Selected open problems for 5010 Fall 2018

1. useful lecture notes about analysis on weighted graphs

2. pure point spectrum on some infinite weakly self-similar graphs

3. resistance estimates on some self-similar graphs, such as the octacarpet

4. spectral decimation and HKE on self-similar weighted graphs (such as pq-model)

5. AC currents on graphs

6. hypergraphs and elements of differential geometry

7. approximations of non-smooth structures by graphs

plan for Friday September 7, 2018

plan for Wednesday September 12, 2018

With some links:

1. useful lecture notes about analysis on weighted graphs

2. pure point spectrum on some infinite weakly self-similar graphs, based on

L. Malozemov and A. Teplyaev, Pure point spectrum of the Laplacians on fractal graphs. Journal of Functional Analysis 129 (1995), 390–405. pdf file

+ long list of other works

3. resistance estimates on some self-similar graphs, such as the octacarpet, based on

A. Brzoska, D.J. Kelleher, H. Panzo, A. Teplyaev, Dual graphs and modified Barlow–Bass resistance estimates for repeated barycentric subdivisions, Discrete & Continuous Dynamical Systems – S February 2019, Volume 12, Issue 1 arXiv:1505.03161

B. Boyle, K. Cekala, D. Ferrone, N. Rifkin and A. Teplyaev, Electrical Resistance of N-gasket Fractal Networks. Pacific Journal of Mathematics 233 (2007), 15–40. pdf file

+ long list of other works

4. spectral decimation and HKE on self-similar weighted graphs (such as pq-model)

U.Andrews, J.P.Chen, G.Bonik, R.W.Martin, A.Teplyaev, Wave equation on one-dimensional fractals with spectral decimation. J. Fourier Anal. Appl. 23 (2017) 994–1027. doi:10.1007/s00041-016-9494-6 arXiv:1505.05855 wave equations animation

J.P.Chen, A.Teplyaev, Singularly continuous spectrum of a self-similar Laplacian on the half-line, Journal of Mathematical Physics, 26 May 2016 Vol.57, Issue 5 DOI: 10.1063/1.4949471 arXiv:1509.08875

Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, Michael Khalil, Poorak Mody, Benjamin Steinhurst, A. Teplyaev Vibration modes of 3n-gaskets and other fractals J. Phys. A: Math. Theor. 41 (2008) 015101 (21pp). pdf file.pdf

Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, Michael Khalil, Poorak Mody, Benjamin Steinhurst, A. Teplyaev Vibration Spectra of Finitely Ramified, Symmetric Fractals Fractals 16 (2008), 243–258. pdf file.pdf

L. Malozemov and A. Teplyaev, Self-similarity, operators and dynamics. Mathematical Physics, Analysis and Geometry 6 (2003), 201–218. pdf file

+ long list of other works

5. AC currents on graphs

Joe P. Chen, Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephew Loew, Alexander Teplyaev, Power dissipation in fractal AC circuits. J. Phys. A: Math. Theor. 50 (2017) 325205 (20pp) doi.org/10.1088/1751-8121/aa7a66 arXiv:1605.03890