# Around Algebra, Combinatorics and Number Theory

## August 14-16, 2018 Chancellor College University of Malawi Zomba, Malawi

We acknowledge the generous financial support of Elsevier, FMSP/AMSP, and of the LMS, the IMU and the AMMSI via the MARM program.

#### Scientific Organisers

Frank Neumann: fn8 at le dot ac dot uk
Ambrus Pál: a dot pal at imperial dot ac dot uk

#### Local Organisers

Patrick Ali: pali at cc dot ac dot mw
Elias Mwakilama: emwakilama at cc dot ac dot mw

#### Schedule

Talks and coffee breaks will be at Lecture Theatre 2, CHANCO campus.

 Tue 14 10:00-11:00 Frank Neumann Hochschild cohomology, differential graded categories and spectral sequences 11:00-11:30 Coffee 11:30-12:00 Isaac Owino Okoth On set partitions and tree-like structures 12:00-14:00 Lunch 14:00-15:00 Hyunsung Kim Data centric security and privacy issues for the Internet of things 15:00-15:30 Coffee

 Wed 15 10:00-11:00 Olga Danilkina Approximations of the solution to the Hodge Laplacian 11:00-11:30 Coffee 11:30-12:00 Isaac Owino Okoth Reachability of vertices in locally oriented trees 12:00-14:00 Lunch 14:00-14:30 Patrick Ali Identities for the minors of the Laplacian, resistance and distance matrices of graphs with arbitrary weights 15:00-15:30 Coffee

 Thu 16 10:00-11:00 Ambrus Pál Absolute Galois groups and their cohomology 11:00-11:30 Coffee 11:30-12:00 John Ogonji Agure On the convexity of Stampfli's numerical range 12:00-14:00 Lunch 14:00-14:30 Kondwani Magamba On the factorisation of polynomials of the form $$cx^{q^s+1}+dx^{q^s}-ax-b\in \mathbb{F}_{q^n}[x]$$ 15:00-15:30 Coffee

Speaker: John Ogonji Agure (Maseno University, Kenya)

Title: On the convexity of Stampfli's numerical range

Abstract: In this talk we investigate the Hilbert space $$\delta$$-numerical range introduced by Stampfli. In particular we prove that this set is convex and that it is also identical to the $$\delta$$-algebra of numerical range.

Speaker: Patrick Ali (University of Malawi, Malawi)

Title: Identities for the minors of the Laplacian, resistance and distance matrices of graphs with arbitrary weights

Abstract: The resistance matrix of a simple connected graph $$G$$ is denoted by $$R_{G}$$ or simply by $$R$$ and is defined by $$R_{G}=(r_{ij})$$, where $$r_{ij}$$ is the resistance distance between the vertices $$i$$ and $$j$$ in $$G$$. In this paper, we consider the resistance matrix of a weighted tree and the resistance matrix of any weighted graph, where the weights are nonzero scalers. We obtain the minors of the Laplacian, resistance and the distance matrices, which are independent of the non-singularity of resistance and distance matrices. Joint work with Fouzul Atik and R. B. Bapat.

Speaker: Olga Danilkina (University of Dodoma, Tanzania)

Title: Approximations of the solution to the Hodge Laplacian

Abstract: In this talk, we introduce an abstract Hilbert space framework that can be applied in stability analysis of FEMs. We start with review of basic definitions of homological algebra and functional analysis such as chain/cochain complexes, closed operators on Hilbert spaces, Hilbert complexes and the Hodge decomposition. Then we define the abstract Hodge Laplacian $$L$$ and its mixed formulation, prove well-posedness and existence of the unique solution to $$Lu=f$$. After that we will discuss approximations of the cohomology spaces, harmonic forms and the solution to the Hodge Laplacian associated to a Hilbert complex by quantities associated to a subcomplex.

Speaker: Hyunsung Kim (Kyungil University, Korea)

Title: Data centric security and privacy issues for the Internet of things

Abstract: In today's interconnected world based on the Internet of things (IoT), security and privacy breach can involve one or more paths to your data. Furthermore, it is no surprise that data breaches are evolving and becoming increasingly more complex. For organisations that provide vital services such as healthcare, transportation or energy, a data failure could be catastrophic. Awareness of the need for better data centric protections has grown in recent years mostly as the result of the data breaches. Data centric security and privacy approach lets you focus on what you really need to protect rather than the information technology (IT) networks, applications and endpoints that keep small amount of your data. Data centric security and privacy allow organisations to overcome the disconnection between IT security technology and the objectives of business strategy by relating security services to the data they implicitly protect. Protecting sensitive data can take advantage of cloud computing, mobile technology and other innovations without placing your data at risk. Traditional security and privacy defences are no match for the mentioned data breaches that circumvent security controls and steal sensitive data. Furthermore, IT or IoT infrastructure and sensitive data are more mobile and ubiquitous as organisations embrace cloud computing, mobility and big data analytics tools. To address the modern threats and IT or IoT trends, we must put our efforts to innovate and devise new security and privacy approaches based on the data centric concern. The purpose of this talk is to provide an overview of my current Korean NRF project titled with "Research on Data Centric Security and Privacy Model for Intelligent Internet of Things" and review some mathematical difficulties on security models and systems for it.

Speaker: Kondwani Magamba (Malawi University of Science and Technology, Malawi)

Title: On the factorisation of polynomials of the form $$cx^{q^s+1}+dx^{q^s}-ax-b\in \mathbb{F}_{q^n}[x]$$

Abstract: Let $$n$$ and $$r\geq 3$$ be prime numbers and $$s$$ be a divisor of $$nr$$. Also, let $$\mathbb F_{q^n}$$ be a finite field with characteristic $$p$$. We count the number of irreducible factors of degree $$r$$ in the factorisation of polynomials of the form $$F_s(x)=cx^{q^s+1}+dx^{q^s}-ax-b \in \mathbb{F}_{q^n}[x]$$ where $$ad-bc \neq 0$$. Joint work with John A. Ryan.

Speaker: Frank Neumann (University of Leicester, United Kingdom)

Title: Hochschild cohomology, differential graded categories and spectral sequences

Abstract: I will give an introduction into differential graded categories which are very useful in algebra, topology and geometry. An important algebraic invariant of dg categories is Hochschild cohomology, introduced first by Hochschild in the 1950s to study deformations of algebras by cohomological means. The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded centre of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general answering a question by Bernstein. To illustrate this, we will discuss coherent sheaves over algebraic curves, as well as topological examples related to free loop spaces.

Speaker: Isaac Owino Okoth (Maseno University, Kenya)

Title: On set partitions and tree-like structures

Abstract: In this talk we provide a proof for a formula that counts the number of connected cycle-free families of $$k$$ set partitions of $$[n]$$ satisfying a certain coherence condition and then establish a bijection between these families and the set of labelled free $$k$$-ay cacti with a given vertex-degree distribution. We also show that the formula counts coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We extend the work to coloured oriented cacti and coloured cacti. The non-crossing counterparts of these graphs are also enumerated.

Speaker: Isaac Owino Okoth (Maseno University, Kenya)

Title: Reachability of vertices in locally oriented trees

Abstract: In this talk, we present some new formulas in the enumeration of labelled trees by paths lengths. We treat labelled trees having their edges oriented from a vertex of lower label towards a vertex of higher label. Among other results, we obtain a counting formula for the number of labelled trees on $$n$$ vertices in which exactly $$k$$ vertices are reachable from a given vertex $$i$$ and also the average number of vertices that are reachable from a specified vertex in labelled trees of order $$n$$ for any large $$n$$. Some known results in the enumeration of labelled trees by the number of sources and sinks also follow from our theorems as corollaries. Reachability of vertices in non-crossing trees with this orientation is also considered.

Speaker: Ambrus Pál (Imperial College London, United Kingdom)

Title: Absolute Galois groups and their cohomology

Abstract: There is a very deep and intricate relation between the arithmetic of fields and the structure of their absolute Galois groups. On the one hand one can read off arithmetic properties of a field from its absolute Galois group, on the other hand using the arithmetic of fields one can strongly curtail the structure of groups which arise as absolute Galois groups. A particularly important invariant of groups is their cohomology; the study of this invariant for absolute Galois groups has been a central object of study in the last few decades. I will describe some of the main results, the outstanding conjectures, and the methods which are used to study them.