Melissa Tacy 

Research interests 

What happens when a quantum butterfly flaps its wings?

Chaos theory tells us that small changes in initial conditions can lead to widely divergent behaviour. But what if your system is a quantum system? The uncertainty principle tells us that you can't know any initial conditions exactly so how can we merge the classical ideas of chaos into the world of quantum uncertainty. This philosophical question underlies my research area.

I study the semiclassical (or high energy limit) of systems and try to find echoes of classical behaviour. I am particularly interested in high energy problems in particular concentration of eigenfunctions, and in the harmonic analysis necessary to study such concentrations.

  In June 2017 I will move the the University of Otago.

Here is my CV (updated 28-11-16)

Papers

Sharp norm estimates for layer potentials and operators at high frequency (with Jeffrey Galkowski and Xiaolong Han), Journal of Functional Analysis, 269(9):2890-2926, 2015
Improvements of eigenfunction estimates on manifolds of non-positive curvature (with Andrew Hassell), Forum Mathematicum, 27(3)1435-1451, 2015
Semiclassical L^p estimates of quasimodes on curved hypersurfaces, (with Andrew Hassell), Journal of Geometric Analysis, 22(1):74-89, 2012
Semiclassical L^p  estimates of quasimodes on submanifolds, Communications in Partial Differential Equations, 35(8):1538-1562, 2010
Strichartz estimates for non-unitary energy bounds and eigenfunction estimates, Proceedings of the 9th International Conference on Mathematical and Numerical Aspects of Waves Propagation (p222-223), Pau, 15-19 June 2009

Preprints

 Equidistribution of random waves on small balls (with Xiaolong Han)
A note on constructing sharp examples for $L^{p}$ norms of eigenfunctions and quasimodes near submanifolds
Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues (with Alex Barnett and Andrew Hassell)
The quantisation of normal velocity does not concentrate on hypersurfaces
L^p bilinear quasimode estimates (with Zihua Guo and Xiaolong Han)

Recent Talks

Applications of semiclassical analysis in PDE
Bilinear L^p estimates for quasimodes
Quantisation and localisation of dynamical observables

You are welcome to use (with attribution) any of the conceptual images I have produced for these talks. Click here to download a zipped directory.

Links to Videos

Here are some links to videos of me giving research talks
Directional localization and toral eigenfunctions
What we can do with waves (as part of the mathematical conversation series at IAS)
Semiclassical eigenfunction estimates
L^p concentration of semiclassical quasimodes

Postal Address 

Mathematical Sciences Institute
Australian National University
Canberra Australia 2601