Final Project Resources

If you open this directory , you will find all the supporting files for a sample one-page outline and a sample final paper on Diffusion Limited Aggregation (1st of the listed topics below), provided by Professor Nestor Guillen who taught this class last year. Note how the sample final paper has an abstract, figures, bibliography, programming code (since the project has a programming aspect). Because the paper had so many figures, to keep the directory ``neat," all figures are in this subdirectory . This subdirectory has to be in the same directory as the TeX sourcecode, so that the TeX compilor knows how to retrieve the figures when needed.

Possible topics for final project using Markov chains.

  • Diffusion Limited Aggregation (chemistry, physics, biology).
    Kesten, Harry. ``Hitting probabilities of random walks on Zd." Stochastic Processes and their Applications 25 (1987): 165-184.
    Daccord, Gerard. ``Chemical dissolution of a porous medium by a reactive fluid." Physical review letters 58.5 (1987): 479.
    Witten Jr, T. A., and Leonard M. Sander. ``Diffusion-limited aggregation, a kinetic critical phenomenon." Physical review letters 47.19 (1981): 1400.

  • Outliers in Markov Chains and applications,
    Chikina, Maria, Alan Frieze, and Wesley Pegden. ``Assessing significance in a Markov chain without mixing." Proceedings of the National Academy of Sciences 114.11 (2017): 2860-2864.

  • Election redistricting (social science):
    Mattingly, J and Vaugh, C. ``Redistricting and the will of the people.'' https://arxiv.org/abs/1410.8796
    Bernstein, Mira, and Moon Duchin. ``A formula goes to court: Partisan gerrymandering and the efficiency gap." Notices of the AMS 64.9 (2017): 1020-1024.

  • Population growth and contagion models (chemistry, biology, physics).
    Renshaw, Eric. Modelling biological populations in space and time. Vol. 11. Cambridge University Press, 1993. (Available in UMass libraries). Chapter 6.
    Scanlon, Todd M., et al. ``Positive feedbacks promote power-law clustering of Kalahari vegetation.'' Nature 449.7159 (2007): 209.
    Dodds, Peter Sheridan, and Duncan J. Watts. ``A generalized model of social and biological contagion." Journal of theoretical biology 232.4 (2005): 587-604..

  • Paretto distributions and their applications (biology, economics,ecology, physics).
    Klass, Oren S., et al. ``The Forbes 400 and the Pareto wealth distribution.'' Economics Letters 90.2 (2006): 290-295.

  • Random walks/web surfing and Google Page Rank (computer science).
    Duhan, Neelam, A. K. Sharma, and Komal Kumar Bhatia. ``Page ranking algorithms: a survey." Advance Computing Conference, 2009. IACC 2009. IEEE International. IEEE, 2009.
    Page, L., Brin, S., Motwani, R., \& Winograd, T. (1999). The PageRank citation ranking: Bringing order to the web. Stanford InfoLab.

  • Cryptography via Markov Chains (computer science):
    The first two sections from: Diaconis, Persi. ``The markov chain monte carlo revolution." Bulletin of the American Mathematical Society 46.2 (2009): 179-205.
    Chen, J and Rosenthal, J. ``Decrypting classical cipher text using Markov chain Monte Carlo.'' Stat Comput (2012) 22:397–413.

  • Hidden Markov Models (computer science, physics, economics, biology):
    Rabiner, Lawrence R., and Biing-Hwang Juang. ``An introduction to hidden Markov models.'' ieee assp magazine 3.1 (1986): 4-16.

  • Markov chain models for markets (economics, finance):
    Norberg, Ragnar. A Markov chain financial market. Centre for Actuarial Studies, Department of Economics, University of Melbourne, 1999.

  • Markov chains and Monte Carlo for stock options (finance):
    Longstaff, F.A.; Schwartz, E.S. (2001). "Valuing American options by simulation: a simple least squares approach". Review of Financial Studies. 14: 113–148.
    Landauskas, M; Valakevicius, E. (2011) ``Modelling of stock prices by the Markov Chain Monte Carlo methods." https://www.mruni.eu/upload/iblock/7ed/Landauskas.pdf


    Here are two websites devoted to the math of gerrymandering. Papers linked from these sites include the Markov Chains and Geometry approaches to the problem.
    Metric geometry and gerrymandering group.
    Quantifying gerrymandering.



    Here is a list of papers using game theory and social choice theory instead of Markov chains, to assess election processes. The first one is a book, and the rest are articles.


    Title: Evaluation and Optimization of Electoral Systems. Authors: Pietro Grilli di Cortona , Cecilia Manzi , Aline Pennisi , Federica Ricca and Bruno Simeone Website: https://epubs.siam.org/doi/book/10.1137/1.9780898719819
    Stackelberg voting games by Conitzer and Xia
    Equilibria of plurality voting with abstentions by Desmedt and Elkind
    Complexity of and Algorithms for Borda manipulation by Davies et al.
    Convergence to Equilibria in Plurality Voting by Meir et al.
    Cloning in Elections: Finding the Possible Winners by Elkind et al.
    Computer-aided Proofs of Arrow's and Other Impossibility Theorems by Tang and Lin
    Coalitional Voting Manipulation: A Game-Theoretic Perspective by Bachrach et al.
    Truth, Justice, and Cake-Cutting by Chen et al.