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:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/1 DESCRIPTION:Title: Algebraic curves and (one of) their applications< /a>\nby Marco Timpanella (Università degli Studi della Campania "Luigi Va nvitelli") as part of Galois geometries and their applications: young semi nars\n\n\nAbstract\nThe foundation of the theory of algebraic curves over the complex field goes back to the Nineteenth century\, and most of this t heory holds true if $\\mathbb{C}$ is replaced by any field of characterist ic zero. However\, significant differences arise in positive characterist ic. One of the main features of algebraic curves in positive characteristi c concerns the fact that they may have much larger automorphism groups (co mpared to their genus) than in the zero characteristic case.\nA part of th is seminar will be dedicated to the description of the relationship betwee n automorphism groups and other birational invariants of an algebraic curv e\, and to the presentation of our main contributions. \n\n\nApart from th eir intrinsic theoretical interest\, algebraic curves over finite fields h ave relevant applications to several areas of Mathematics. \nIn particular \, in the last decades\, methods of Algebraic Geometry have been prominent in Coding Theory with the so-called AG codes.\nIn fact\, the essential in gredients for the computation of the parameters of AG codes are Riemann-Ro ch spaces and Weierstrass semigroups. We will give an overview of this top ic and present some recent results.\n \n\n\nJoint work with Massimo Giulie tti\, Gábor Korchmáros\, Stefano Lia\, Gábor Nagy.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Anina Gruica (Eindhoven University of Technology) DTSTART:20210503T140000Z DTEND:20210503T150000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/2 DESCRIPTION:Title: The Sparseness of MRD Codes\nby Anina Gruica (Eindhoven University of Technology) as part of Galois geometries and thei r applications: young seminars\n\n\nAbstract\nAn open question in coding t heory 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 family of subspaces over a finite field in terms of the cardinality of the family a nd its intersection structure. Upper and lower bounds for this number will be derived\, along with their asymptotic versions as the field size tends to infinity.\nBy specializing these results to matrix spaces\, one obtain s upper and lower bounds for the number of MRD codes. In particular\, I wi ll show that MRD codes are sparse for almost all parameter sets as the fie ld size grows. The new results in this talk are joint work with Alberto Ra vagnani.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Giusy Monzillo (Università degli Studi della Basilicata) DTSTART:20210708T140000Z DTEND:20210708T150000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/3 DESCRIPTION:Title: Pseudo-ovals of elliptic quadrics as Delsarte des igns of association schemes\nby Giusy Monzillo (Università degli Stud i della Basilicata) as part of Galois geometries and their applications: y oung seminars\n\n\nAbstract\n(Joint work with John Bamberg and Alessandro Siciliano)\nA pseudo-oval of a finite projective space over a finite fiel d of odd order q is a configuration of equidimensional subspaces that is e ssentially equivalent to a translation generalised quadrangle of order $(q ^n\,q^n)$ and a Laguerre plane of order $q^n$ (for some $n$). In setting o ut a programme to construct new generalised quadrangles\, Shult and Thas a sked whether there are pseudo-ovals consisting only of lines of an ellipti c quadric $Q^-(5\,q)$\, non-equivalent to the classical example\, a so-cal led 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 our work is on characterisations of pseudo-conics from an algebraic combinatorial p oint 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 inte resting five-class association scheme.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Adriaensen (Vrije Universiteit Brussel) DTSTART:20211005T140000Z DTEND:20211005T150000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/4 DESCRIPTION:Title: Erdős-Ko-Rado theorems for ovoidal circle geomet ries and polynomials over finite fields\nby Sam Adriaensen (Vrije Univ ersiteit Brussel) as part of Galois geometries and their applications: you ng seminars\n\n\nAbstract\nGiven an incidence structure (P\, B)\, we say t hat a family F contained in B is intersecting if any two elements of F sha re at least one point. There have been ample investigations into the size and structure of the largest intersecting families in a wide variety of in cidence structures. We say that an incidence structure satisfies the stron g 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 this pro blem in ovoidal circle geometries. They arise by taking a quadratic surfac e Q in PG(3\,q) (which is a slight generalisation of a classical polar spa ce) 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 Möb ius planes of even order greater than two\, and in ovoidal Laguerre planes . As a corollary\, the strong EKR property also holds for polynomials of b ounded degree over a finite field.\n\nThe proof is an illustration of the beautiful marriage of Erdős-Ko-Rado problems and algebraic graph theory.\ n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonardo Landi (Danmarks Tekniske Universitet) DTSTART:20211109T150000Z DTEND:20211109T160000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/5 DESCRIPTION:Title: Galois subcovers of the two Skabelund maximal cur ves\nby Leonardo Landi (Danmarks Tekniske Universitet) as part of Galo is geometries and their applications: young seminars\n\n\nAbstract\nIn 201 6 D. Skabelund constructed two maximal curves over finite fields as cyclic covers of the Suzuki and Ree curves. The two curves have been later inves tigated by M. Giulietti\, M. Montanucci\, L. Quoos and G. Zini\, who deter mined the full automorphism group and computed the genera of many Galois s ubcovers of the two curves. This talk will give an overview of a recent wo rk\, in collaboration with P. Beelen and M. Montanucci\, in which we compl eted the classification of all Galois subcovers of the two Skabelund maxim al curves. The talk will focus on some of the techniques involved in the g enus computation of such Galois subcovers\, that lead to obtain new values in the spectrum of genera of maximal curves.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Jean Paul Zerafa (Comenius University) DTSTART:20220111T150000Z DTEND:20220111T160000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/6 DESCRIPTION:Title: Snarks and perfect matchings\nby Jean Paul Ze rafa (Comenius University) as part of Galois geometries and their applicat ions: young seminars\n\n\nAbstract\nSnarks\, which for us represent bridge less cubic graphs which are not 3-edge-colourable (Class II)\, are crucial when considering conjectures about bridgeless cubic graphs\, and\, if suc h statements are true for snarks\, then they would be true for all bridgel ess cubic graphs. One such conjecture which is known for its simple statem ent\, but still indomitable after half a century\, is the Berge-Fulkerson Conjecture which states that every bridgeless cubic graph $G$ admits six p erfect matchings such that every edge in $G$ is contained in exactly two o f these six perfect matchings. In this talk we discuss two other related a nd well-known conjectures about bridgeless cubic graphs\, both consequence s of the Berge-Fulkerson Conjecture which are still very much open: the Fa n-Raspaud Conjecture (1994) and the $S_{4}$-Conjecture (Mazzuoccolo\, 2013 ).\n\nGiven the obstacles encountered when dealing with such problems\, ma ny have considered trying to bridge the gap between Class I and Class II b ridgeless cubic graphs by looking at invariants that measure how far Class II bridgeless cubic graphs are from being Class I. This is done in an att empt to further refine the class of snarks\, and thus\, enlarging the 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 matchings (not necessarily distinct) needed to be added to a bridgeless cubic graph such that the resulting multigraph is Class I. We show that the Petersen graph is\, in some sense\, the only obstruction for a bridgeless cubic gra ph to have a finite value for the parameter studied. We also relate this p arameter to already well-studied concepts: the excessive index\, and the l ength of a shortest cycle cover of a bridgeless cubic graph. \n\nThe above is joint work with Edita Máčajová\, Giuseppe Mazzuoccolo and Vahan Mkr tchyan.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Giovanni Longobardi (University of Padova) DTSTART:20220301T150000Z DTEND:20220301T160000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/7 DESCRIPTION:Title: Scattered linear sets in a finite projective line \, translation planes and hyper-reguli of $\\mathbb F_{q^t}^2$\nby Gio vanni Longobardi (University of Padova) as part of Galois geometries and t heir applications: young seminars\n\n\nAbstract\nIn [2]\, G. Lunardon and O. Polverino show that the point set of a scattered \n$\\mathbb F_q$-linea r set of rank $t$ in PG$(1\,q^t)$\, also called maximum scattered linear s et\n(MSLS for short)\, is a derivable partial spread of the $\\mathbb F_q$ -vector space $\\mathbb F_{q^t}^2$\n(elsewhere such structures are also ca lled hyper-reguli).\nHence any MSLS gives rise to a non-Desarguesian trans lation plane.\nIn the case dealt with in [2]\, the authors obtain an Andr \\'e plane.\nIn this talk\, a quasifield associated with any MSLS will 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 and only if they are related to $\\mathbb F_q$-subspaces $U$ and $U'$ of \n$\\mathb b F_{q^t}^2$\nbelonging to the same orbit under the action of $\\Gamma\\ma thrm L(2\,q^t)$.\nAs a consequence\, any MSLS $L_U$ gives rise to a set of \npairwise nonisomorphic translation planes whose size is the $\\Gamma\\ma thrm L$-class of $L_U$\,\nas defined in [1].\n\nThis is a joint work with V. Casarino and C. Zanella.\n\n[1] B. Csajbók\, G. Marino\, O. Polverino: \nClasses and equivalence of linear sets in PG(1\,q^n)\,\nJ. Combin. Theor y Ser. A 157 (2018)\, 402-426.\n\n\n[2] G. Lunardon\, O. Polverino:\nBlock ing sets and derivable partial spreads\,\nJ. Algebraic Combin. 14 (2001)\, 49-56.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Mannaert (Vrije Universiteit Brussel) DTSTART:20220524T150000Z DTEND:20220524T160000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/8 DESCRIPTION:Title: On Cameron-Liebler sets of k-spaces in finite pro jective spaces (Part II)\nby Jonathan Mannaert (Vrije Universiteit Bru ssel) as part of Galois geometries and their applications: young seminars\ 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 they appea r to be scarce. These objects can be generalized naturally to Cameron-Lieb ler sets of k-spaces in PG(n\,q)\, which inherit many properties. It is kn own that these sets of k-spaces are also examples of sets arising from Boo lean degree 1 functions. Each Cameron-Liebler set of k-spaces has a parame ter x.\nConditions on this parameter yield non-existence results. In 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-trivial exam ples of Cameron-Liebler sets of k-spaces. The main techniques we apply ari se from generalizaing techniques used in [2]. In his thesis\, Drudge wante d to classify Cameron-Liebler line classes in PG(n\,q) using their interse ction with subspaces. In [1]\, these concepts where generalized to Cameron -Liebler sets of k-spaces and they also improve the known results obtained 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 i n subspaces and non-existence conditions.\nDes. Codes Cryptogr.\, 90: 633 –651\, 2022.\n\n[2] K. Drudge.\nExtremal sets in projective and polar sp aces.\nPhD thesis\, The University of West Ontario\, London\, Canada\, 199 8.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Lara Vicino (Technical University of Denmark) DTSTART:20221109T150000Z DTEND:20221109T160000Z DTSTAMP:20260404T084337Z 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 L ara Vicino (Technical University of Denmark) as part of Galois geometries 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 projective \, geometrically irreducible\, non-singular algebraic curve defined over $ \\mathbb{F}_{q^2}$ such that the number of its $\\mathbb{F}_{q^2}$-rationa l points attains the Hasse-Weil upper bound.\n$\\mathbb{F}_{q^2}$-maximal curves\, especially those with large genus\, are of particular interest in coding theory since they give rise to excellent AG codes.\n\nIt is well k nown 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 Hermitian curve. The first and the second largest genera of $\\mathbb{F}_{q^2}$-max imal 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 l argest 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 exactly on e curve up to $\\mathbb{F}_{q^2}$-isomorphism. \n\n\n\nIn this talk\, I wi ll present our results on the Weierstrass semigroups at the $\\mathbb{F}_{ q^2}$-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 curve known to have genus equal to $g_3$. One of the surprising results is that 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\nJoint work with Peter Beelen and Maria Montanucci.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Lins Denaux (University of Ghent) DTSTART:20230125T150000Z DTEND:20230125T160000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/10 DESCRIPTION:Title: On higgledy-piggledy sets and the André/Bruck-B ose representation\nby Lins Denaux (University of Ghent) as part of Ga lois geometries and their applications: young seminars\n\n\nAbstract\nIn t his talk\, we focus on higgledy-piggledy sets of $k$-subspaces in $\\textn ormal{PG}(N\,q)$\, i.e. sets of projective subspaces that are "well-spread -out".\nMore precisely\, the set of intersection points of these $k$-subsp aces 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 the se $k$-subspaces forms a \\emph{strong blocking set} w.r.t. $(N-k)$-subspa ces.\nNaturally\, one would like to find a higgledy-piggledy set consistin g of a small number of $k$-subspaces.\n\nAlthough these combinatorial sets of subspaces are sporadically mentioned in older works\, only since $2014 $ researchers have started to investigate these sets as a main point of in terest.\nIn this talk\, we aim to discuss the state of the art concerning this special type of subspace sets.\nMoreover\, we want to present some re cent results\, some of which are joint work with Jozefien D'haeseleer and Geertrui Van de Voorde and concerns a higgledy-piggledy plane set in $\\ma thrm{PG}(5\,q)$. The proof of its existence relies on the field reduction\ , linear sets and the Andr\\'e/Bruck-Bose representation of the projective plane $\\mathrm{PG}(2\,q^3)$.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Francisco Galluccio (Universidad Nacional del Litoral) DTSTART:20230426T140000Z DTEND:20230426T150000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/11 DESCRIPTION:Title: LRC codes and a construction from towers of func tion fields\nby Francisco Galluccio (Universidad Nacional del Litoral) as part of Galois geometries and their applications: young seminars\n\n\n Abstract\nIn this work\, we construct sequences of locally recoverable AG codes arising from function fields. In particular we study this constructi on for a tower of function fields and give bounds for the parameters of th e obtained codes. We will show an example from a tower over $\\mathbb{F}_ {q^2}$ for any odd $q$\, defined by Garcia and Stichtenoth\, and we will s how that the bound is sharp for the first code in the sequence.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Stefano Lia (University College Dublin) DTSTART:20231122T120000Z DTEND:20231122T130000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/12 DESCRIPTION:Title: Tensor representation of semifields and commutin g polarities\nby Stefano Lia (University College Dublin) as part of Ga lois geometries and their applications: young seminars\n\n\nAbstract\nWe u se the cyclic model for threefold tensors to investigate a geometric inter pretation of finite semifields. In general dimension\, this leads to a bou nd for a semifield invariant called BEL-rank. In the two dimensional case\ , this approach incorporates a new proof\, purely geometrical\, of a class ical result by Dickson. Also\, it naturally defines a pair of commuting po larities 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://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Ermes Franch (University of Bergen) DTSTART:20240202T150000Z DTEND:20240202T160000Z DTSTAMP:20260404T084337Z UID:CombinatoricsAndAlgebraicGeometr/13 DESCRIPTION:Title: Bounded Degree-Low Rank Parity Check Codes\n by Ermes Franch (University of Bergen) as part of Galois geometries and th eir applications: young seminars\n\n\nAbstract\nLow Rank Parity Check Code s (LRPC codes) are the rank-metric analogue of low density parity check co des.\nWe introduced a new constrain on the support of the parity check mat rix. 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 dimension $2$ ) correspond to BD-LRPC of bounded degree $2$.\nThanks to the special stru cture of the subspace $V_{a\,d}$\, we proved that BD-LRPC codes with bound ed 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 condition $n − k \\geq dr$ required for the standard LRPC codes.\nThe probability 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 sh own 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 probability\, u niquely decodable with the decoding radius $\\rho = r/n$ attaining the Sin gleton bound $1 − R$.\n LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicGeomet r/13/ END:VEVENT END:VCALENDAR