Alexander G. Melnikov

Office: INMS 3.05
Address: The Institute of Natural and Mathematical Sciences, Private Bag 102 904 NSMC Albany 0745 AUCKLAND
Email: alexander.g.melnikov@gmail.com

Welcome to my professional personal webpage. I am a Senior Lecturer at Massey University, Auckland. My research area is computability theory and its applications to algebra, logic, and computer science.

My area is computable mathematics. It blends the abstract theory with computation with mathematical logic, algebra, topology, and theoretical computer science. In pure mathematics we often manipulate with abstract objects that exist because if they did not then we'd arrive at a contradiction. In computable mathematics we look at whether such objects have an algorithmically effective presentation. Indeed, such an effectivasation often becomes a necessity if one needs to apply mathematics in computational practice.

For example, we look at abstract groups and ask which ones can be represented by a computer program. A group is an abstraction for symmetry, and it is heavily used throughout science (physics, chemistry, etc.). To analyse the computational content of abstract objects, we usually use Turing machines or finite automata, or some other models of computation. We sometimes restrict resources of our computation (e.g. put bounds on space or the runtime). However, in computable mathematics we are mostly interested whether an object is computable in principle, i.e. without any restrictions. It is still unclear why, but many algorithms that we develop for purely theoretical purpuses are actually feasibly (i.e. efficiently) computable. This phenomenon can be partially explained by so-called generic computability and parametrised copmplexity but we are getting off the subject.

Investigations in our field have deep connections with other topics of mathematics. Examples include the famous Godel's Incompleteness Theorem in logic, Higman's Embedding Theorem in group theory, the old results of van der Waerden on explicit fields from algebraic geometry, the famous works of Mal'cev on applied universal algebra and Turing on analysis and analytic number theory, and many other results. On the other hand, abstract computability theory research provides a theoretical base for more applied studies. In fact, basic results of effective algebra are commonly used in the standard computational applications including MathLab (e.g., factorisation algorithms for polynomial rings).

In 2006 I graduated from The Department of Mechanics and Mathematics, Novosibirsk State University, at the age of 21. For those unfamiliar with the post-Soviet hierarchy of universities, this is one of the top two or three mathematics departments in Russia, and definitely the top logic school in Russia. I had the highest average score among all the 2006 graduates, which was the highest possible 5.0 out of 5.0. I thought maybe it was a sign, and I should try going to academia. Very soon I realised that doing research has almost nothing in common with undergraduate studies, but it was too late; I got addicted to research. In 2008 I defended a MSc thesis in Mathematics and Theoretical Computer Science under the supervision of the famous Sergey S. Goncharov. In 2008, I moved to New Zealand to do a PhD with Bakhadyr Khoussainov and Andre Nies at The University of Auckland.

I much enjoyed working with both advisors who are excellent world-class mathematicians. Also, my wife Tanya and I greatly appreciated the prestigious University of Auckland International Doctoral Scholarship which both of us had during our PhD study in Auckland. Back then the University of Auckland had -- perhaps, still has -- a highly bizarre restriction on the size of a PhD thesis. It had to be at most 200 pages long, including references. I had no life during my PhD, and I ended up having enough material for almost two such theses. Sergey S. Goncharov suggested me to write and submit a Candidate of Science dissertation based on the extra material, and I did. Although it may sound like a smart decision, I would not recommend writing two theses simultaneously to anyone who wants to have some life beyond their office.

Shortly after my CSc defence in Russia, and a few months before my PhD confirmation in Auckland, I joined Nanyang Technological University as a postdoc. Tanya was a postdoc there as well; that was a perfect solution to the undecidable two-body problem! At NTU I worked mainly with Keng Meng Ng. We still collaborate a lot to this day. Since I was already working in Singapore, my PhD confirmation exam had to be done via Skype. The connection was terrible, but otherwise it was an interesting and memorable experience.

While I was working in Singapore I kept looking for job opportunities in New Zealand. This is the country that I love; I was dreaming of coming back. After one year in Singapore, I returned to New Zealand to work with Rod Downey and Noam Greenberg at Victoria University of Wellington. Both of them are world-class mathematicians, having had infinitely many Marsden grants and uncountably many postdocs who themselves are now professors at top universities around the globe.

As much as I enjoyed being a postdoc, it was the time for me to start worrying about a permanent job. Sadly, it was absolutely clear that New Zealand did not have enough jobs in mathematics, let alone pure mathematics. I applied for all of them, where "all'' means "two'', and I was lucky to get an offer from Massey. Around the same time I got a postdoc offer from UC Berkeley, which is considered by many the top logic school in the US. Thanks to Gaven Martin, Massey let me go to the US. After working with the gifted Antonio Montalban in California for about a year, I returned to New Zealand in 2015 to join Massey, Auckland. I am still here and I am not planning on moving in the foreseeable future, at least as of the date below.