On my decision to study automatic groups

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In this post I would like to relate some of the reasons why I decided to study automatic groups near the end of my PhD term. I have always been fascinated about patterns occurring in universe. I marvel at the objects of ever-increasing complexities populating our universe. Probably, that is the reason why I chose to study mathematics. For me mathematics is a study of patterns. But these patterns are not limited to our physical universe. Most of the time, they live in some abstract universes. In those universes, patterns live in different continents which can interpreted as fields like Algebra, Analysis, Geometry, Topology, Mathematical Logic and much more. Some patterns occur at different continents and travel back and forth along some bridges. One of the most famous among such bridges is Langlands program. Pondering on such bridges gives me a deep sense of wonder.

In my third year, I mainly focused on a notion of algorithmic randomness. A randomness occurs when some patterns are not extractable by computational machines. These machines could be Turing machines, pushdown automata or even finite automata. I studied randomness with respect to finite automata. First I focused on randomness of infinite binary sequences using the measure-typicalness paradigm. Those studies culminated in the preprint titled 'Automatic Randomness Tests'. Then I focused my attention on randomness of formal languages. The standard approach was to transform given formal language to an infinite binary sequence in some canonical way and measure randomness of the sequence. I used ideas form Algorithmic learning theory and Algorithmic randomness to define randomness for formal languages. The preprint is finished, hopefully I can post it onto my webpage soon.

When I was approaching my fourth year, my supervisor, Frank Stephan, suggested for me to study automatic groups. These are groups which could be presented in automata-theoretic framework. I found the idea great for few reasons. First, groups are closely related to symmetries of various spaces. Having been studying randomness for whole year, it would be a good idea to study objects which are epitomes of symmetry. Secondly, automatic groups are closely related to Geometric group theory (GGT). In fact the domain automatic groups was heavily investigated by GGT community. Having been working in mainly analytic and symbolic domains, switching to something geometric feels exciting. One should realize that I have to learn lots of new things from geometry, so that I might stand a chance of having anything publishable at the end of my investigations. However, even half-publishable content together with my previous works should make a valid thesis, I hope. With these thoughts in my mind, I am starting a journey towards understanding automatic groups and their geometries. 

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