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BEGIN:VEVENT
SUMMARY:Daniele Bartoli (Università degli Studi di Perugia)
DTSTART:20200914T140000Z
DTEND:20200914T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/1
DESCRIPTION:Title: Curves over finite fields and polynomial problems<
/a>\nby Daniele Bartoli (Università degli Studi di Perugia) as part of Ga
lois geometries and their applications eseminars\n\nLecture held in Google
Meet.\n\nAbstract\nAlgebraic curves over finite fields are not only inter
esting objects from a theoretical point of view\, but they also have deep
connections with different areas of mathematics and combinatorics.\nIn fac
t\, they are important tools when dealing with\, for instance\, permutatio
n polynomials\, APN functions\, planar functions\, exceptional polynomials
\, scattered polynomials.\nIn this talk I will present some applications o
f algebraic curves to the above mentioned objects.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alberto Ravagnani (Eindhoven University of Technology)
DTSTART:20200930T140000Z
DTEND:20200930T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/2
DESCRIPTION:Title: Network Coding\, Rank-Metric Codes\, and Rook Theo
ry\nby Alberto Ravagnani (Eindhoven University of Technology) as part
of Galois geometries and their applications eseminars\n\nLecture held in G
oogle Meet.\n\nAbstract\nIn this talk\, I will first propose an introducti
on to network coding and its methods. In particular\, I will explain how c
odes with the rank metric naturally arise as a solution to the problem of
error amplification in communication networks (no prerequisite in informat
ion theory is needed for this part). \n\nThe second part of the talk conce
ntrates instead on the mathematical structure of codes with the rank metri
c and its connection with topics in contemporary combinatorics. More preci
sely\, I will present a link between rank-metric codes and q-rook polynomi
als\, showing how this connection plays a role in the theory of MacWilliam
s identities for the rank metric.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Lavrauw (Sabanci University)
DTSTART:20201023T120000Z
DTEND:20201023T130000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/3
DESCRIPTION:Title: On linear systems of conics over finite fields
\nby Michel Lavrauw (Sabanci University) as part of Galois geometries and
their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract\n
A form on an $n$-dimensional projective space ${\\mathbb{P}}^n$ is a homog
eneous polynomial in $n+1$ variables. The forms of degree $d$ on ${\\mathb
b{P}}^n$ comprise a vector space $W$ of dimension ${n+d}\\choose{d}$. Subs
paces of the projective space ${\\mathbb{P}} W$ are called linear systems
of hypersurfaces of degree $d$.\nThe problem of classifying linear systems
consists of determining the orbits of such subspaces under the induced ac
tion of the projectivity group of ${\\mathbb{P}}^n$ on ${\\mathbb{P}}W$. I
n this talk we will focus on linear systems of quadratic forms on ${\\math
bb{P}}^2$ over finite fields. We will give an overview of what is known an
d explain some of the recent results. This is based on joint work with T.
Popiel and J. Sheekey.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Montanucci (Technical University of Denmark)
DTSTART:20201125T150000Z
DTEND:20201125T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/4
DESCRIPTION:Title: Maximal curves over finite fields\nby Maria Mo
ntanucci (Technical University of Denmark) as part of Galois geometries an
d their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract
\nAlgebraic curves over a finite field $\\mathbb{F}_q$ and their function
fields have been a source of great fascination for number theorists and ge
ometers alike\, ever since the seminal work of Hasse and Weil in the 1930s
and 1940s. \nMany important and fruitful ideas have arisen out of this ar
ea\, where number theory and algebraic geometry meet. For a long time\, th
e study of algebraic curves and their function fields was the province of
pure mathematicians. But then\, in a series of three papers in the period
1977-1982\, Goppa found important applications of algebraic curves over fi
nite fields to coding theory. \n\nThe key point of Goppa's construction is
that the code parameters are essentially expressed in terms of arithmetic
and geometric features of the curve\, such as the number $N_q$ of $\\math
bb{F}_q$-rational points and the genus $g$.\n\nGoppa codes with good param
eters are constructed from curves with large $N_q$ with respect to their g
enus $g$. \nGiven a smooth projective\, algebraic curve of genus $g$ over
$\\mathbb{F}_q$\, an upper bound for $N_q$ is a corollary to the celebrate
d Hasse-Weil Theorem\,\n$$N_q \\leq q+ 1 + 2g\\sqrt{q}.$$\nCurves attainin
g this bound are called $\\mathbb{F}_q$-maximal. The Hermitian curve $\\ma
thcal{H}$\, that is\, the plane projective curve with equation \n$$X^{\\sq
rt{q}+1}+Y^{\\sqrt{q}+1}+Z^{\\sqrt{q}+1}= 0\,$$\nis a key example of an $\
\mathbb{F}_q$-maximal curve\, as it is the unique curve\, up to isomorphis
m\, 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 attributed to
Serre that any curve which is $\\mathbb{F}_q$-covered by an $\\mathbb{F}_q
$-maximal curve is still $\\mathbb{F}_q$-maximal. In particular\, quotient
curves of $\\mathbb{F}_q$-maximal curves are $\\mathbb{F}_q$-maximal. Man
y examples of $\\mathbb{F}_q$-maximal curves have been constructed as quot
ient curves $\\mathcal{X}/G$ of the Hermitian/Ree/Suzuki curve $\\mathcal{
X}$ under the action of subgroups $G$ of the full automorphism group of $\
\mathcal{X}$.\nIt is a challenging problem to construct maximal curves tha
t cannot be obtained in this way for some $G$. \n\nIn this presentation\,
we will describe our main contributions to the theory of maximal curves ov
er finite fields.\nIn particular\, the following topics will be discussed:
\n\n- how can we decide whether a given $\\mathbb{F}_q$-maximal curve 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 of the p
ossible genera of $\\mathbb{F}_q$-maximal curves\, especially quotients of
$\\mathcal{H}$\;\n\n- Weierstrass semigroups on maximal curves.\n\nJoint
work with: Daniele Bartoli\, Peter Beelen\, Massimo Giulietti\, Leonardo L
andi\, Vincenzo Pallozzi Lavorante\, Luciane Quoos\, Fernando Torres\, Gio
vanni Zini.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bence Csajbók (MTA-ELTE Geometric and Algebraic Combinatorics Res
earch Group)
DTSTART:20201218T140000Z
DTEND:20201218T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/5
DESCRIPTION:Title: Combinatorially defined point sets of finite Desar
guesian planes\nby Bence Csajbók (MTA-ELTE Geometric and Algebraic Co
mbinatorics Research Group) as part of Galois geometries and their applica
tions 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 $k$-secant of $S
$\, if it meets $S$ in exactly $k$ points. Many of the famous objects of $
\\mathrm{PG}(2\,q)$ have the property that each of their points is inciden
t with the same number of $k$-secants\, for every integer $k$. For example
arcs\, unitals\, subplanes\, maximal arcs and Korchmáros-Mazzocca arcs a
re such objects. In my talk I will present some characterization results o
f point sets with this property.\n\nI will also introduce the following pr
oblem 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\, if more
than half of the lines with slope $d$ meet $M$ in the same number of point
s modulo $p$. We will call this number the typical intersection number at
$(d)$. The rest of the affine lines with slope $d$ will be called renitent
. Note that we allow different uniform directions to have different typica
l intersection numbers. I will show structural properties of the renitent
lines\, in particular I will show that they are contained in some low degr
ee algebraic curves of the dual plane.\n\nThe talk is based on joint works
with Simeon Ball\, Péter Sziklai and Zsuzsa Weiner.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cicero Carvalho (Universidade Federal de Uberlandia)
DTSTART:20210209T130000Z
DTEND:20210209T140000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/7
DESCRIPTION:Title: On certain pairs of primitive elements on finite f
ields\nby Cicero Carvalho (Universidade Federal de Uberlandia) as part
of Galois geometries and their applications eseminars\n\nLecture held in
Google Meet.\n\nAbstract\nIn this talk we would like to present some resul
ts on the existence of pairs of elements in a finite field\, where the fi
rst element is either primitive or primitive and normal over a subfield\,
and the second element is primitive and a rational function of the first o
ne. \n\nThis is based on joint works with J.P Guardieiro\, V. Neumann and
G. Tizziotti.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro Neri (Technical University of Munich)
DTSTART:20200706T130000Z
DTEND:20200706T140000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/8
DESCRIPTION:Title: Defining Reed--Muller codes in the rank metric: th
e Alon--Füredi theorem for endomorphisms\nby Alessandro Neri (Technic
al University of Munich) as part of Galois geometries and their applicatio
ns eseminars\n\nLecture held in Google Meet.\n\nAbstract\nCodes in the ran
k metric have gained a huge interest in the last years\, due to their appl
ications to network coding and cryptography. The most celebrated family of
rank-metric codes is given by Gabidulin codes. It is well-known that they
can be seen as analogues of Reed-Solomon codes in classical coding theory
\, which are codes constructed from spaces of univariate polynomials. The
generalization of Reed-Solomon codes to multivariate polynomials lead to t
he family of Reed-Muller codes. In the last years\, several researchers tr
ied to adapt a Reed-Muller-type construction in the rank metric setting\,
unfortunately without success. Hence\, finding such a construction has bee
n an open problem for several years.\n\nWe observed that the main obstruct
ion for constructing Reed-Muller codes in the rank metric was the impossib
ility to have abelian Galois extensions which are not cyclic\, when dealin
g 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 the analo
gues of the celebrated Alon-Füredi theorem and of the Schwartz-Zippel lem
ma for endomorphisms. These results provide nontrivial lower bounds on the
rank of a linear endomorphism and are of independent interest. Moreover\,
they allow to show that we can construct rank-metric codes that share the
same parameters with classical Reed-Muller codes. Central tool for this a
pproach is the Dickson matrix associated to an endomorphism\, which we car
efully investigate.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alain Couvreur (INRIA)
DTSTART:20210118T140000Z
DTEND:20210118T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/9
DESCRIPTION:Title: On the hardness of the code equivalence problem in
rank metric\nby Alain Couvreur (INRIA) as part of Galois geometries a
nd their applications eseminars\n\nLecture held in Google Meet.\n\nAbstrac
t\nIn this talk\, we discuss the code equivalence problem in rank metric.
For $\\mathbb{F}_{q^m}$-linear codes\, which is the most commonly studied
case of rank metric codes\, we prove that the problem can be solved in pol
ynomial case with an algorithm which is "worst case". On the other hand\,
the problem can be stated for general matrix spaces. In this situation\, w
e are able to prove that this problem is at least as hard as the monomial
equivalence for codes endowed with the Hamming metric.\n\nThis is a common
work with Thomas Debris Alazard and Philippe Gaborit.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giuseppe Mazzuoccolo (University of Verona)
DTSTART:20210316T150000Z
DTEND:20210316T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/10
DESCRIPTION:Title: How many lines of the Fano plane do we need to co
lor a cubic graph?\nby Giuseppe Mazzuoccolo (University of Verona) as
part of Galois geometries and their applications eseminars\n\nLecture held
in Google Meet.\n\nAbstract\nThe problem of establishing the number of pe
rfect matchings necessary\nto cover the edge-set of a cubic bridgeless gra
ph is related to a long standing conjecture in graph theory attributed to
Berge and Fulkerson. \nIt turns out that such a problem can be nicely des
cribed in term of colorings of the edge-set of the graph by using as color
s 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 projective
space $PG(n\,2)$\, a $T$-coloring of a cubic graph $G$ is a coloring of t
he edges of $G$ by points of $PG(n\,2)$ such that the three colors occurri
ng at any vertex form a line in $T$.\nIn the first part of the talk we pre
sent the main problem in its original formulation and we show the connecti
on with $T$-colorings.\nThen\, we present some recent results (see [2]) on
a minimum possible counterexample for the Berge-Fulkerson Conjecture.\n\n
[1] E. Máčajová\, M. Škoviera\, Fano colourings of cubic graphs and th
e Fulkerson Conjecture\, Theor. Comput. Sci. 349 (2005) 112-- 120.\n\n[2]
E. Máčajová\, G. Mazzuoccolo\, Reduction of the Berge-Fulkerson conject
ure to cyclically 5-edge-connected snarks\, Proc. Amer. Math. Soc. 148 (20
20)\, 4643--4652.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Sheekey (University College Dublin)
DTSTART:20210420T140000Z
DTEND:20210420T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/11
DESCRIPTION:Title: The tensor rank of semifields of order 81\nby
John Sheekey (University College Dublin) as part of Galois geometries and
their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract\
nTensor products of vector spaces are fundamental objects in mathematics.
The tensor product of two vector spaces can be studied using matrices\, an
d this case is well-understood\; the rank can be calculated easily\, and e
quivalence corresponds precisely with rank. However for higher order tenso
rs\, problems such as calculating the rank or determining equivalence beco
mes very difficult.\n\nThe case of the tensor product of three isomorphic
vector spaces corresponds to algebras in which multiplication is not assum
ed to be associative. In this case\, the tensor rank gives an important me
asure of the complexity of the multiplication in the corresponding algebra
. For the case of a finite semifield (i.e. a not-necessarily associative d
ivision algebras)\, lower bounds can be obtained using results from linear
codes\, while for field extensions upper bounds can be obtained via polyn
omial interpolation and algebraic geometry.\n\nIn this talk we will survey
these problems and present new results where we determine the tensor rank
of all finite semifields of order 81. In particular we show that some sem
ifields of order 81 have lower tensor rank than the field of order 81\, th
e first known example of such a phenomenon.\n\nThis is joint work with Mic
hel Lavrauw.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gary McGuire (University College Dublin)
DTSTART:20210615T140000Z
DTEND:20210615T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/12
DESCRIPTION:Title: Linear Fractional Transformations and Irreducible
Polynomials over Finite Fields\nby Gary McGuire (University College D
ublin) as part of Galois geometries and their applications eseminars\n\nLe
cture held in Google Meet.\n\nAbstract\nWe will discuss polynomials over a
finite field where linear fractional transformations permute the roots. F
or subgroups $G$ of $\\mathrm{PGL}(2\,q)$ we will demonstrate some connect
ions between the field of $G$-invariant rational functions and factorizati
ons of certain polynomials into irreducible polynomials over $\\mathbb{F}_
q$. Some unusual patterns in the factorizations are explained by this conn
ection.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Buratti (University of Perugia)
DTSTART:20210511T140000Z
DTEND:20210511T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/13
DESCRIPTION:Title: Designs over finite fields by difference methods<
/a>\nby Marco Buratti (University of Perugia) as part of Galois geometries
and their applications eseminars\n\nLecture held in Google Meet.\n\nAbstr
act\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-designs. Of
course\, a particular attention will be given to the renowned 2-analog of
a 2-(13\,3\,1) design found by Braun\, Etzion\, Östergård\, Vardy and W
assermann.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mattheus (Vrije Universiteit Brussel)
DTSTART:20210720T140000Z
DTEND:20210720T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/14
DESCRIPTION:Title: Eigenvalues of oppositeness graphs and Erdős-Ko-
Rado for flags\nby Sam Mattheus (Vrije Universiteit Brussel) as part o
f Galois geometries and their applications eseminars\n\nLecture held in Go
ogle Meet.\n\nAbstract\nOver the last few years\, Erdős-Ko-Rado theorems
have been found in many different geometrical contexts including for examp
le sets of subspaces in projective or polar spaces. A recurring theme thro
ughout these theorems is that one can find sharp upper bounds by applying
the Delsarte-Hoffman coclique bound to a matrix belonging to the relevant
association scheme. In the aforementioned cases\, the association schemes
turn out to be commutative\, greatly simplifying the matter. However\, whe
n we do not consider subspaces of a certain dimension but more general fla
gs\, we lose this property. In this talk\, we will explain how to overcome
this problem\, using a result originally due to Brouwer. This result\, wh
ich has seemingly been flying under the radar so far\, allows us to find e
igenvalues of oppositeness graphs and derive sharp upper bounds for EKR-se
ts of certain flags in projective spaces and general flags in polar spaces
and exceptional geometries. We will show how Chevalley groups\, buildings
\, Iwahori-Hecke algebras and representation theory tie into this story an
d discuss their connections to the theory of non-commutative association s
chemes.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Pott (Otto von Guericke University)
DTSTART:20211019T140000Z
DTEND:20211019T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/16
DESCRIPTION:Title: Vectorial bent functions and beyond\nby Alexa
nder Pott (Otto von Guericke University) as part of Galois geometries and
their applications eseminars\n\nLecture held in Google Meet.\n\nAbstract\n
A 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 $2^{n-m}$ solutions.\nI
t 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 vectorial bent
functions\, in particular:\n\n- Classification of $(6\,3)$-vectorial bent
functions [1].\n\n- Number of quadratic $(n\,2)$-vectorial bent functions
[3].\n\nDue to the bound $m\\leq k$\, one may ask which functions are \ncl
ose to vectorial bent functions if $m>k$. In [2]\nwe determined the maxim
um 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 th
e non-bent functions form a vector space \n(bent complement). This has bee
n later generalized to\nfunctions $F:\\F_2^{2k}\\to\\F_2^{m}$ [4].\n\nI wi
ll briefly report about recent progress on such MNBC functions\n(joint wor
k 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 f
unction\, IEEE Trans. Inform. Theory\, 67 (2021)\, pp. 1027–1037.\n\n[2]
A. Pott\, E. Pasalic\, A. Muratović-Ribić\, and S. Bajrić\, On the max
imum number of bent components of vectorial functions\, IEEE Trans. Inform
. Theory\, 64 (2018)\, pp. 403–411.\n\n[3] A. Pott\, K.-U. Schmidt\, and
Y. Zhou\, Pairs of quadratic forms over finite fields\, Electron. J. Comb
in.\, 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)-functions
with maximal number of bent components\, Des. Codes Cryptogr.\, 88 (2020)
\, pp. 2171–2186.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Panario (Carleton University)
DTSTART:20211214T150000Z
DTEND:20211214T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/18
DESCRIPTION:Title: The dynamics of iterating functions over finite f
ields\nby Daniel Panario (Carleton University) as part of Galois geome
tries and their applications eseminars\n\nLecture held in Google Meet.\n\n
Abstract\nWhen we iterate functions over finite structures\, there is an\n
underlying natural functional graph. For a function $f$ over\na finite fie
ld $\\mathbb{F}_q$\, this graph has $q$ nodes and\na directed edge from ve
rtex $a$ to vertex $b$ if and only if\n$f(a)=b$. It is well known\, combin
atorially\, that functional\ngraphs are sets of connected components\, com
ponents are \ndirected cycles of nodes\, and each of these nodes is the ro
ot \nof a directed tree from leaves to its root.\n\nThe study of iteration
s of functions over a finite field and\ntheir corresponding functional gra
phs is a growing area of\nresearch\, in part due to their applications in
cryptography\nand integer factorization methods like Pollard rho algorithm
.\n\nSome functions over finite fields when iterated present strong\nsymme
try properties. These symmetries allow mathematical proofs\nof some dynami
cal properties such as period and preperiod of a\ngeneric element\, (avera
ge) ``rho length'' (number of iterations\nuntil a cycle is formed)\, numbe
r of connected components\, cycle\nlengths\, and permutational properties
(including the cycle \ndecomposition).\n\nWe survey the main problems addr
essed in this area so far.\nWe exemplify by describing the functional grap
h of Chebyshev \npolynomials over a finite field. We use the structural re
sults\nto obtain estimates for the average rho length\, average number\nof
connected components and the expected value for the period\nand preperiod
of iterating Chebyshev polynomials over finite \nfields. We conclude pro
viding a list of open problems. \n\nBased on joint works with Rodrigo Mart
ins (UTFPR\, Brazil)\,\nClaudio Qureshi (UdelaR\, Uruguay) and Lucas Reis
(UFMG\, Brazil).\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martino Borello (Université Paris 8)
DTSTART:20220201T150000Z
DTEND:20220201T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/19
DESCRIPTION:Title: Small strong blocking sets and their coding theor
etical counterparts\nby Martino Borello (Université Paris 8) as part
of Galois geometries and their applications eseminars\n\nLecture held in G
oogle Meet.\n\nAbstract\nStrong blocking sets are sets of points in the pr
ojective space such that the intersection with each hyperplane spans the h
yperplane. They have been defined first in Davydov\, Giulietti\, Marcugini
\, Pambianco\, 2011\, in relation to covering codes\, and reintroduced lat
er as generator sets in Fancsali\, Sziklai\, 2014 and as cutting blocking
sets in Bonini\, Borello\, 2021\, in relation with minimal codes. In Alfar
ano\, Borello\, Neri\, 2019 and independently in Tang\, Qiu\, Liao\, Zhou\
, 2019\, it has been shown that strong blocking sets are the geometric cou
nterparts of such codes. From their definition\, it is clear that adding a
point to a strong blocking set maintains the property of being strong\, s
o that strong blocking sets of small cardinality are the most interesting
ones. In the coding theoretical language\, this is equivalent to have a sh
ort minimal code. A natural question is then how small a strong blocking s
et in a projective space of a given dimension can be.\n\n In the talk\, we
will illustrate these connections\, together with some bounds on their pa
rameters 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://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentina Pepe (Sapienza Università di Roma)
DTSTART:20220322T160000Z
DTEND:20220322T170000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/20
DESCRIPTION:Title: The geometry of extremal Cayley graphs\nby Va
lentina Pepe (Sapienza Università di Roma) as part of Galois geometries a
nd their applications eseminars\n\nLecture held in Google Meet.\n\nAbstrac
t\nThe geometric aspect of extremal Cayley graphs is highlighted\, providi
ng a different proof of known results and giving a new perspective on how
to tackle such problems.\nSome new results about extremal pseudrandom tria
ngle free graphs are also presented.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Massimo Giulietti (Università degli Studi di Perugia)
DTSTART:20220426T150000Z
DTEND:20220426T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/21
DESCRIPTION:Title: Algebraic curves with many automorphisms\nby
Massimo Giulietti (Università degli Studi di Perugia) as part of Galois g
eometries and their applications eseminars\n\nLecture held in Google Meet.
\n\nAbstract\nThe Hurwitz upper bound on the size of the $\\mathbb{K}$-aut
omorphism group Aut($\\mathcal{C}$) of an algebraic curve $\\mathcal{C}$ o
f genus $g$ greater than $1$ defined over a field $\\mathbb{K}$ of zero ch
aracteristic is $84(g-1)$. \nIn positive characteristic $p$\, algebraic cu
rves can have many more automorphisms than expected from the Hurwitz bound
. \nThere even exist algebraic curves of arbitrarily large genus $g$ with
more than $16g^4$ automorphisms. Besides the genus\, an important invaria
nt for curves in positive characteristic is the $p$-rank of the curve\, wh
ich 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 of algeb
raic curves with a very large automorphism group invariably have zero $p$-
ranks.\nSeveral results on the interaction between the automorphism group\
, the genus and the $p$-rank of a curve can be found in the literature. In
this talk we survey some reults on the following issues that have been ob
tained in the last decade:\n\n(i) Upper bounds on the size of Aut($\\mathc
al{C}$) depending on g and the structure of Aut($\\mathcal{C}$).\n\n(ii) T
he 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\nSo
me applications to maximal curves over finite fields are also discussed.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan De Beule (Jan.De.Beule@vub.be)
DTSTART:20220510T150000Z
DTEND:20220510T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/22
DESCRIPTION:Title: On Cameron-Liebler sets of k-spaces in finite pro
jective spaces (Part I)\nby Jan De Beule (Jan.De.Beule@vub.be) as part
of Galois geometries and their applications eseminars\n\nLecture held in
Google Meet.\n\nAbstract\nThis is part 1 (of 2) of a double talk together
with Jonathan Mannaert. Cameron-Liebler line classes in a finite 3-dimensi
onal space PG(3\,q) originate from the study by Cameron and Liebler in 198
2 of groups of collineations with equally many orbits on the points and th
e lines of PG(3\,q). These objects have some interesting equivalent charac
terizations\, and are examples of Boolean functions of degree one. One of
the main properties of this set is that these line classes admit a paramet
er x\, which can be used to classify or exclude examples. In this talk\, w
e focus on these objects from a geometric perspective\, and report on seve
ral existence and non-existence results\, including a recent so-called mod
ular equality for the parameter of Cameron-Liebler line classes in finite
n-dimensional projective spaces found in [2] for n odd. This modular equal
ity 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-Liebler se
ts of k-spaces in PG(n\,q).\nDes. Codes Cryptogr.\, 87(8):1839--1856\, 201
9.\n\n[2] J. De Beule and J. Mannaert.\nA modular equality for Cameron-Lie
bler line classes in projective and affine spaces of odd dimension.\nSubmi
tted.\n\n[3] A. L. Gavrilyuk and K. Metsch.\nA modular equality for Camero
n-Liebler line classes.\nJ. Combin. Theory Ser. A\, 127:224--242\, 2014.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oriol Serra (Universitat Politècnica de Catalunya)
DTSTART:20221019T140000Z
DTEND:20221019T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/23
DESCRIPTION:Title: Sidon spaces\nby Oriol Serra (Universitat Pol
itècnica de Catalunya) as part of Galois geometries and their application
s eseminars\n\nLecture held in Google Meet.\n\nAbstract\nMotivated by a pr
oblem related to difference sets\, Hou\, Leu and Xiang introduced in 2002
a linear version of the classical theorem of Kneser in additive combinato
rics\, where sets are replaced by subspaces and cardinalities by dimensio
ns. A nice feature of the linear version is that\, via Galois extensions\
, it provides an alternate proof of the original version. This openned a t
rend to prove extensions of theorems in additive combinatorics to their li
near analogues. The talk will focuss on one of these extensions\, the Vosp
er theorem\, which gives rise to the notion of Sidon spaces. This notion t
urned out to find interesting applications in coding theory.\n\nThis is jo
int work with Christine Bachoc and Gilles Zémor\, with a nice simplificat
ion by Chiara Castello.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zsuzsa Weiner (ELKH-ELTE GAC and Prezi.com)
DTSTART:20221207T131500Z
DTEND:20221207T141500Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/24
DESCRIPTION:Title: Consequences of a resultant-like theorem in Galoi
s geometries\nby Zsuzsa Weiner (ELKH-ELTE GAC and Prezi.com) as part o
f Galois geometries and their applications eseminars\n\nLecture held in Go
ogle Meet.\n\nAbstract\nWith Tamás Szőnyi we proved the following theore
m on two variable polynomials\, see [8]\, [7]\, [5]. The present form of t
he theorem is due to Tamás Héger [3]. \n\nTheorem\n Let $f\,g \\i
n \\F[X\,Y]$ be polynomials over the arbitrary field $\\F$.\n Assume that
the coefficient of $X^{\\deg f}$ in $f$ is not $0$ and for $y\\in \\F$ pu
t $k_y = \\deg \\gcd(f(X\,y)\,g(X\,y))$. Then for any $y_0 \\in \\F$\n \\
[\n \\sum_{y\\in \\F}(k_y-k_{y_0})^+ \\leq (\\deg f - k_{y_0})(\\deg g -
k_{y_0}).\n \\]\n\nHere $\\alpha^+=\\max\\{0\,\\alpha\\}$. Note that $g$
can be the zero polynomial as well\, in that case $\\deg f=k_y=k_{y_0}$ an
d the lemma claims the trivial $0 \\leq 0$. \n\n\nIn my talk\, I will 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 survey o
n these results\, my aim is to show the part of the proofs in detail where
we gain benefit from this theorem. I will talk about an upper bound on th
e number of lines that may intersect a point set in $\\mathrm{PG}(2\,q)$ [
4]\; about the possible sizes of the second largest minimal blocking sets
in PG$(2\,q)$\, $q$ square [6]\; about codewords generated by the lines 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 u
nitals which are codewords are necessarily Hermitian.\n\n \n\n[1] A. Blokh
uis\, A.E. Brouwer\, H. Wilbrink: Hermitian unitals are code words\, Discr
ete 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 g
raph theoretic aspects of finite geometries\, PhD Thesis\, Eötvös Lorán
d 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. Combin. T
heory Ser. A 118:7 pp. 2066-2070 (2011). \n\n\n[5] T. Szőnyi\, Zs. Weine
r: Stability of $k$ mod $p$ multisets and small weight codewords of the co
de 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. Weiner: La
rge blocking sets in $\\mathrm{PG}(2\, q^2)$\, Finite Fields Appl.\, to ap
pear.\n\n[7] Zs. Weiner: On $(k\,p^e)$-arcs in Desarguesian planes\, Finit
e Fields Appl. 10 (2004)\, 390-404. \n\n[8] Zs. Weiner: Geometric and alge
braic methods in Galois-geometries\, PhD Thesis\, Eötvös Loránd Univers
ity\, 2002\, https://web.cs.elte.hu/~weiner/main_jav.pdf\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heide Gluesing-Luerssen (University of Kentucky)
DTSTART:20230301T150000Z
DTEND:20230301T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/25
DESCRIPTION:Title: Properties of the Direct Sum of q-Matroids\nb
y Heide Gluesing-Luerssen (University of Kentucky) as part of Galois geome
tries and their applications eseminars\n\nLecture held in Google Meet.\n\n
Abstract\nAfter a brief introduction of $q$-matroids and their relevance f
or 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 cryptomor
phisms. While these are non-trivial results\, they all form quite natural
$q$-analogues of the corresponding cryptomorphisms for (classical) matroid
s. We will then turn to the direct sum of $q$-matroids\, which was introdu
ced in 2021 by Ceria/Jurrius. It turns out that the definition as well as
the properties of the direct sum are significantly different from those fo
r matroids. After discussing the construction of the direct sum\, we will
report on properties where the theory diverges the most from that of matro
ids. Thereafter\, we will turn to a result where the theory\, surprisingly
\, meets that of matroids. Indeed\, the direct sum behaves very naturally
with respect to cyclic flats. This allows us to show that every $q$-matroi
d can be decomposed into irreducible ones and to characterize irreducibili
ty.\n\nThis is joint work with Benjamin Jany.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Sudakov (ETH Zurich)
DTSTART:20230308T150000Z
DTEND:20230308T160000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/26
DESCRIPTION:Title: Evasive sets\, covering by subspaces\, and point-
hyperplane incidences\nby Benjamin Sudakov (ETH Zurich) as part of Gal
ois 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 $k$-di
mensional affine subspace contains at most $c$ elements of $S$.\nBy a simp
le averaging argument\, the maximum size of a $(k\,c)$-subspace evasive se
t is at most $c |F|^{d-k}$.\nIn this talk we discuss the construction of e
vasive sets\, matching this bound.\n\nThe existence of optimal evasive set
s has several interesting consequences in combinatorial geometry.\nUsing i
t we determine the minimum number of $k$-dimensional linear hyperplanes ne
eded to cover the grid $[n]^{d}$.\nThis extends the work by Balko\, Cibulk
a\, and Valtr\, and settles a problem proposed by Brass\, Moser\, and Pach
.\nFurthermore\, we improve the best known lower bound on the maximum numb
er of incidences between points and hyperplanes\nin dimension $d$ assuming
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://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilaria Cardinali (Università degli studi di Siena)
DTSTART:20230524T140000Z
DTEND:20230524T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/27
DESCRIPTION:Title: Grassmannian of codes\nby Ilaria Cardinali (U
niversità degli studi di Siena) as part of Galois geometries and their ap
plications 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 poi
nts all the $[n\,k]$-linear codes having minimum dual Hamming weight at le
ast $t+1$ 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 at least
$t+1$.\n Let $\\Lambda_t(n\,k)$ be the collinearity graph of $\\mathcal{
P}_t(n\,k).$ 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 codes hav
ing 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 its relat
ion with well-studied configurations of points of a projective space such
as the saturated sets. In particular\, I will characterize the set of isol
ated vertices of $\\Lambda_t(n\,k)$ and for $t=1$ and $t=2$\, necessary a
nd sufficient conditions for $\\Lambda_t(n\,k)$ \n to be connected will be
provided.\n Finally\, these results will be applied to the geometry ${\\
mathcal P}_t(n\,k)$\n in order to study its projective embeddability by m
eans of the\n Plücker map.\n\n\n\n[1] I. Cardinali and L. Giuzzi\, Gra
ssmannians of codes\, submitted.\n\n\n[2] M. Kwiatkowski\, M. Pankov\, O
n the distance between linear codes\, Finite Fields Appl. 39 (2016)\, 251
-263.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mattheus (University of California\, San Diego and Vrije Unive
rsiteit Brussel)
DTSTART:20230927T140000Z
DTEND:20230927T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/28
DESCRIPTION:Title: The asymptotics of r(4\,t)\nby Sam Mattheus (
University of California\, San Diego and Vrije Universiteit Brussel) as pa
rt of Galois geometries and their applications eseminars\n\nLecture held i
n Zoom.\n\nAbstract\nFor integers $s\,t \\geq 2$\, the Ramsey numbers $r(s
\,t)$ denote the \nminimum $N$ such that every $N$-vertex graph contains e
ither 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 s
hows\n\n$r(4\,t)=\\Omega\\Bigl(\\frac{t^3}{\\log^4 \\! t}\\Bigr)$ as $t \\
rightarrow \\infty$.\n\n\n\nThis determines $r(4\,t)$ up to a factor of or
der $\\log^2 \\! t$\, and \nsolves a conjecture of Erdős. Moreover\, I wi
ll discuss some \nsubsequent work with David Conlon\, Dhruv Mubayi and Jac
ques Verstraete \nshowing the need for good constructions\, possibly comin
g from finite \ngeometry.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Calderbank (Duke University)
DTSTART:20231025T140000Z
DTEND:20231025T150000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/29
DESCRIPTION:Title: Back to the Future\nby Robert Calderbank (Duk
e University) as part of Galois geometries and their applications eseminar
s\n\nLecture held in Zoom.\n\nAbstract\nThis talk tells the history of cod
ing theory through the lens of Reed Muller codes. In the beginning\, there
were no computers\, and coding theory was the mathematics of sphere packi
ng. This was a golden time for algebraic coding\, with the discovery of Re
ed Muller and Reed Solomon codes. As everyday computers became more powerf
ul coding theory changed character and focused on iterative algorithms. To
day with quantum computers on the horizon\, Reed Muller codes are back in
fashion.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Bamberg ((University of Western Australia)
DTSTART:20231220T100000Z
DTEND:20231220T110000Z
DTSTAMP:20260422T225721Z
UID:CombinatoricsAndAlgebraicCurves/30
DESCRIPTION:Title: Foundations of hyperbolic geometry\nby John B
amberg ((University of Western Australia) as part of Galois geometries and
their applications eseminars\n\nLecture held in Zoom.\n\nAbstract\nThe in
dependent discovery by Lobachevsky and Bolyai of hyperbolic geometry in th
e 1830's was followed by slow acceptance of the subject from the 1860'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 work intr
oducing the "calculus of ends"\, introduced an axiomatisation for hyperbol
ic plane geometry by adding a hyperbolic parallel axiom to the axioms for
plane absolute geometry. In 1938\, Karl Menger (of the famous Vienna Circl
e) made the important discovery that in hyperbolic geometry the concepts o
f betweenness and equidistance can be defined in terms of point-line incid
ence. Since an axiom system obtained by replacing all occurrences of betwe
enness and equidistance with their definitions in terms of incidence would
look highly unnatural\, Menger and his students looked for a more natural
axiom system. In particular\, Helen Skala showed in 1992 that there is a
set of axioms whose models are the classical hyperbolic planes over Euclid
ean fields\, and her axioms were the first that contained only first order
statements. This talk will be on joint work with Tim Penttila (Emeritus\,
University of Adelaide) where we endeavour to simplify Skala's axioms and
retain a characterisation of the classical hyperbolic planes.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsAndAlgebraicCurves
/30/
END:VEVENT
END:VCALENDAR