BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Marco Timpanella (Università degli Studi della Campania "Luigi Va nvitelli") DTSTART:20210302T150000Z DTEND:20210302T160000Z DTSTAMP:20260404T110832Z UID:CombinatoricsAndAlgebraicGeometr/1 DESCRIPTION:Title: Algebraic curves and (one of) their applic ations\nby Marco Timpanella (Università degli Studi della Campania "L uigi Vanvitelli") as part of Galois geometries and their applications: you ng seminars\n\n\nAbstract\nThe foundation of the theory of algebraic curve s over the complex field goes back to the Nineteenth century\, and most of this theory holds true if $\\mathbb{C}$ is replaced by any field of chara cteristic zero. However\, significant differences arise in positive chara cteristic. One of the main features of algebraic curves in positive charac teristic concerns the fact that they may have much larger automorphism gro ups (compared to their genus) than in the zero characteristic case.\nA par t of this seminar will be dedicated to the description of the relationship between automorphism groups and other birational invariants of an algebra ic curve\, and to the presentation of our main contributions. \n\n\nApart from their intrinsic theoretical interest\, algebraic curves over finite f ields have relevant applications to several areas of Mathematics. \nIn par ticular\, in the last decades\, methods of Algebraic Geometry have been pr ominent in Coding Theory with the so-called AG codes.\nIn fact\, the essen tial ingredients for the computation of the parameters of AG codes are Rie mann-Roch spaces and Weierstrass semigroups. We will give an overview of t his topic and present some recent results.\n \n\n\nJoint work with Massimo Giulietti\, Gábor Korchmáros\, Stefano Lia\, Gábor Nagy.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Anina Gruica (Eindhoven University of Technology) DTSTART:20210503T140000Z DTEND:20210503T150000Z DTSTAMP:20260404T110832Z UID:CombinatoricsAndAlgebraicGeometr/2 DESCRIPTION:Title: The Sparseness of MRD Codes\nby Anina Gruica (Eindhoven University of Technology) as part of Galois geometries a nd their applications: young seminars\n\n\nAbstract\nAn open question in c oding theory asks whether or not MRD codes with the rank metric are dense as the field size tends to infinity. For answering this question\, I will discuss the problem of estimating the number of common complements of a fa mily of subspaces over a finite field in terms of the cardinality of the f amily and its intersection structure. Upper and lower bounds for this numb er will be derived\, along with their asymptotic versions as the field siz e tends to infinity.\nBy specializing these results to matrix spaces\, one obtains upper and lower bounds for the number of MRD codes. In particular \, I will show that MRD codes are sparse for almost all parameter sets as the field size grows. The new results in this talk are joint work with Alb erto Ravagnani.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Giusy Monzillo (Università degli Studi della Basilicata) DTSTART:20210708T140000Z DTEND:20210708T150000Z DTSTAMP:20260404T110832Z UID:CombinatoricsAndAlgebraicGeometr/3 DESCRIPTION:Title: Pseudo-ovals of elliptic quadrics as Delsa rte designs of association schemes\nby Giusy Monzillo (Università deg li Studi della Basilicata) as part of Galois geometries and their applicat ions: young seminars\n\n\nAbstract\n(Joint work with John Bamberg and Ales sandro Siciliano)\nA pseudo-oval of a finite projective space over a fini te field of odd order q is a configuration of equidimensional subspaces th at is essentially equivalent to a translation generalised quadrangle of or der $(q^n\,q^n)$ and a Laguerre plane of order $q^n$ (for some $n$). In se tting out a programme to construct new generalised quadrangles\, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric $Q^-(5\,q)$\, non-equivalent to the classical example\, a so-called pseudo-conic. To date\, every known pseudo-oval of lines of $Q^ -(5\,q)$ is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the perspective property\, and ou r work is on characterisations of pseudo-conics from an algebraic combinat orial point of view. In particular\, we show that pseudo-ovals in $Q^-(5\, q)$ and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Adriaensen (Vrije Universiteit Brussel) DTSTART:20211005T140000Z DTEND:20211005T150000Z DTSTAMP:20260404T110832Z UID:CombinatoricsAndAlgebraicGeometr/4 DESCRIPTION:Title: Erdős-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields\nby Sam Adriaensen (Vri je Universiteit Brussel) as part of Galois geometries and their applicatio ns: young seminars\n\n\nAbstract\nGiven an incidence structure (P\, B)\, w e say that a family F contained in B is intersecting if any two elements o f F share at least one point. There have been ample investigations into th e size and structure of the largest intersecting families in a wide variet y of incidence structures. We say that an incidence structure satisfies th e strong EKR property if all intersecting families of maximum size consist of all the blocks through a fixed point.\n\nIn this talk I will discuss t his problem in ovoidal circle geometries. They arise by taking a quadratic surface Q in PG(3\,q) (which is a slight generalisation of a classical po lar space) and taking the plane sections with every plane that intersects Q in an oval. I will discuss the proof that the strong EKR property holds in Möbius planes of even order greater than two\, and in ovoidal Laguerre planes. As a corollary\, the strong EKR property also holds for polynomia ls of bounded degree over a finite field.\n\nThe proof is an illustration of the beautiful marriage of Erdős-Ko-Rado problems and algebraic graph t heory.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonardo Landi (Danmarks Tekniske Universitet) DTSTART:20211109T150000Z DTEND:20211109T160000Z DTSTAMP:20260404T110832Z UID:CombinatoricsAndAlgebraicGeometr/5 DESCRIPTION:Title: Galois subcovers of the two Skabelund maxi mal curves\nby Leonardo Landi (Danmarks Tekniske Universitet) as part of Galois geometries and their applications: young seminars\n\n\nAbstract\ nIn 2016 D. Skabelund constructed two maximal curves over finite fields as cyclic covers of the Suzuki and Ree curves. The two curves have been late r investigated by M. Giulietti\, M. Montanucci\, L. Quoos and G. Zini\, wh o determined the full automorphism group and computed the genera of many G alois subcovers of the two curves. This talk will give an overview of a re cent work\, in collaboration with P. Beelen and M. Montanucci\, in which w e completed the classification of all Galois subcovers of the two Skabelun d maximal curves. The talk will focus on some of the techniques involved i n the genus computation of such Galois subcovers\, that lead to obtain new values in the spectrum of genera of maximal curves.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Jean Paul Zerafa (Comenius University) DTSTART:20220111T150000Z DTEND:20220111T160000Z DTSTAMP:20260404T110832Z UID:CombinatoricsAndAlgebraicGeometr/6 DESCRIPTION:Title: Snarks and perfect matchings\nby Jean Paul Zerafa (Comenius University) as part of Galois geometries and their a pplications: young seminars\n\n\nAbstract\nSnarks\, which for us represent bridgeless cubic graphs which are not 3-edge-colourable (Class II)\, are crucial when considering conjectures about bridgeless cubic graphs\, and\, if such statements are true for snarks\, then they would be true for all bridgeless cubic graphs. One such conjecture which is known for its simple statement\, but still indomitable after half a century\, is the Berge-Ful kerson Conjecture which states that every bridgeless cubic graph $G$ admit s six perfect matchings such that every edge in $G$ is contained in exactl y two of these six perfect matchings. In this talk we discuss two other re lated and well-known conjectures about bridgeless cubic graphs\, both cons equences of the Berge-Fulkerson Conjecture which are still very much open: the Fan-Raspaud Conjecture (1994) and the $S_{4}$-Conjecture (Mazzuoccolo \, 2013).\n\nGiven the obstacles encountered when dealing with such proble ms\, many have considered trying to bridge the gap between Class I and Cla ss II bridgeless cubic graphs by looking at invariants that measure how fa r Class II bridgeless cubic graphs are from being Class I. This is done in an attempt to further refine the class of snarks\, and thus\, enlarging t he set of cubic graphs for which such conjectures can be verified. In this spirit we consider a parameter which gives the least number of perfect ma tchings (not necessarily distinct) needed to be added to a bridgeless cubi c graph such that the resulting multigraph is Class I. We show that the Pe tersen graph is\, in some sense\, the only obstruction for a bridgeless cu bic graph to have a finite value for the parameter studied. We also relate this parameter to already well-studied concepts: the excessive index\, an d the length of a shortest cycle cover of a bridgeless cubic graph. \n\nTh e above is joint work with Edita Máčajová\, Giuseppe Mazzuoccolo and Va han Mkrtchyan.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Giovanni Longobardi (University of Padova) DTSTART:20220301T150000Z DTEND:20220301T160000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/7 DESCRIPTION:Title: Scattered linear sets in a finite projecti ve line\, translation planes and hyper-reguli of $\\mathbb F_{q^t}^2$\ nby Giovanni Longobardi (University of Padova) as part of Galois geometrie s and their applications: young seminars\n\n\nAbstract\nIn [2]\, G. Lunard on and O. Polverino show that the point set of a scattered \n$\\mathbb F_q $-linear set of rank $t$ in PG$(1\,q^t)$\, also called maximum scattered l inear set\n(MSLS for short)\, is a derivable partial spread of the $\\math bb F_q$-vector space $\\mathbb F_{q^t}^2$\n(elsewhere such structures are also called hyper-reguli).\nHence any MSLS gives rise to a non-Desarguesia n translation plane.\nIn the case dealt with in [2]\, the authors obtain an Andr\\'e plane.\nIn this talk\, a quasifield associated with any MSLS w ill be exhibited.\nOur main contribution is to prove that two translation planes associated with two MSLSs\n$L_U$ and $L_{U'}$ are\nisomorphic if an d only if they are related to $\\mathbb F_q$-subspaces $U$ and $U'$ of \n$ \\mathbb F_{q^t}^2$\nbelonging to the same orbit under the action of $\\Ga mma\\mathrm L(2\,q^t)$.\nAs a consequence\, any MSLS $L_U$ gives rise to a set of\npairwise nonisomorphic translation planes whose size is the $\\Ga mma\\mathrm L$-class of $L_U$\,\nas defined in [1].\n\nThis is a joint wor k with V. Casarino and C. Zanella.\n\n[1] B. Csajbók\, G. Marino\, O. Pol verino:\nClasses and equivalence of linear sets in PG(1\,q^n)\,\nJ. Combin . Theory Ser. A 157 (2018)\, 402-426.\n\n\n[2] G. Lunardon\, O. Polverino: \nBlocking sets and derivable partial spreads\,\nJ. Algebraic Combin. 14 ( 2001)\, 49-56.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Mannaert (Vrije Universiteit Brussel) DTSTART:20220524T150000Z DTEND:20220524T160000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/8 DESCRIPTION:Title: On Cameron-Liebler sets of k-spaces in fin ite projective spaces (Part II)\nby Jonathan Mannaert (Vrije Universit eit Brussel) as part of Galois geometries and their applications: young se minars\n\n\nAbstract\nThis is part 2 of a double talk together with Jan De Beule. Cameron-Liebler line classes in PG(n\,q) are well studied objects due to several equivalent definitions and interesting properties\, yet the y appear to be scarce. These objects can be generalized naturally to Camer on-Liebler sets of k-spaces in PG(n\,q)\, which inherit many properties. I t is known that these sets of k-spaces are also examples of sets arising f rom Boolean degree 1 functions. Each Cameron-Liebler set of k-spaces has a parameter x.\nConditions on this parameter yield non-existence results. I n this talk we focus on general non-existence results for these sets of k- spaces\, we do this by proving a lower bound on the parameter of non-trivi al examples of Cameron-Liebler sets of k-spaces. The main techniques we ap ply arise from generalizaing techniques used in [2]. In his thesis\, Drudg e wanted to classify Cameron-Liebler line classes in PG(n\,q) using their intersection with subspaces. In [1]\, these concepts where generalized to Cameron-Liebler sets of k-spaces and they also improve the known results o btained for Cameron-Liebler line classes in sufficiently large projective spaces.\n\n[1] J. De Beule\, J. Mannaert and L. Storme.\nCameron-Liebler k -sets in subspaces and non-existence conditions.\nDes. Codes Cryptogr.\, 9 0: 633–651\, 2022.\n\n[2] K. Drudge.\nExtremal sets in projective and po lar spaces.\nPhD thesis\, The University of West Ontario\, London\, Canada \, 1998.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Lara Vicino (Technical University of Denmark) DTSTART:20221109T150000Z DTEND:20221109T160000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/9 DESCRIPTION:Title: Weierstrass semigroups at the $\\mathbb{F} _{q^2}$-rational points of a maximal curve with the third largest genus\nby Lara Vicino (Technical University of Denmark) as part of Galois geom etries and their applications: young seminars\n\n\nAbstract\nAn $\\mathbb{ F}_{q^2}$-maximal curve $\\mathcal{X}$ of genus $g$ is defined to be a pro jective\, geometrically irreducible\, non-singular algebraic curve defined over $\\mathbb{F}_{q^2}$ such that the number of its $\\mathbb{F}_{q^2}$- rational points attains the Hasse-Weil upper bound.\n$\\mathbb{F}_{q^2}$-m aximal curves\, especially those with large genus\, are of particular inte rest in coding theory since they give rise to excellent AG codes.\n\nIt is well known that\, for an $\\mathbb{F}_{q^2}$-maximal curve $\\mathcal{X}$ \, $g(\\mathcal{X}) \\leq q(q - 1)/2$ and that it reaches this upper bound if and only if $\\mathcal{X}$ is $\\mathbb{F}_{q^2}$-isomorphic to the He rmitian curve. The first and the second largest genera of $\\mathbb{F}_{q^ 2}$-maximal curves are known\, and they are realized by exactly one curve up to $\\mathbb{F}_{q^2}$-isomorphism\, but the same is not clear for the third largest genus. Its value is known to be equal to $g_3=\\lfloor (q^2- q+4)/6 \\rfloor$\, but it is still unclear whether this is realized by exa ctly one curve up to $\\mathbb{F}_{q^2}$-isomorphism. \n\n\n\nIn this talk \, I will present our results on the Weierstrass semigroups at the $\\math bb{F}_{q^2}$-rational points of the curve $\\mathcal{X}_3: x^{(q+1)/3} + x ^{2(q+1)/3} + y^{q+1} = 0$\, with $q \\equiv 2 \\pmod 3$\, which is a curv e known to have genus equal to $g_3$. One of the surprising results is tha t there are roughly $(q+1)/3$ possible different semigroups\, although not all of them may occur for a given $q$. Moreover\, the curve $\\mathcal{X} _3$ has many non-$\\mathbb{F}_{q^2}$-rational Weierstrass points.\n\n\nJoi nt work with Peter Beelen and Maria Montanucci.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Lins Denaux (University of Ghent) DTSTART:20230125T150000Z DTEND:20230125T160000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/10 DESCRIPTION:Title: On higgledy-piggledy sets and the André/ Bruck-Bose representation\nby Lins Denaux (University of Ghent) as par t of Galois geometries and their applications: young seminars\n\n\nAbstrac t\nIn this talk\, we focus on higgledy-piggledy sets of $k$-subspaces in $ \\textnormal{PG}(N\,q)$\, i.e. sets of projective subspaces that are "well -spread-out".\nMore precisely\, the set of intersection points of these $k $-subspaces with any $(N-k)$-subspace $\\kappa$ of $\\textnormal{PG}(N\,q) $ spans $\\kappa$ itself.\nIn other words\, the set of points in the union of these $k$-subspaces forms a \\emph{strong blocking set} w.r.t. $(N-k)$ -subspaces.\nNaturally\, one would like to find a higgledy-piggledy set co nsisting of a small number of $k$-subspaces.\n\nAlthough these combinatori al sets of subspaces are sporadically mentioned in older works\, only sinc e $2014$ researchers have started to investigate these sets as a main poin t of interest.\nIn this talk\, we aim to discuss the state of the art conc erning this special type of subspace sets.\nMoreover\, we want to present some recent results\, some of which are joint work with Jozefien D'haesele er and Geertrui Van de Voorde and concerns a higgledy-piggledy plane set i n $\\mathrm{PG}(5\,q)$. The proof of its existence relies on the field red uction\, linear sets and the Andr\\'e/Bruck-Bose representation of the pro jective plane $\\mathrm{PG}(2\,q^3)$.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Francisco Galluccio (Universidad Nacional del Litoral) DTSTART:20230426T140000Z DTEND:20230426T150000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/11 DESCRIPTION:Title: LRC codes and a construction from towers of function fields\nby Francisco Galluccio (Universidad Nacional del L itoral) as part of Galois geometries and their applications: young seminar s\n\n\nAbstract\nIn this work\, we construct sequences of locally recovera ble AG codes arising from function fields. In particular we study this con struction for a tower of function fields and give bounds for the parameter s of the obtained codes. We will show an example from a tower over $\\mat hbb{F}_{q^2}$ for any odd $q$\, defined by Garcia and Stichtenoth\, and we will show that the bound is sharp for the first code in the sequence.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Stefano Lia (University College Dublin) DTSTART:20231122T120000Z DTEND:20231122T130000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/12 DESCRIPTION:Title: Tensor representation of semifields and c ommuting polarities\nby Stefano Lia (University College Dublin) as par t of Galois geometries and their applications: young seminars\n\n\nAbstrac t\nWe use the cyclic model for threefold tensors to investigate a geometri c interpretation of finite semifields. In general dimension\, this leads t o a bound for a semifield invariant called BEL-rank. In the two dimensiona l case\, this approach incorporates a new proof\, purely geometrical\, of a classical result by Dickson. Also\, it naturally defines a pair of commu ting polarities in PG$(3\,q^2)$. Using these polarities\, we construct new quasi-polar spaces.\n\nThis is a joint work with John Sheekey.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Ermes Franch (University of Bergen) DTSTART:20240202T150000Z DTEND:20240202T160000Z DTSTAMP:20260404T110833Z UID:CombinatoricsAndAlgebraicGeometr/13 DESCRIPTION:Title: Bounded Degree-Low Rank Parity Check Code s\nby Ermes Franch (University of Bergen) as part of Galois geometries and their applications: young seminars\n\n\nAbstract\nLow Rank Parity Che ck Codes (LRPC codes) are the rank-metric analogue of low density parity c heck codes.\nWe introduced a new constrain on the support of the parity ch eck matrix. In particular we require that the parity check matrix has its support in the $\\mathbb{F}_q$-linear space $V_{a\,d} = \\langle 1\,a\,a^2 \,...\,a^{d-1}\\rangle$.\nIt is easy to show that LRPC codes of density $2 $ (i.e. LRPC such that the support of their parity check matrix has dimens ion $2$) correspond to BD-LRPC of bounded degree $2$.\nThanks to the speci al structure of the subspace $V_{a\,d}$\, we proved that BD-LRPC codes wit h bounded degree $d$ can uniquely correct errors of rank weight $r$ when $ n − k \\geq r + u$ for certain $u \\geq 1$\, in contrast to the conditio n $n − k \\geq dr$ required for the standard LRPC codes.\nThe probabilit y of failure of the algorithm we propose is exponential in $q^{-u+1}$.\nAs the code length $n$ approaches infinity\, when $n/m \\rightarrow 0$\, it is shown that $u$ can be chosen as certain constant\, which indicates that the BD-LRPC codes with a code rate of $R$ can be\, with a high probabilit y\, uniquely decodable with the decoding radius $\\rho = r/n$ attaining th e Singleton bound $1 − R$.\n LOCATION:https://stable.researchseminars.org/talk/CombinatoricsAndAlgebrai cGeometr/13/ END:VEVENT END:VCALENDAR