Research

Research Interests: Mathematical physics on fractals and graphs; Quantum computing and information theory; Spectral theory on weakly self-similar systems and quasicrystals; Random Schrödinger Operators.

Koch Snowflake with boundary and interior energies

This research project is joint work with L. Rogers, A. Teplyaev, M. Gabbard, and C. Lima and was supported in part by NSF DMS-1659643 and DMS-1613025. The project results are published in Fractals in engineering: theoretical aspects and numerical approximations, (eds) M. Lancia, A. Rozanova-Pierrat, SEMA SIMAI Springer Series, vol 8, Springer, Cham (2021).
DOI:10.1007/978-3-030-61803-2_4

Spectral decimation of a self-similar version of almost Mathieu-type operators

This research project is joint work with R. Balu, K. Okoudjou, A. Teplyaev and was supported in part by ARO grant W911NF1910366, the National Science Foundation under Grant No. DMS-1814253, NSF DMS grant 1613025, and by the Simons Foundation. A preprint is available under: arXiv:2105.09896

Solitons, Lax Pairs, Hamiltonian systems, Toda lattices on weighted graded graphs

This research project is joint work with M. Derevyagin, G. Dunne, and A. Teplyaev and supported in part by the University of Connecticut Research Excellence Program, by DOE DE-SC0010339, NSF DMS 1613025, and NSF DMS 2008844. The project results are published in Journal of Mathematical Physics 62, 042204 (2021). DOI: 10.1063/5.0025475

Spectra of perfect state transfer Hamiltonians on fractal-like graphs

This research project is joint work with M. Derevyagin, G. Dunne, and A. Teplyaev and is supported in part by the University of Connecticut Research Excellence Program, by DOE DE-SC0010339 and NSF DMS 1613025. The project results are published in Journal of Physics A: Mathematical and Theoretical 54(12), 125301 (2021). DOI: 10.1088/1751-8121/abc4b9

Perfect quantum state transfer on diamond fractal graphs

This research project is joint work with M. Derevyagin, G. Dunne, and A. Teplyaev and was supported in part by the University of Connecticut Research Excellence Program, DOE DE-SC0010339, and NSF DMS 1613025. The project results are published in Quantum Information Processing 19, 328 (2020). DOI: 10.1007/s11128-020-02828-w

Gaps labeling theorem for the Bubble-diamond self-similar graphs

This research project is joint work with L. Rogers, A. Teplyaev, E. Melville, and N. Nagabandi. This research was supported in part by the University of Connecticut Research Excellence Program, by DOE grant DE-SC0010339, and by NSF DMS grants 1613025 and 2008844. The work of G. Mograby was additionally supported by ARO grant W911NF1910366. (In preparation).

Spectral decimation of piecewise centrosymmetric Jacobi operators

This research project is joint work with R. Balu, K. Okoudjou, A. Teplyaev and was supported in part by ARO grant W911NF1910366, the National Science Foundation under Grant No. DMS-1814253, NSF DMS grant 1613025, and by the Simons Foundation (In preparation).

Harmonic gradients on higher dimensional Sierpinski Gaskets

This research project is joint work with L. Rogers, L. Brown, G. Ferrer, and K. Sangam and was supported in part by NSF DMS-1659643 and DMS-1613025. The project results are published in Fractals Vol. 28, No. 06, 2050108 (2020)
DOI:
10.1142/S0218348X2050108X

Spectra of three-peg Hanoi towers graphs

This research project is joint work with L. Rogers, B. Hungar, M. Phelps, and J. Wheeler and was supported in part by NSF Grants DMS-1659643 and DMS-1613025. A preprint is available under:
arXiv:2107.02697