In this project, we are investigating a version of Fried's conjecture, relating analytic torsion to the Ruelle zeta function, that incorporates symmetries.
Equivariant version of Fried’s conjecture
Analytic torsion was introduced by Ray and Singer as a way to understand Reidemeister-Franz torsion using analytical methods. Cheeger and Müller independently proved that these two concepts are equal. The Ruelle dynamical zeta function is a topological tool used to count closed curves of flows on compact manifolds. The Fried conjecture states that, for certain types of flows, the Ruelle dynamical zeta function has a specific value at zero, and the absolute value of this number equals analytic torsion.
With Hemanth Saratchandran, we define equivariant versions of analytic torsion and of the Ruelle dynamical zeta function, which incorporate group actions. This leads to the research question: under what conditions is the equivariant version of Fried’s conjecture true?
Extend fundamental mathematics
With this research, we hope to extend fundamental mathematical theories to include symmetries, potentially leading to new insights and deeper understanding of the relationships between topology, geometry, and analysis.