# Research

My research background is in Abstract Algebra, and may be broadly described as studying algebraic structures to understand their complexity as well as their interaction with various non-algebraic subjects.

Below you will find some details about my work, as well as short descriptions about some other research areas I am interested in.

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Dualizability of Algebras from Congruence-Permutable Varieties

Topological Algebra

Square Packing

Dualisability of Automatic Algebras

Empathy Development in Infancy

**Dualizability of Algebras from Congruence-Permutable Varieties**

Introduction

While duality theory for lattices and their generalizations is well developed and widely used, the dualizability of groups and related algebras still needs to be explored in more depth. Duality theory in general aims to represent abstract algebraic structures using functions on a topological space oftenâ€¨enriched with a relational and/or operational structure. Apart from making the elements of the algebra more concrete, this representation allows us to solve algebraic questions using topological methods.

These techniques originally developed from representation results for Boolean algebras (Stone), distributive lattices (Birkhoff), and abelian groups (Pontryagin). In 1983, Davey and Werner introduced the notion of natural duality of a finite algebra, generalizing and combining these concepts. This project aims to extend previous findings to more general classes of congruence-permutable varieties, and in particular establish whether dualizability is decidable in Malcev varieties, building upon recent results characterizing the dualizable groups (Quackenbush, Szabo; Nickodemus) and dualizable commutative rings with unity (Clark, Idziak, Sabourin, Szabo, Willard). The project also addresses two questions raised by Willard in 2007 at the International Conference on Order, Algebra and Logics.

Duality

A finite algebra is dualizable if it can be associated with a topological structure on the same underlying set, its “alter ego", such that each member of the quasi-variety generated by the algebra is in a canonical sense isomorphic to the algebra of continuous homomorphisms from a member of the topological quasi-variety generated by the alter ego into the alter ego itself. In such a situation, a duality exists between the quasi-variety generated by the algebra and the topological quasi-varietyâ€¨generated by the alter ego.

The motivation behind duality theory is that a duality allows one to translate algebraic questions into the potentially simpler, and usually more intuitive setting of a topological structure.

Known results

Dualizability (i.e. the existence of a duality) has been completely characterized for the following classes of algebras

- Algebras from congruence-distributive varieties (Davey, Heindorf, McKenzie; one direction following from earlier results by Baker, Pixley)
- Groups (Quackenbush, Szabo; Nickodemus)
- Commutative rings with unity (Clark et al)
- Graph algebras (Davey et al)

Partial results have been obtained for other classes, such as:

- Rings with or without unity (Szabo)
- Automatic algebras (Bentz, Davey, Pitkethly, and Willard).

Duality of Malcev Algebras

Many aspects of Malcev algebras and congruence-permutable varieties are considered to be well known due to a large amount of previous results and their closeness to classical algebras.

In contrast, the dualizability question for Malcev algebras is surprisingly poorly understood. While the corresponding questions have been solved for algebras from congruence-distributive varieties (which include all lattice based algebras), complete results for Malcev algebras are restricted to relatively small classes. The aim of the project is to close this knowledge gap.

Acknowledgments

This project is funded by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. PCOFUND-GA-2009-246542 and from the Foundation for Science and Technology of Portugal.

**Additional Research **

**Topological Algebra**

An algebra and a topology on the same underlying set are considered to be compatible if the algebraic operations are continuous with respect to the topology. Under this correspondence, the algebraic structure places restrictions on the topological one and vice versa. This raises the question to what extent algebraic properties can be brought into correspondence with topological ones. I have investigated several topological properties and was able to find algebraic characterizations for two important implications in a particular class of algebras. In particular, one of these results amounts to establishing an algebraic version of the Hausdorff property in this particular context. Some of these methods were adopted by researchers in theoretical computer science to establish properties of the Lambda-calculus. My current research centre on the question under what algebraic conditions two topological properties can be forced to coincide

**Square Packing**

A packing of a collection of Euclidean geometrical objects in another object (all of the same dimension and with sufficiently well-behaved boundaries) assigns to each “packed object” a subset of the “containing object”, its placement, such that each packed object and its placement are congruent and the placements have pairwise disjoint interiors. Two packing are identified if they can be connected continuously by a collection of other packings, or if they can be obtained through symmetries of the containing object or permutations of congruent placements. A packing problem is concerned with the most efficient packing possible in one particular setting, i.e. find the smallest object (among various options) in which a particular collection of object can be packed, or conversely find the largest collection (of a certain kind) that can be packed into a given object. Here largest and smallest need to be defined in a reasonable way, although in general efficiency is related to the area or volume of “unused” or “wasted” space.

Packing problems have several real-life applications, from minimal energy states to the optimal use of material in manufacturing to optimization questions in coding theory. Moreover, proofing the optimality of a packing (or sometimes even the existence) is a serious mathematical challenge even when the number of objects is low.

While the majority of results in this area study circles or spheres, we are concerned with squares, which introduce more algebraic and geometrical aspects to the problem. In addition, the basic techniques used on squares have generalizations to arbitrary convex polygons.

**Dualisability of Automatic Algebras**

Consider a finite algebra M and a structure N with potentially partial operations and relations having the same underlying set as M and being endowed with the discrete topology. M and N yield a duality if in a canonical sense, each member of the quasi-variety generated M is isomorphic to the algebra of all continuous homomorphisms from a member of the topological quasi-variety generated by N into N itself. For example, the two-element Boolean algebra and the two-element discrete topological space yield a duality between Boolean algebras and Boolean spaces. The presence of a duality allows one to translate algebraic questions into the (potentially simpler) setting of the topological structure. I am part of a research project that has successfully classified the dualisability of various families of algebras derived from determinist automaton. A paper on the result is in preparation. This is joint work with Brian Davey, Jane Pitkethly (both from the School of Engineering and Mathematical Sciences, La Trobe University), and Ross Willard (Department of Pure Mathematics, University of Waterloo).

**Empathy Development in Infancy**

Affect sharing or emotional contagion is one important component of empathy, already present in the very first few days of life. One of the hypothesized mechanisms underlying affect sharing involves shared representations between one’s own experience of emotion and perceiving the same emotion in another. It has been shown that adults manifest mimicry of pupil size when they see sad faces. That is, their pupil diameter increases when they perceive a sad face with large pupils, and decreases when the face they see has a sad expression and small pupil size. We are investigating if this effect is present in infants as well. The unique data produced by these experiments requires intelligent extraction of relevant information that is not available in standard software, and which is not feasible by manual work alone due to size considerations. I am writing custom software to address these problems. This is joint work with Elena Geangu (Psychology Department, Durham University), Petra Hauf (Department of Psychology, St. Francis Xavier University), and Rishi Bhardwaj (Psychology Department, Durham University).