Interview with Karol Kozlowski, invited speaker at ICM 2022

Link to the virtual ICM 2022 talks

What is your field of study?

I carry research in mathematical physics. It is a branch of mathematics that aims to rigorously establish results and conjectures taking their root in physics’ problems. I am mainly interested in a sub-branch of mathematical physics called quantum integrable models. This refers to a specific class of models that can be solved explicitly and which arise in one-dimensional quantum physics or two-dimensional statistical physics. This exact solvability means that many observables of physical interest can be computed in closed form by using ingenious algebraic structures present at the root of these models. As such, they offer an unprecedented possibility of studying a vast amount of phenomena that are still barely understood on a satisfactory level of rigour, in particular those related to phase transitions and especially to the arisal of a universal behaviour. I work on these questions whose resolution combines a rich arsenal of mathematical tools. I specialise in developing methods of asymptotic analysis of all kinds that can be applied to this kind of question.

What led you to take up mathematics?

At the beginning of my academic studies, I was spellbound by the utter simplicity of understanding reality – or at least certain aspects thereof – that is offered by physics through a mathematisation of reality. The finer or more complete a physical theory is, the more advanced the mathematical concepts required to formulate and manipulate it become evoled. Upon pushing my mathematical education further, I understood that what I enjoyed most was the interface between physics and mathematics where one seeks to obtain, in full rigour, results that can be applied to a given physics issued problem. In particular, I was amazed by the beautiful vastness of tools and concepts that one is bound to handle so as to bring the resolution of a physics’s problem to the end.

What is a truly significant result for you? Or an "elegant" solution? 

I do trust that a result is significant when it produces an important modification of a discipline's established preconceptions. This means that, significance not only demands that the result has been long awaited, or that its establishing required some ingenious or  technically challenging proof, but also that the result was not necessarily expected, or that it was even unthinkable given the conception of things that preceded it.

I think that in order to consider a result to be deep, the latter it must open up to a plethora of applications, and allow one to go beyond the borders of what was possible before its existence.

The elegance of a proof, just as the very concept of beauty, obviously depends on the person studying it. In particular, the broadness, level and kind of mathematical knowledge – in the sense of the field in which one specialises – will play a great role in the appreciation of this elegance.

In my case, I find a solution to be elegant if it underscores the synergy of the mathematical arsenal it employed. In other words, this refers to results that involve the imbrication of several concepts that are a priori quite distant from one another.