In his doctoral thesis, Mykola Pochekai has used ideas from Hodge theory to prove new cases of genus-one mirror symmetry. Hodge theory is also used for the seemingly unrelated mathematical field of combinatorics.
The main project of the thesis is about a new type of symmetry between algebraic complex geometry and symplectic geometry. The connection between the two fields was found in the area of string theory in the 1980s. Since then, several different approaches to making this connection mathematically rigorous have been developed. The thesis provides new examples of the phenomenon of genus-one mirror symmetry, although much more remains to be explained.
– It is not obvious that this type of connection should really exist. I have chosen the approach of algebraic geometry, but you could go from physics or analysis as well and apply the techniques to the same phenomena. Mirror symmetry is one of the five or six big topics in mathematics just now, you could say that it is fashionable, but I think there is a good reason for this. There are a lot of interpretations since we need to understand these phenomena and conceptualise them. In my opinion it gives additional value that it has originated in physics, it could be part of a larger story that we have not yet discovered.
Intersectional mathematics
The projects in Mykola’s thesis are mathematically intersectional, but belong mostly to algebraic geometry, dealing with notions of dimensions, symmetries and equalities. There are also elements of complex geometry, and techniques from arithmetic geometry, thoric geometry and partial differential equations. In the second part of the thesis Mykola use the same kind of theory, called Hodge theory, in an entirely different mathematical field: combinatorics. Here, new methods are developed for proving combinatorial conjectures for some specific cases of polytopes, which are higher-dimensional polyhedron bodies.
In school, Mykola was mostly interested in programming. He participated in several school competitions and thought he would become a programmer. But when he started to study applied mathematics, as part of his programming studies, he found that he liked the theoretical parts more. Mykola asked his linear algebra teacher to give him a project and was thus introduced to operator algebras, and started to learn more about deep and theoretical mathematics.
Next stop South Korea
After his undergraduate education Mykola applied for as many PhD positions as he could. He was accepted at the University of Gothenburg for an analysis position and started his work, but when his supervisor moved and he had to choose another one, he decided to change the subject as well into algebraic geometry. That the thesis would be about mirror symmetry was more like a coincidence, but Mykola will continue in that direction in a three-year postdoc position at the Centre for Geometry and Physics at the Institute for Basic Science, South Korea. Many of its researchers work on mirror symmetry, and Mykola’s thesis project relates to what they do.
When Mykola moved to Gothenburg he knew almost nothing about its universities, but found the conditions to be surprisingly good. Not least, the complex geometry research group is unexpectedly powerful, with many publications in top ranked journals. That the Department of Mathematical Sciences belongs to two different universities was hard to grasp and has the consequence of many systems to be mastered, often with two accounts, one for each university. On the other hand, the conditions regarding office space and work equipment are very good. It is maybe not very common that the supervisor and examiner are different people, but Mykola finds the divided responsibilities to be a good thing. He also likes the teaching experience included in the five-year position.
– I begin my postdoc on December 1, so this is my last month in Gothenburg. But I hope I will be able to come back as a guest, to keep the contacts and to further the work that we have accomplished.
Mykola Pochekai will defend his PhD thesis Hodge Theory in Combinatorics and Mirror Symmetry on November 14 at 13.15 in lecture hall Pascal, Hörsalsvägen 1. Supervisor is Dennis Eriksson, assistant supervisors are Christian Johansson and Magnus Goffeng.
- Doctoral Student, Algebra and Geometry, Mathematical Sciences