Mathematical Research Seminar - Archive
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Recently, we have investigated a new environment for the conjecture - that of stacks - the geometric objects generalizing varieties. This perspective has led to unexpected applications beyond its original scope, offering insights into phenomena related to the occurrence of Galois extensions—a topic that, at first glance, seems unrelated to counting solutions of polynomial equations.
The starting point is the use of auxiliary graphs, the so called hexagon graphs, associated with Cayley maps of $(2,s,3)$-groups. Large induced trees in these auxiliary graphs provide a route to Hamilton cycles/paths in the original Cayley graphs. I will then explain why quartic one-regular graphs are central in a possible generalization of this approach to $(2,s,4)$-groups.
In the final part, I will discuss the Feng--Xu normality theorem for quartic Cayley graphs on regular $p$-groups and propose a CFSG-free proof in the special case where one generator has order $p$.
This leads naturally to broader questions about direct proofs in algebraic combinatorics, including primitive groups of degree $2p$ and the Polycirculant conjecture.
Cubic Surfaces are a classical area of study. In this talk we will discuss some recent results by Anton Betten and Fatma Karaoglu, and show how they give rise to interesting geometric consequences for a particular family of cubic surfaces over finite fields with characteristic 2.
ZOOM link: https://upr-si.zoom.us/j/65127426860?pwd=kYjS6yFt6yeUcYAGGPcnJShbb5UI7r.1
We investigate finite semigroups equipped with a preorder relation that is compatible with the semigroup operation and we develop an enumeration methodology based on automorphism group actions. For a fixed finite semigroup S, the automorphism group Aut(S) acts naturally on the set Pre(S) of compatible preorders via pushforward; the orbits of this action correspond exactly to isomorphism classes of preordered semigroups. Using known classifications of semigroups of small order, together with exhaustive generation of compatible preorders, our method yields a complete enumeration of preordered semigroups up to order five. Finally, this framework opens a computational window toward the study of more complex algebraic systems, particularly EL-hyperstructures arising in hypercompositional algebra, where preordered semigroups serve as a fundamental tool for their construction.
ZOOM link: https://upr-si.zoom.us/j/65127426860?pwd=kYjS6yFt6yeUcYAGGPcnJShbb5UI7r.1
