# Research

A brief description of research projects in which I am involved

## Higher Arity Science

The order of a relation or the number of arguments of function is called its arity. Most of modern mathematics and science uses arity 2, i.e. binary operations and relations, but there is incresing evidence of the limitations of these tools to describe emergent behaviour in complex systems. The abstract study of arity can be understood as the investigation of the "atoms of emergece".

## Generalizations of Associativity in Ternary Structures

Filippov algebras, the n-ary generalization of (binary) Lie algebras, are not well understood as infinitesimal objects of global counterparts, in the same way that Lie algebras are infinitesimal Lie groups. In an attempt to develop some intuition for ternary Filippov algebras, I identified a set of axioms for a ternary operation that generalizes associativity which, curiously, had not been noted in the mathematical literature before. I call this property trisociativity. It turns out that cubic matrices carry a natural ternary products that seem to be somewhat related to trisociativity.

## Foundations of Phase Space Physics

What are the converging and diverging features of classical, quantum and statistical mechanics? Symplectic and contact geometry seem to be unifying themes. How can we find a complete characterization of obserbables and dynamics, both quantum and classical, in terms of categories of phase spaces and associated algebraic structures? My PhD thesis contains a review of these topics and some tentative approaches to the categorization of phase space theories.

## Unit-Free and Dimensioned approaches to Jacobi Geometry

In an attempt to integrate physical dimensions and units of measurement into geometric mechanics explicitly, I found a surprisingly transparent interpretation of Jacobi geometry as the unit-free counterpart of Poisson geometry. This is explained in detail in my paper on this topic. This lead, in turn, to the identification of new algebraic structures that carry a mathematical structure that formally generalizes the notion of physical dimension. These structures are first introduced in my PhD thesis under the name 'dimensioned structures' and are further studied in my paper on this topic.

## Cognitiveness as a Measurable Property of a System

Is the degree to which a system is considered cognitive, its 'cognitiveness', something that can be theoretically parametrised and experimentally measured? In this talk I present some preliminary thoughts on how relatively recent advancements in the fields of animal cognition, AI development and ant colony psychology suggest common themes between seemingly dissimilar systems. As a project living at the intersection of cognitive science, complexity science and physics this will require a long and careful study of the existing literature and the development of formal (mathematical) frameworks that aid in the integration of the knowledge from many different branches.

## Foundations of Relativity and Quantum Observers

Classical relativity theory seems to be naturally connected with the classical phase spaces since one may recover Euclidean configuration spaces in the context of the causal structure of Minkowski spacetimes. Is there a direct analogue for quantum phase spaces? In other words, can we find a geometric object where quantum observers can be axiomatized, the principle of relativity implemented and that naturally contains projective Hilbert spaces (quantum phase spaces)? This may lead to a notion of 'quantum relativity' that is entirely unrelated to 'quantum gravity' which may, nonetheless, shed some light on the conundrum posed by the latter.

## Deep Themes in Mathematics and Physics (with Michael Atiyah)

During the years of my PhD in Edinburgh I was fortunate to work as Michael Atiyah's personal research assistant. I am very happy to have accompanied Michael during his last few years. His mind was nowhere near the acuteness and agility of years past but the passion and drive for mathematics, science and beauty was very much still there. In the many long conversations I shared with Michael, touching on the deep connections between algebra, geometry and physics, I could glimpse at a mind that once probably had one of the most comprehensive vantage point of modern mathematics. A small collaboration on the application of K-theory to knots followed from this conversations.

## Verbal Cognition vs Visual Cognition

Most intellectual work is encapsulated in symbolic languages, typically verbal ones, i.e. written word. This contrasts with how we seem to comprehend shape and spatial structure instinctively and to a high degree of sophistication without any use of verbal language. I am developing a comparative cognitive study, to be conducted with human subjects, that will test the suitability of verbal language for the efficient description of shape.