Arithmetic and Complex Algebraic Geometry

Mathematical sciences


Host Department
Department of Mathematics



Research Topic B

Abstract of the proposed UNIPhD research project


The Deligne-Mumford compactified moduli space that classifies stable algebraic curves of genus g with n marked points is a fundamental object in algebraic geometry and its cohomology ring contains information required to formulate intersection theory. Since the cohomology ring is far too complicated to investigate, we are interested in the tautological ring, which is the smallest subring of the cohomology ring that is constructed using only natural geometrical objects of the moduli space. The tautological ring is generated by a finite set of additive generators defined by combinations of psi- and kappa-classes on their corresponding strata. The main goal is to find all linear relations among these additive generators; such relations are called tautological relations.
There has been significant progress in constructing these tautological relations, the most renowned of which are the Pixton’s relations, proved by Pandharipande-Pixton-Zvonkine in 2015. For this project, we are focusing on a conjectural set of tautological relations which implies the strong DR/DZ equivalence conjecture. The strong DR/DZ equivalence conjecture, formulated by Buryak-Dubrovin-Guéré-Rossi, states that for any semi-simple cohomological field theory (CohFT), the Dubrovin-Zhang (DZ) and Double- Ramification (DR) integrable hierarchies are related by a normal Miura transformation, which the authors completely identify in terms of the partition function of the CohFT. The conjecture holds for simpler CohFTs and for specific varieties of a Gromov-Witten theory. We would like to provide more evidence for and eventually prove the DR/DZ equivalence conjecture. It would also be interesting to investigate the connection between Pixton’s relations and the DR/DZ conjectural. Furthermore, generalizations of DR hierarchies and Givental-Teleman’s classification by considering partial CohFTs and F-CohFTs are not studied in depth. We would like to expand and develop this theory.



Short bio

I was born in Malaysia and am currently a PhD student in mathematics at the Tullio Levi-Civita department, specializing in arithmetic and complex algebraic geometry. I was previously a master’s student in mathematical physics at the Universit´e de Bourgogne, where I worked on cohomological field theory and Givental’s theory under the guidance of Prof Guido Carlet. I received my bachelor’s degree in physics from University of Tunku Abdul Rahman, where I focused my bachelor thesis on the differential geometric structures of general relativity and cosmology. I’ve always had a fascination in the mathematical foundations that underpin modern theoretical physics.
My current research goal is to better comprehend the enigmatic relationship between enumerative geometry and integrable systems. It is an exciting topic to study as it offers powerful mathematical results as well as important implications for string theory and quantum gauge theory. In my spare time, I enjoy discussing and reading about mathematics, physics and philosophy, as I find it fun to break down abstract concepts and integrate them into the real world. Aside from intellectual pursuits, I enjoy going on group treks, participating in sports, meeting people from different cultures, and attending retreats.

Topic assigned to
Ishan Singh


Project documents