Dr Melissa TacyOffice:
** Melissa left the Department in 2020 **
My research lies within the intersection of microlocal analysis, semiclassical analysis and harmonic analysis. I am particularly interested in problems relating to the area of quantum chaos. The driving question of quantum chaos asks what echoes of classical dynamics (particularly chaotic dynamics) are apparent in high energy quantum systems. A major focus of this field is the study of high energy eigenfunctions, or quasimodes (approximate eigenfunctions), of operators such as the Laplace-Beltrami operator. Such functions arise as stationary states of the Schrodinger equation where their eigenvalue is interpreted as the energy of that state.
Here are links to some of my most recent talks
Eigenfunction concentration and its connection to geometry, Banff workshop, Probing the Earth and the Universe with Microlocal Analysis, April 2019.
Does it matter what we randomise?, Banff workshop, Around Quantum Chaos, July 2018.
Equidistribution of random waves on shrinking balls IAS emerging topics workshop, Nodal sets of Eigenfunctions, February 2017.
- M. Tacy, L^p estimates for joint quasimodes of semiclassical pseudodifferential operators whose characteristic sets have kth order contact, arXiv:1909.12559.
- M. Tacy, Stationary phase type estimates for low symbol regularity, arXiv:1902.02409.
- X. Han, M. Tacy, Equidistribution of random waves on small balls, In press, Communications in Mathematical Physics, arXiv1611.05983.
- M. Tacy, L^p estimates for joint quasimodes of semiclassical pseudodifferential operators, Israel Journal of Mathematics, 232(1):401–425, 2019 arXiv version.
- Z. Guo, X. Han, M. Tacy, L^p bilinear quasimode esimates, The Journal of Geometric Analysis, 29(3):2242–2289, 2019, arXiv version.
- A. Barnett, A. Hassell, M. Tacy, Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues, Duke Mathematical Journal 167(16):3059–3114, 2018, arXiv version
- M. Tacy, The quantisation of normal velocity does not concentrate on hypersurfaces, Communications in PDE 42(11):1749-1780, 2017 arXiv version.
- M. Tacy, A note on constructing sharp examples for L^p norms of eigenfunctions and quasimodes near submanifolds, Proceedings of the AMS 146(7):29-9-2924, 2018 arXiv version.
- X. Han, M. Tacy, (appendix by J. Galkowski), Sharp norm estimates of layer potentials and operators at high frequency, Journal of Functional Analysis, 269(9):2890-2926, 2015 arXiv version.
- A. Hassell, M. Tacy, Improvement of eigenfunction estimates on manifolds of nonpositive curvature, Forum Mathematicum, 27(3):1435-1451, 2015 arXiv version.
- A. Hassell, M. Tacy, Semiclassical Lp Estimates of Quasimodes on Curved Hypersurfaces, Journal of Geometric Analysis, 22(1):74-89, 2012 arXiv version.
- M. Tacy, Semiclassical L^p estimates of quasimodes on submanifolds, Communications in Partial Differential Equations, 35(8):1538-1562, 2010 arXiv version.
- M Tacy, Strichartz estimates for non-unitary energy bounds and eigenfunction estimates, Proceedings of the 9th International Conference on Mathematical and Numerical Aspects of Waves Propagation, 222-223, 2009.