Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

Which rational double points occur on del Pezzo surfaces?

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.

Identically Self-Dual Matroids

<p>In this thesis we focus on identically self-dual matroids and their minors. We show that every sparse paving matroid is a minor of an identically self-dual sparse paving matroid. The same result is true if the property sparse paving is replaced with the property of representability and more specifically, F-representable where F is a field of characteristic 2, an algebraically closed field, or equal to GF(p) for a prime p = 3 (mod 4). We extend a result of Lindstrom [11] saying that no identically self-dual matroid is regular and simple. We assert that this also applies to all matroids which can be obtained by contracting an identically self-dual matroid. Finally, we present a characterisation of identically self-dual frame matroids and prove that the class of self-dual matroids is not axiomatisable.</p>

Identically Self-Dual Matroids

<p>In this thesis we focus on identically self-dual matroids and their minors. We show that every sparse paving matroid is a minor of an identically self-dual sparse paving matroid. The same result is true if the property sparse paving is replaced with the property of representability and more specifically, F-representable where F is a field of characteristic 2, an algebraically closed field, or equal to GF(p) for a prime p = 3 (mod 4). We extend a result of Lindstrom [11] saying that no identically self-dual matroid is regular and simple. We assert that this also applies to all matroids which can be obtained by contracting an identically self-dual matroid. Finally, we present a characterisation of identically self-dual frame matroids and prove that the class of self-dual matroids is not axiomatisable.</p>

Classification of Nilpotent Lie Superalgebras of Multiplier-Rank Less than or Equal to 6

In this paper, we classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank less than or equal to 6 over an algebraically closed field of characteristic zero. We also determine the covers of all the nilpotent Lie superalgebras mentioned above.

Low-dimensional commutative power-associative superalgebras

The aim of this work is to provide a concrete list of non-isomorphic commutative power-associative superalgebras up to dimension 4 over an algebraically closed field of characteristic prime to 30. As a byproduct, we exhibit an example of a simple non-Jordan power-associative superalgebra whose even part is not semisimple.

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)−12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

We investigate the deformations of pairs [Formula: see text], where [Formula: see text] is a line bundle on a smooth projective variety [Formula: see text], defined over an algebraically closed field [Formula: see text] of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair [Formula: see text] is homotopy abelian whenever [Formula: see text] has trivial canonical bundle, and so these deformations are unobstructed.

An Application of Finite Groups to Hopf algebras

Kaplansky’s famous conjectures about generalizing results from groups to Hopf al-gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.