BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Daniele Bartoli (Università degli Studi di Perugia) DTSTART:20200914T140000Z DTEND:20200914T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/1 DESCRIPTION:Title: Curves over finite fields and polynomial pr oblems\nby Daniele Bartoli (Università degli Studi di Perugia) as par t of Galois geometries and their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract\nAlgebraic curves over finite fields are not onl y interesting objects from a theoretical point of view\, but they also hav e deep connections with different areas of mathematics and combinatorics.\ nIn fact\, they are important tools when dealing with\, for instance\, per mutation polynomials\, APN functions\, planar functions\, exceptional poly nomials\, scattered polynomials.\nIn this talk I will present some applica tions of algebraic curves to the above mentioned objects.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Alberto Ravagnani (Eindhoven University of Technology) DTSTART:20200930T140000Z DTEND:20200930T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/2 DESCRIPTION:Title: Network Coding\, Rank-Metric Codes\, and Ro ok Theory\nby Alberto Ravagnani (Eindhoven University of Technology) a s part of Galois geometries and their applications eseminars\n\nLecture he ld in Google Meet.\n\nAbstract\nIn this talk\, I will first propose an int roduction to network coding and its methods. In particular\, I will explai n how codes with the rank metric naturally arise as a solution to the prob lem of error amplification in communication networks (no prerequisite in i nformation theory is needed for this part). \n\nThe second part of the tal k concentrates instead on the mathematical structure of codes with the ran k metric and its connection with topics in contemporary combinatorics. Mor e precisely\, I will present a link between rank-metric codes and q-rook p olynomials\, showing how this connection plays a role in the theory of Mac Williams identities for the rank metric.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Michel Lavrauw (Sabanci University) DTSTART:20201023T120000Z DTEND:20201023T130000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/3 DESCRIPTION:Title: On linear systems of conics over finite fie lds\nby Michel Lavrauw (Sabanci University) as part of Galois geometri es and their applications eseminars\n\nLecture held in Google Meet.\n\nAbs tract\nA form on an $n$-dimensional projective space ${\\mathbb{P}}^n$ is a homogeneous polynomial in $n+1$ variables. The forms of degree $d$ on ${ \\mathbb{P}}^n$ comprise a vector space $W$ of dimension ${n+d}\\choose{d} $. Subspaces of the projective space ${\\mathbb{P}} W$ are called linear s ystems of hypersurfaces of degree $d$.\nThe problem of classifying linear systems consists of determining the orbits of such subspaces under the ind uced action of the projectivity group of ${\\mathbb{P}}^n$ on ${\\mathbb{P }}W$. In this talk we will focus on linear systems of quadratic forms on $ {\\mathbb{P}}^2$ over finite fields. We will give an overview of what is k nown and explain some of the recent results. This is based on joint work w ith T. Popiel and J. Sheekey.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Maria Montanucci (Technical University of Denmark) DTSTART:20201125T150000Z DTEND:20201125T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/4 DESCRIPTION:Title: Maximal curves over finite fields\nby M aria Montanucci (Technical University of Denmark) as part of Galois geomet ries and their applications eseminars\n\nLecture held in Google Meet.\n\nA bstract\nAlgebraic curves over a finite field $\\mathbb{F}_q$ and their fu nction fields have been a source of great fascination for number theorists and geometers alike\, ever since the seminal work of Hasse and Weil in th e 1930s and 1940s. \nMany important and fruitful ideas have arisen out of this area\, where number theory and algebraic geometry meet. For a long ti me\, the study of algebraic curves and their function fields was the provi nce of pure mathematicians. But then\, in a series of three papers in the period 1977-1982\, Goppa found important applications of algebraic curves over finite fields to coding theory. \n\nThe key point of Goppa's construc tion is that the code parameters are essentially expressed in terms of ari thmetic and geometric features of the curve\, such as the number $N_q$ of $\\mathbb{F}_q$-rational points and the genus $g$.\n\nGoppa codes with goo d parameters are constructed from curves with large $N_q$ with respect to their genus $g$. \nGiven a smooth projective\, algebraic curve of genus $g $ over $\\mathbb{F}_q$\, an upper bound for $N_q$ is a corollary to the ce lebrated Hasse-Weil Theorem\,\n$$N_q \\leq q+ 1 + 2g\\sqrt{q}.$$\nCurves a ttaining this bound are called $\\mathbb{F}_q$-maximal. The Hermitian curv e $\\mathcal{H}$\, that is\, the plane projective curve with equation \n$$ X^{\\sqrt{q}+1}+Y^{\\sqrt{q}+1}+Z^{\\sqrt{q}+1}= 0\,$$\nis a key example o f an $\\mathbb{F}_q$-maximal curve\, as it is the unique curve\, up to iso morphism\, attaining the maximum possible genus $\\sqrt{q}(\\sqrt{q}-1)/2$ of an $\\mathbb{F}_q$-maximal curve. Other important examples of maximal curves are the Suzuki and the Ree curves.\nIt is a result commonly attribu ted to Serre that any curve which is $\\mathbb{F}_q$-covered by an $\\math bb{F}_q$-maximal curve is still $\\mathbb{F}_q$-maximal. In particular\, q uotient curves of $\\mathbb{F}_q$-maximal curves are $\\mathbb{F}_q$-maxim al. Many examples of $\\mathbb{F}_q$-maximal curves have been constructed as quotient curves $\\mathcal{X}/G$ of the Hermitian/Ree/Suzuki curve $\\m athcal{X}$ under the action of subgroups $G$ of the full automorphism grou p of $\\mathcal{X}$.\nIt is a challenging problem to construct maximal cur ves that cannot be obtained in this way for some $G$. \n\nIn this presenta tion\, we will describe our main contributions to the theory of maximal cu rves over finite fields.\nIn particular\, the following topics will be dis cussed:\n\n- how can we decide whether a given $\\mathbb{F}_q$-maximal cur ve is a quotient of the Hermitian curve?\n\n- further examples of maximal curves that are not quotient of the Hermitian curve\;\n\n- determination o f the possible genera of $\\mathbb{F}_q$-maximal curves\, especially quoti ents of $\\mathcal{H}$\;\n\n- Weierstrass semigroups on maximal curves.\n\ nJoint work with: Daniele Bartoli\, Peter Beelen\, Massimo Giulietti\, Leo nardo Landi\, Vincenzo Pallozzi Lavorante\, Luciane Quoos\, Fernando Torre s\, Giovanni Zini.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Bence Csajbók (MTA-ELTE Geometric and Algebraic Combinatorics Res earch Group) DTSTART:20201218T140000Z DTEND:20201218T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/5 DESCRIPTION:Title: Combinatorially defined point sets of finit e Desarguesian planes\nby Bence Csajbók (MTA-ELTE Geometric and Algeb raic Combinatorics Research Group) as part of Galois geometries and their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract\nLet $S $ be a point set of $\\mathrm{PG}(2\,q)$. A line $m$ is called a $k$-secan t of $S$\, if it meets $S$ in exactly $k$ points. Many of the famous objec ts of $\\mathrm{PG}(2\,q)$ have the property that each of their points is incident with the same number of $k$-secants\, for every integer $k$. For example arcs\, unitals\, subplanes\, maximal arcs and Korchmáros-Mazzocca arcs are such objects. In my talk I will present some characterization re sults of point sets with this property.\n\nI will also introduce the follo wing problem of a similar flavour. \n\nLet $M$ be a point set of $\\mathrm {AG}(2\,q)$\, $q=p^n$\, $p$ prime\, and call a direction $(d)$ uniform\, i f more than half of the lines with slope $d$ meet $M$ in the same number o f points modulo $p$. We will call this number the typical intersection num ber at $(d)$. The rest of the affine lines with slope $d$ will be called r enitent. Note that we allow different uniform directions to have different typical intersection numbers. I will show structural properties of the re nitent lines\, in particular I will show that they are contained in some l ow degree algebraic curves of the dual plane.\n\nThe talk is based on join t works with Simeon Ball\, Péter Sziklai and Zsuzsa Weiner.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Cicero Carvalho (Universidade Federal de Uberlandia) DTSTART:20210209T130000Z DTEND:20210209T140000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/7 DESCRIPTION:Title: On certain pairs of primitive elements on f inite fields\nby Cicero Carvalho (Universidade Federal de Uberlandia) as part of Galois geometries and their applications eseminars\n\nLecture h eld in Google Meet.\n\nAbstract\nIn this talk we would like to present som e results on the existence of pairs of elements in a finite field\, where the first element is either primitive or primitive and normal over a subf ield\, and the second element is primitive and a rational function of the first one. \n\nThis is based on joint works with J.P Guardieiro\, V. Neuma nn and G. Tizziotti.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandro Neri (Technical University of Munich) DTSTART:20200706T130000Z DTEND:20200706T140000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/8 DESCRIPTION:Title: Defining Reed--Muller codes in the rank met ric: the Alon--Füredi theorem for endomorphisms\nby Alessandro Neri ( Technical University of Munich) as part of Galois geometries and their app lications eseminars\n\nLecture held in Google Meet.\n\nAbstract\nCodes in the rank metric have gained a huge interest in the last years\, due to the ir applications to network coding and cryptography. The most celebrated fa mily of rank-metric codes is given by Gabidulin codes. It is well-known th at they can be seen as analogues of Reed-Solomon codes in classical coding theory\, which are codes constructed from spaces of univariate polynomial s. The generalization of Reed-Solomon codes to multivariate polynomials le ad to the family of Reed-Muller codes. In the last years\, several researc hers tried to adapt a Reed-Muller-type construction in the rank metric set ting\, unfortunately without success. Hence\, finding such a construction has been an open problem for several years.\n\nWe observed that the main o bstruction for constructing Reed-Muller codes in the rank metric was the i mpossibility to have abelian Galois extensions which are not cyclic\, when dealing with finite fields. Motivated by this intuition\, in this talk we switch to general infinite fields\, and present the theory of rank-metric codes over arbitrary Galois extension. In the abelian case\, we derive th e analogues of the celebrated Alon-Füredi theorem and of the Schwartz-Zip pel lemma for endomorphisms. These results provide nontrivial lower bounds on the rank of a linear endomorphism and are of independent interest. Mor eover\, they allow to show that we can construct rank-metric codes that sh are the same parameters with classical Reed-Muller codes. Central tool for this approach is the Dickson matrix associated to an endomorphism\, which we carefully investigate.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Alain Couvreur (INRIA) DTSTART:20210118T140000Z DTEND:20210118T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/9 DESCRIPTION:Title: On the hardness of the code equivalence pro blem in rank metric\nby Alain Couvreur (INRIA) as part of Galois geome tries and their applications eseminars\n\nLecture held in Google Meet.\n\n Abstract\nIn this talk\, we discuss the code equivalence problem in rank m etric. For $\\mathbb{F}_{q^m}$-linear codes\, which is the most commonly s tudied case of rank metric codes\, we prove that the problem can be solved in polynomial case with an algorithm which is "worst case". On the other hand\, the problem can be stated for general matrix spaces. In this situat ion\, we are able to prove that this problem is at least as hard as the mo nomial equivalence for codes endowed with the Hamming metric.\n\nThis is a common work with Thomas Debris Alazard and Philippe Gaborit.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Giuseppe Mazzuoccolo (University of Verona) DTSTART:20210316T150000Z DTEND:20210316T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/10 DESCRIPTION:Title: How many lines of the Fano plane do we nee d to color a cubic graph?\nby Giuseppe Mazzuoccolo (University of Vero na) as part of Galois geometries and their applications eseminars\n\nLectu re held in Google Meet.\n\nAbstract\nThe problem of establishing the numbe r of perfect matchings necessary\nto cover the edge-set of a cubic bridgel ess graph is related to a long standing conjecture in graph theory attribu ted to Berge and Fulkerson. \nIt turns out that such a problem can be nic ely described in term of colorings of the edge-set of the graph by using a s colors the points of suitable configurations in $PG(2\,2)$ and $PG(3\,2) $ (see [1]). \nMore precisely\, given a set $T$ of lines in the finite pro jective space $PG(n\,2)$\, a $T$-coloring of a cubic graph $G$ is a colori ng of the edges of $G$ by points of $PG(n\,2)$ such that the three colors occurring at any vertex form a line in $T$.\nIn the first part of the talk we present the main problem in its original formulation and we show the c onnection with $T$-colorings.\nThen\, we present some recent results (see [2]) on a minimum possible counterexample for the Berge-Fulkerson Conjectu re.\n\n[1] E. Máčajová\, M. Škoviera\, Fano colourings of cubic graphs and the Fulkerson Conjecture\, Theor. Comput. Sci. 349 (2005) 112-- 120.\ n\n[2] E. Máčajová\, G. Mazzuoccolo\, Reduction of the Berge-Fulkerson conjecture to cyclically 5-edge-connected snarks\, Proc. Amer. Math. Soc. 148 (2020)\, 4643--4652.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/10/ END:VEVENT BEGIN:VEVENT SUMMARY:John Sheekey (University College Dublin) DTSTART:20210420T140000Z DTEND:20210420T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/11 DESCRIPTION:Title: The tensor rank of semifields of order 81< /a>\nby John Sheekey (University College Dublin) as part of Galois geometr ies and their applications eseminars\n\nLecture held in Google Meet.\n\nAb stract\nTensor products of vector spaces are fundamental objects in mathem atics. The tensor product of two vector spaces can be studied using matric es\, and this case is well-understood\; the rank can be calculated easily\ , and equivalence corresponds precisely with rank. However for higher orde r tensors\, problems such as calculating the rank or determining equivalen ce becomes very difficult.\n\nThe case of the tensor product of three isom orphic vector spaces corresponds to algebras in which multiplication is no t assumed to be associative. In this case\, the tensor rank gives an impor tant measure of the complexity of the multiplication in the corresponding algebra. For the case of a finite semifield (i.e. a not-necessarily associ ative division algebras)\, lower bounds can be obtained using results from linear codes\, while for field extensions upper bounds can be obtained vi a polynomial interpolation and algebraic geometry.\n\nIn this talk we will survey these problems and present new results where we determine the tens or rank of all finite semifields of order 81. In particular we show that s ome semifields of order 81 have lower tensor rank than the field of order 81\, the first known example of such a phenomenon.\n\nThis is joint work w ith Michel Lavrauw.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Gary McGuire (University College Dublin) DTSTART:20210615T140000Z DTEND:20210615T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/12 DESCRIPTION:Title: Linear Fractional Transformations and Irre ducible Polynomials over Finite Fields\nby Gary McGuire (University Co llege Dublin) as part of Galois geometries and their applications eseminar s\n\nLecture held in Google Meet.\n\nAbstract\nWe will discuss polynomials over a finite field where linear fractional transformations permute the r oots. For subgroups $G$ of $\\mathrm{PGL}(2\,q)$ we will demonstrate some connections between the field of $G$-invariant rational functions and fact orizations of certain polynomials into irreducible polynomials over $\\mat hbb{F}_q$. Some unusual patterns in the factorizations are explained by th is connection.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Marco Buratti (University of Perugia) DTSTART:20210511T140000Z DTEND:20210511T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/13 DESCRIPTION:Title: Designs over finite fields by difference m ethods\nby Marco Buratti (University of Perugia) as part of Galois geo metries and their applications eseminars\n\nLecture held in Google Meet.\n \nAbstract\nAt the kind request of the organizers\, I will try to give an outline of how difference methods allow to obtain some q-analogs of 2-desi gns. Of course\, a particular attention will be given to the renowned 2-an alog of a 2-(13\,3\,1) design found by Braun\, Etzion\, Östergård\, Vard y and Wassermann.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Mattheus (Vrije Universiteit Brussel) DTSTART:20210720T140000Z DTEND:20210720T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/14 DESCRIPTION:Title: Eigenvalues of oppositeness graphs and Erd ős-Ko-Rado for flags\nby Sam Mattheus (Vrije Universiteit Brussel) as part of Galois geometries and their applications eseminars\n\nLecture hel d in Google Meet.\n\nAbstract\nOver the last few years\, Erdős-Ko-Rado th eorems have been found in many different geometrical contexts including fo r example sets of subspaces in projective or polar spaces. A recurring the me throughout these theorems is that one can find sharp upper bounds by ap plying the Delsarte-Hoffman coclique bound to a matrix belonging to the re levant association scheme. In the aforementioned cases\, the association s chemes turn out to be commutative\, greatly simplifying the matter. Howeve r\, when we do not consider subspaces of a certain dimension but more gene ral flags\, we lose this property. In this talk\, we will explain how to o vercome this problem\, using a result originally due to Brouwer. This resu lt\, which has seemingly been flying under the radar so far\, allows us to find eigenvalues of oppositeness graphs and derive sharp upper bounds for EKR-sets of certain flags in projective spaces and general flags in polar spaces and exceptional geometries. We will show how Chevalley groups\, bu ildings\, Iwahori-Hecke algebras and representation theory tie into this s tory and discuss their connections to the theory of non-commutative associ ation schemes.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Pott (Otto von Guericke University) DTSTART:20211019T140000Z DTEND:20211019T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/16 DESCRIPTION:Title: Vectorial bent functions and beyond\nb y Alexander Pott (Otto von Guericke University) as part of Galois geometri es and their applications eseminars\n\nLecture held in Google Meet.\n\nAbs tract\nA function $F:\\F_2^n\\to \\F_2^m$ is called vectorial bent\nif $F (x+a)+F(x)=b$ \nfor all $a\\ne 0$ and all $b$ has exactly $2^{n-m}$ soluti ons.\nIt is well known that $n=2k$ must be even and that $m\\le k$.\nIn my talk\, I will address some problems about the classification\nof vectoria l bent functions\, in particular:\n\n- Classification of $(6\,3)$-vectoria l bent functions [1].\n\n- Number of quadratic $(n\,2)$-vectorial bent fun ctions [3].\n\nDue to the bound $m\\leq k$\, one may ask which functions a re \nclose to vectorial bent functions if $m>k$. In [2]\nwe determined th e maximum number of bent functions that may occur as\ncomponent functions of $F:\\F_2^{2k}\\to\\F_2^{2k}$. It turns out that \nthis maximum is $2^k$ and the non-bent functions form a vector space \n(bent complement). This has been later generalized to\nfunctions $F:\\F_2^{2k}\\to\\F_2^{m}$ [4].\ n\nI will briefly report about recent progress on such MNBC functions\n(jo int work with Bapić\, Pasalic and Polujan). \n\nReferences\n\n[1] A. A. Polujan and A. Pott\, On design-theoretic aspects of Boolean and vectorial bent function\, IEEE Trans. Inform. Theory\, 67 (2021)\, pp. 1027–1037. \n\n[2] A. Pott\, E. Pasalic\, A. Muratović-Ribić\, and S. Bajrić\, On the maximum number of bent components of vectorial functions\, IEEE Trans. Inform. Theory\, 64 (2018)\, pp. 403–411.\n\n[3] A. Pott\, K.-U. Schmid t\, and Y. Zhou\, Pairs of quadratic forms over finite fields\, Electron. J. Combin.\, 23 (2016)\, pp. Paper 2.8\, 13.\n\n[4] L. Zheng\, J. Peng\, H . Kan\, Y. Li\, and J. Luo\, On constructions and properties of (n\, m)-fu nctions with maximal number of bent components\, Des. Codes Cryptogr.\, 88 (2020)\, pp. 2171–2186.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Panario (Carleton University) DTSTART:20211214T150000Z DTEND:20211214T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/18 DESCRIPTION:Title: The dynamics of iterating functions over f inite fields\nby Daniel Panario (Carleton University) as part of Galoi s geometries and their applications eseminars\n\nLecture held in Google Me et.\n\nAbstract\nWhen we iterate functions over finite structures\, there is an\nunderlying natural functional graph. For a function $f$ over\na fin ite field $\\mathbb{F}_q$\, this graph has $q$ nodes and\na directed edge from vertex $a$ to vertex $b$ if and only if\n$f(a)=b$. It is well known\, combinatorially\, that functional\ngraphs are sets of connected component s\, components are \ndirected cycles of nodes\, and each of these nodes is the root \nof a directed tree from leaves to its root.\n\nThe study of it erations of functions over a finite field and\ntheir corresponding functio nal graphs is a growing area of\nresearch\, in part due to their applicati ons in cryptography\nand integer factorization methods like Pollard rho al gorithm.\n\nSome functions over finite fields when iterated present strong \nsymmetry properties. These symmetries allow mathematical proofs\nof some dynamical properties such as period and preperiod of a\ngeneric element\, (average) ``rho length'' (number of iterations\nuntil a cycle is formed)\ , number of connected components\, cycle\nlengths\, and permutational prop erties (including the cycle \ndecomposition).\n\nWe survey the main proble ms addressed in this area so far.\nWe exemplify by describing the function al graph of Chebyshev \npolynomials over a finite field. We use the struct ural results\nto obtain estimates for the average rho length\, average num ber\nof connected components and the expected value for the period\nand pr eperiod of iterating Chebyshev polynomials over finite \nfields. We concl ude providing a list of open problems. \n\nBased on joint works with Rodri go Martins (UTFPR\, Brazil)\,\nClaudio Qureshi (UdelaR\, Uruguay) and Luca s Reis (UFMG\, Brazil).\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Martino Borello (Université Paris 8) DTSTART:20220201T150000Z DTEND:20220201T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/19 DESCRIPTION:Title: Small strong blocking sets and their codin g theoretical counterparts\nby Martino Borello (Université Paris 8) a s part of Galois geometries and their applications eseminars\n\nLecture he ld in Google Meet.\n\nAbstract\nStrong blocking sets are sets of points in the projective space such that the intersection with each hyperplane span s the hyperplane. They have been defined first in Davydov\, Giulietti\, Ma rcugini\, Pambianco\, 2011\, in relation to covering codes\, and reintrodu ced later as generator sets in Fancsali\, Sziklai\, 2014 and as cutting bl ocking sets in Bonini\, Borello\, 2021\, in relation with minimal codes. I n Alfarano\, Borello\, Neri\, 2019 and independently in Tang\, Qiu\, Liao\ , Zhou\, 2019\, it has been shown that strong blocking sets are the geomet ric counterparts of such codes. From their definition\, it is clear that a dding a point to a strong blocking set maintains the property of being str ong\, so that strong blocking sets of small cardinality are the most inter esting ones. In the coding theoretical language\, this is equivalent to ha ve a short minimal code. A natural question is then how small a strong blo cking set in a projective space of a given dimension can be.\n\n In the ta lk\, we will illustrate these connections\, together with some bounds on t heir parameters and with some constructions of small strong blocking sets. At the end\, we will describe some perspectives and analogues in the rank metric.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Valentina Pepe (Sapienza Università di Roma) DTSTART:20220322T160000Z DTEND:20220322T170000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/20 DESCRIPTION:Title: The geometry of extremal Cayley graphs \nby Valentina Pepe (Sapienza Università di Roma) as part of Galois geome tries and their applications eseminars\n\nLecture held in Google Meet.\n\n Abstract\nThe geometric aspect of extremal Cayley graphs is highlighted\, providing a different proof of known results and giving a new perspective on how to tackle such problems.\nSome new results about extremal pseudrand om triangle free graphs are also presented.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Massimo Giulietti (Università degli Studi di Perugia) DTSTART:20220426T150000Z DTEND:20220426T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/21 DESCRIPTION:Title: Algebraic curves with many automorphisms\nby Massimo Giulietti (Università degli Studi di Perugia) as part of G alois geometries and their applications eseminars\n\nLecture held in Googl e Meet.\n\nAbstract\nThe Hurwitz upper bound on the size of the $\\mathbb{ K}$-automorphism group Aut($\\mathcal{C}$) of an algebraic curve $\\mathca l{C}$ of genus $g$ greater than $1$ defined over a field $\\mathbb{K}$ of zero characteristic is $84(g-1)$. \nIn positive characteristic $p$\, algeb raic curves can have many more automorphisms than expected from the Hurwit z bound. \nThere even exist algebraic curves of arbitrarily large genus $ g$ with more than $16g^4$ automorphisms. Besides the genus\, an important invariant for curves in positive characteristic is the $p$-rank of the cur ve\, which is the integer $c$ such that the Jacobian of $\\mathcal{C}$ has $p^c$ $p$-torsion points. It turns out that the most anomalous examples o f algebraic curves with a very large automorphism group invariably have ze ro $p$-ranks.\nSeveral results on the interaction between the automorphism group\, the genus and the $p$-rank of a curve can be found in the literat ure. In this talk we survey some reults on the following issues that have been obtained in the last decade:\n\n(i) Upper bounds on the size of Aut($ \\mathcal{C}$) depending on g and the structure of Aut($\\mathcal{C}$).\n\ n(ii) The possibilities for Aut($\\mathcal{C}$) when the $p$-rank is $0$.\ n\n(iii) Upper bounds on the size of $d$-subgroups of Aut($\\mathcal{C}$). \n\nSome applications to maximal curves over finite fields are also discu ssed.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Jan De Beule (Jan.De.Beule@vub.be) DTSTART:20220510T150000Z DTEND:20220510T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/22 DESCRIPTION:Title: On Cameron-Liebler sets of k-spaces in fin ite projective spaces (Part I)\nby Jan De Beule (Jan.De.Beule@vub.be) as part of Galois geometries and their applications eseminars\n\nLecture h eld in Google Meet.\n\nAbstract\nThis is part 1 (of 2) of a double talk to gether with Jonathan Mannaert. Cameron-Liebler line classes in a finite 3- dimensional space PG(3\,q) originate from the study by Cameron and Liebler in 1982 of groups of collineations with equally many orbits on the points and the lines of PG(3\,q). These objects have some interesting equivalent characterizations\, and are examples of Boolean functions of degree one. One of the main properties of this set is that these line classes admit a parameter x\, which can be used to classify or exclude examples. In this t alk\, we focus on these objects from a geometric perspective\, and report on several existence and non-existence results\, including a recent so-cal led modular equality for the parameter of Cameron-Liebler line classes in finite n-dimensional projective spaces found in [2] for n odd. This modula r equality is a natural generalization of the modular equality found in [3 ].\n\n\n\n[1] A. Blokhuis\, M. De Boeck\, and J. D'haeseleer.\nCameron-Lie bler sets of k-spaces in PG(n\,q).\nDes. Codes Cryptogr.\, 87(8):1839--185 6\, 2019.\n\n[2] J. De Beule and J. Mannaert.\nA modular equality for Came ron-Liebler line classes in projective and affine spaces of odd dimension. \nSubmitted.\n\n[3] A. L. Gavrilyuk and K. Metsch.\nA modular equality for Cameron-Liebler line classes.\nJ. Combin. Theory Ser. A\, 127:224--242\, 2014.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Oriol Serra (Universitat Politècnica de Catalunya) DTSTART:20221019T140000Z DTEND:20221019T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/23 DESCRIPTION:Title: Sidon spaces\nby Oriol Serra (Universi tat Politècnica de Catalunya) as part of Galois geometries and their appl ications eseminars\n\nLecture held in Google Meet.\n\nAbstract\nMotivated by a problem related to difference sets\, Hou\, Leu and Xiang introduced in 2002 a linear version of the classical theorem of Kneser in additive co mbinatorics\, where sets are replaced by subspaces and cardinalities by d imensions. A nice feature of the linear version is that\, via Galois exte nsions\, it provides an alternate proof of the original version. This open ned a trend to prove extensions of theorems in additive combinatorics to t heir linear analogues. The talk will focuss on one of these extensions\, t he Vosper theorem\, which gives rise to the notion of Sidon spaces. This n otion turned out to find interesting applications in coding theory.\n\nThi s is joint work with Christine Bachoc and Gilles Zémor\, with a nice simp lification by Chiara Castello.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Zsuzsa Weiner (ELKH-ELTE GAC and Prezi.com) DTSTART:20221207T131500Z DTEND:20221207T141500Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/24 DESCRIPTION:Title: Consequences of a resultant-like theorem i n Galois geometries\nby Zsuzsa Weiner (ELKH-ELTE GAC and Prezi.com) as part of Galois geometries and their applications eseminars\n\nLecture hel d in Google Meet.\n\nAbstract\nWith Tamás Szőnyi we proved the following theorem on two variable polynomials\, see [8]\, [7]\, [5]. The present fo rm of the theorem is due to ‪Tamás Héger‬ [3]. \n\nTheorem\n Let $f \,g \\in \\F[X\,Y]$ be polynomials over the arbitrary field $\\F$.\n Assu me that the coefficient of $X^{\\deg f}$ in $f$ is not $0$ and for $y\\in \\F$ put $k_y = \\deg \\gcd(f(X\,y)\,g(X\,y))$. Then for any $y_0 \\in \\F $\n \\[\n \\sum_{y\\in \\F}(k_y-k_{y_0})^+ \\leq (\\deg f - k_{y_0})(\\d eg g - k_{y_0}).\n \\]\n\nHere $\\alpha^+=\\max\\{0\,\\alpha\\}$. Note th at $g$ can be the zero polynomial as well\, in that case $\\deg f=k_y=k_{y _0}$ and the lemma claims the trivial $0 \\leq 0$. \n\n\nIn my talk\, I wi ll show several examples (old and new) on how this theorem can be used in finite geometry\, mostly in PG$(2\,q)$. I do not intend to cover a full s urvey on these results\, my aim is to show the part of the proofs in detai l where we gain benefit from this theorem. I will talk about an upper boun d on the number of lines that may intersect a point set in $\\mathrm{PG}(2 \,q)$ [4]\; about the possible sizes of the second largest minimal blockin g sets in PG$(2\,q)$\, $q$ square [6]\; about codewords generated by the l ines of PG$(2\,q)$ [5]. I will also present a natural generalisation (see [2]) of a nice lemma which helped Blokhuis\, Brouwer and Wilbrink to prove that unitals which are codewords are necessarily Hermitian.\n\n \n\n[1] A . Blokhuis\, A.E. Brouwer\, H. Wilbrink: Hermitian unitals are code words\ , Discrete Math. 97 (1991)\, 63-68.\n\n\n[2] B. Csajbók\, P. Sziklai\, Zs .Weiner: Renitent lines\, submitted.\n\n \n[3] T. ‪Héger‬: Some graph theoretic aspects of finite geometries\, PhD Thesis\, Eötvös Loránd University\, 2013\, http://heger.web.elte.hu//publ/HTdiss-e.pdf\n \n \n[4] T. Szőnyi\, Zs. Weiner: Proof of a conjecture of Metsch\, J. Co mbin. Theory Ser. A 118:7 pp. 2066-2070 (2011). \n\n\n[5] T. Szőnyi\, Zs . Weiner: Stability of $k$ mod $p$ multisets and small weight codewords of the code generated by the lines of $\\mathrm{PG}(2\,q)$\, J.\\ Combin.\\ Theory Ser.\\ A} {\\bf 157} (2018)\, 321--333.\n\n[6] T. Szőnyi\, Zs. Wei ner: Large blocking sets in $\\mathrm{PG}(2\, q^2)$\, Finite Fields Appl.\ , to appear.\n\n[7] Zs. Weiner: On $(k\,p^e)$-arcs in Desarguesian planes\ , Finite Fields Appl. 10 (2004)\, 390-404. \n\n[8] Zs. Weiner: Geometric a nd algebraic methods in Galois-geometries\, PhD Thesis\, Eötvös Loránd University\, 2002\, https://web.cs.elte.hu/~weiner/main_jav.pdf\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Heide Gluesing-Luerssen (University of Kentucky) DTSTART:20230301T150000Z DTEND:20230301T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/25 DESCRIPTION:Title: Properties of the Direct Sum of q-Matroids \nby Heide Gluesing-Luerssen (University of Kentucky) as part of Galoi s geometries and their applications eseminars\n\nLecture held in Google Me et.\n\nAbstract\nAfter a brief introduction of $q$-matroids and their rele vance for rank-metric codes we will survey some of the main results in the still young theory of $q$-matroids. They comprise an extensive list of cr yptomorphisms. While these are non-trivial results\, they all form quite n atural $q$-analogues of the corresponding cryptomorphisms for (classical) matroids. We will then turn to the direct sum of $q$-matroids\, which was introduced in 2021 by Ceria/Jurrius. It turns out that the definition as w ell as the properties of the direct sum are significantly different from t hose for matroids. After discussing the construction of the direct sum\, w e will report on properties where the theory diverges the most from that o f matroids. Thereafter\, we will turn to a result where the theory\, surpr isingly\, meets that of matroids. Indeed\, the direct sum behaves very nat urally with respect to cyclic flats. This allows us to show that every $q$ -matroid can be decomposed into irreducible ones and to characterize irred ucibility.\n\nThis is joint work with Benjamin Jany.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Benjamin Sudakov (ETH Zurich) DTSTART:20230308T150000Z DTEND:20230308T160000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/26 DESCRIPTION:Title: Evasive sets\, covering by subspaces\, and point-hyperplane incidences\nby Benjamin Sudakov (ETH Zurich) as part of Galois geometries and their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract\nGiven positive integers $k\\leq d$ and a finite field $F$\, a set $S\\subset F^{d}$ is $(k\,c)$-subspace evasive\nif every $k$-dimensional affine subspace contains at most $c$ elements of $S$.\nBy a simple averaging argument\, the maximum size of a $(k\,c)$-subspace eva sive set is at most $c |F|^{d-k}$.\nIn this talk we discuss the constructi on of evasive sets\, matching this bound.\n\nThe existence of optimal evas ive sets has several interesting consequences in combinatorial geometry.\n Using it we determine the minimum number of $k$-dimensional linear hyperpl anes needed to cover the grid $[n]^{d}$.\nThis extends the work by Balko\, Cibulka\, and Valtr\, and settles a problem proposed by Brass\, Moser\, a nd Pach.\nFurthermore\, we improve the best known lower bound on the maxim um number of incidences between points and hyperplanes\nin dimension $d$ a ssuming their incidence graph avoids the complete bipartite graph $K_{t\,t }$ for some large constant $t=t(d)$.\n\nJoint work with Istvan Tomon.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Ilaria Cardinali (Università degli studi di Siena) DTSTART:20230524T140000Z DTEND:20230524T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/27 DESCRIPTION:Title: Grassmannian of codes\nby Ilaria Cardi nali (Università degli studi di Siena) as part of Galois geometries and t heir applications eseminars\n\nLecture held in Zoom.\n\nAbstract\nIn this talk I will consider the point line-geometry $\\mathcal{P}_t(n\,k)$ having as points all the $[n\,k]$-linear codes having minimum dual Hamming weigh t at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $ X\\cap Y$ is a $[n\,k-1]$-linear code having minimum dual Hamming weight a t least $t+1$.\n Let $\\Lambda_t(n\,k)$ be the collinearity graph of $\\m athcal{P}_t(n\,k).$ Then $\\Lambda_t(n\,k)$ is a subgraph of the Grassmann graph and also a subgraph of the graph $\\Delta_t(n\,k)$ of the linear co des having minimum dual Hamming weight at least $t+1$ introduced in [2].\n \n I will investigate the structure of $\\Lambda_t(n\,k)$ focusing on it s relation with well-studied configurations of points of a projective spac e such as the saturated sets. In particular\, I will characterize the set of isolated vertices of $\\Lambda_t(n\,k)$ and for $t=1$ and $t=2$\, nece ssary and sufficient conditions for $\\Lambda_t(n\,k)$ \n to be connected will be provided.\n Finally\, these results will be applied to the geomet ry ${\\mathcal P}_t(n\,k)$\n in order to study its projective embeddabili ty by means of the\n Plücker map.\n\n\n\n[1] I. Cardinali and L. Giuzz i\, Grassmannians of codes\, submitted.\n\n\n[2] M. Kwiatkowski\, M. Pan kov\, On the distance between linear codes\, Finite Fields Appl. 39 (2016 )\, 251-263.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Mattheus (University of California\, San Diego and Vrije Unive rsiteit Brussel) DTSTART:20230927T140000Z DTEND:20230927T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/28 DESCRIPTION:Title: The asymptotics of r(4\,t)\nby Sam Mat theus (University of California\, San Diego and Vrije Universiteit Brussel ) as part of Galois geometries and their applications eseminars\n\nLecture held in Zoom.\n\nAbstract\nFor integers $s\,t \\geq 2$\, the Ramsey numbe rs $r(s\,t)$ denote the \nminimum $N$ such that every $N$-vertex graph con tains either a clique of \norder $s$ or an independent set of order $t$. \ nI will give an overview of recent work\, joint with Jacques Verstraete\, which shows\n\n$r(4\,t)=\\Omega\\Bigl(\\frac{t^3}{\\log^4 \\! t}\\Bigr)$ a s $t \\rightarrow \\infty$.\n\n\n\nThis determines $r(4\,t)$ up to a facto r of order $\\log^2 \\! t$\, and \nsolves a conjecture of Erdős. Moreover \, I will discuss some \nsubsequent work with David Conlon\, Dhruv Mubayi and Jacques Verstraete \nshowing the need for good constructions\, possibl y coming from finite \ngeometry.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Calderbank (Duke University) DTSTART:20231025T140000Z DTEND:20231025T150000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/29 DESCRIPTION:Title: Back to the Future\nby Robert Calderba nk (Duke University) as part of Galois geometries and their applications e seminars\n\nLecture held in Zoom.\n\nAbstract\nThis talk tells the history of coding theory through the lens of Reed Muller codes. In the beginning\ , there were no computers\, and coding theory was the mathematics of spher e packing. This was a golden time for algebraic coding\, with the discover y of Reed Muller and Reed Solomon codes. As everyday computers became more powerful coding theory changed character and focused on iterative algorit hms. Today with quantum computers on the horizon\, Reed Muller codes are b ack in fashion.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/29/ END:VEVENT BEGIN:VEVENT SUMMARY:John Bamberg ((University of Western Australia) DTSTART:20231220T100000Z DTEND:20231220T110000Z DTSTAMP:20260404T110742Z UID:CombinatoricsAndAlgebraicCurves/30 DESCRIPTION:Title: Foundations of hyperbolic geometry\nby John Bamberg ((University of Western Australia) as part of Galois geometr ies and their applications eseminars\n\nLecture held in Zoom.\n\nAbstract\ nThe independent discovery by Lobachevsky and Bolyai of hyperbolic geometr y in the 1830's was followed by slow acceptance of the subject from the 18 60's on\, with the publications of relevant parts of the correspondence of Gauss. A new phase was entered from 1903\, when David Hilbert\, in his wo rk introducing the "calculus of ends"\, introduced an axiomatisation for h yperbolic plane geometry by adding a hyperbolic parallel axiom to the axio ms for plane absolute geometry. In 1938\, Karl Menger (of the famous Vienn a Circle) made the important discovery that in hyperbolic geometry the con cepts of betweenness and equidistance can be defined in terms of point-lin e incidence. Since an axiom system obtained by replacing all occurrences o f betweenness and equidistance with their definitions in terms of incidenc e would look highly unnatural\, Menger and his students looked for a more natural axiom system. In particular\, Helen Skala showed in 1992 that ther e is a set of axioms whose models are the classical hyperbolic planes over Euclidean fields\, and her axioms were the first that contained only firs t order statements. This talk will be on joint work with Tim Penttila (Eme ritus\, University of Adelaide) where we endeavour to simplify Skala's axi oms and retain a characterisation of the classical hyperbolic planes.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cCurves/30/ END:VEVENT END:VCALENDAR