turkmath.org

The theory of partitions has been a vibrant and highly active subject of study since Euler. The problems from this topic usually lie in between the number theory and combinatorics, and problems have been a great source of research and progress in many fields spanning from algebraic geometry to modular forms and special functions. In this talk, we will start with a brief introduction to the theory of partitions. We will move on to a combinatorics result (Capparelli's partition theorems) that is rooted from vertex operator algebras in the elementary frame of partitions and q-series. We will discuss recent developments on these theorems by Kanade--Russell and Kursungoz. Then we will move onto a refinement of their approaches to present the first group of polynomial identities that imply Capparelli's partition theorems. We will finish the talk by presenting multiple analytic implications of the results coming from this study and the developed techniques. This research is artly joint with Alexander Berkovich.