Nalini Anantharaman is a french mathematician working in the fields of dynamical systems, partial differential equations and mathematical physics.

Her early works deal with the counting of closed geodesics on hyperbolic surfaces in a given homology class. She gave a full asymptotic expansion for the counting function following dynamical methods introduced by D. Dolgopyat, when the homology class lies in the interior of the set of winding cycles of the invariant measures of the geodesic flow. She also gave estimates when the homology class is in the boundary of this set. In that case, the problem is connected to the zero temperature limit in the theory of Markov processes and maximizing measures in the Aubry-Mather theory of Lagrangian systems.

She then got interested in semi-classical analysis and used entropy methods to study the weak limits of the sequence of probability measures $$ |\psi_k|^2 d\hbox{vol}$$ where $\psi_k$ are the eigenfunctions of the Laplacian defined on a negatively curved compact manifold. The quantum unique ergodicity conjecture asserts that the sequence should converge to the Liouville measure after a suitable lift of the measures to the unit tangent bundle of the manifold. She showed that any cluster points of the sequence must have positive entropy, thus ruling out a convergence to the Dirac mass on some closed orbit. So this is a remarkable application of ergodic theory to the study of the linear wave equation and Schrodinger equation. See a survey of P. Sarnak for additional details.

More recently, she studied related problems on billiards and regular graphs. Interesting pictures of cardioid billiards can be found in her joint work with Arnd Backer. Her webpage contains a few of her lectures in video format.