous models, respectively, with total size \(4n/3\). We have constructions for both the homogeneous and heterogeneous models so that the maximum size of these subsequences are at most \(2(2-\sqrt 2)n+o(n) < 4n/3\).

The talk will be in Hungarian.

## Irregular event: Double blocking sets of size 3q-1 in finite projective planes of order q, in particular when q is prime

A double blocking set in a finite projective plane is a set of points which meets every line in at least two points. A trivial example for a double blocking set is the union of three non-concurrent lines (that is, a triangle), which is of size \(3q\) if the order of the plane is \(q\).

It is particularly interesting to find double blocking sets in \(\mathrm{PG}(2,q)\) (the classical projective plane coordinatized by the finite field of \(q\) elements) of size less than \(3q\) when \(q\) is a prime, in which case the only example known up to 2019 was one of size \(38 = 3q-1\) in \(\mathrm{PG}(2,13)\) found by Braun, Kohnert and Wassermann (2005). In [B. Csajbók, T. Héger: Double blocking sets of size \(3q-1\) in \(\mathrm{PG}(2,q)\), Eur. J. Comb., 2019] we studied in detail a particular construction idea which led to examples of double blocking sets of size \(3q-1\) in \(\mathrm{PG}(2,q)\) for \(q=\)13, 16, 19, 25, 27, 31, 37 and 43 (found by computer search) and to the resolution of two minor conjectures of R. Hill from 1984. In the talk we will discuss these results.

## No seminar due to public holiday

## No seminar

## SAGE minicourse - Part 4

Introductory four-part course to SageMath.

## SAGE minicourse - Part 3, video session

Due to the uncertainity caused by the pandemic, Part 3 of the Sage Minicourse is offered as a video recording. Details can be found on the mailing list.

## No seminar due to spring break

## SAGE minicourse - Part 2

Introductory four-part course to SageMath.

# Important note:

Due to the pandemic, seminar talks from March 6, 2020, are given online. The invitation link will be sent via the seminars' mailing list by Zoltán Lóránt Nagy or the speaker.

## Véges síkok kiegyensúlyozott felső kromatikus számáról

Egy hipergraf kiegyensulyozott felso kromatikus szama, bk, a maximalis szinszam a kovetkezo tulajdonsagokkal rendelkezo csucsszinezesekre nezve:

(i) minden szinosztaly egyforma v. majdnem egyforma meretu (meretbeli elteres legfeljebb 1 lehet)

(ii) nincsen 'rainbow' hiperel, vagyis minden hiperel tartalmaz azonos szinu csucspart.

Eloadasunkban a projektiv sikok egyeneseinekre nezve mutatunk be a bk parameterre vonatkozo korlatokat, hangsullyal a nem-desargues-i esetre.

## SAGE minicourse - Part 1

Introductory four-part course to SageMath.

## Problems in Euclidean Distance Geometry Arising from Practical Applications

Euclidean distance geometry, that is the study of Euclidean geometry based on the concept of distance, is of current interest in several practical applications, such as molecular biology, wireless sensor networks, statics, data visualization and robotics. In this talk we show how introductory algebraic geometry can be used as an effective tool for the solution of certain problems in Euclidean distance geometry.

## Special event: An introduction to elliptic curves over finite fields and cryptography (minicourse)

Elliptic curves are certain algebraic curves that arise naturally when studying many classical problems and they provide a link between number theory, geometry and algebra.

Their surprising abelian group structure makes them prolific in number theory, with applications to famous conjectures such as Fermat’s Last Theorem, and in cryptography, with encryption standards employed by the
biggest computer companies such as Google and Facebook. Also, although the study of elliptic curves dates back to the ancient greeks, there are still many open problems such as the famous Birch-Swinnerton-Dyer conjecture, one of the seven millenium problems.

This minicourse will focus on studying elliptic curves over finite fields and their applications to cryptography.