Research

My research is in arithmetic dynamics — the study of number-theoretic and arithmetic-geometric questions that arise from iterating dominant self-maps of projective varieties — and more broadly in number theory and arithmetic geometry. Recurring themes in my work include moduli spaces of dynamical systems, post-critically finite maps, Galois theory of iterated preimages (arboreal representations), Belyi maps, and dynamical Mahler measure.

For the complete and current list, see Google Scholar, arXiv, or ORCID.

Preprints and papers

  • P. H. Lee, M. Manes, N. X. Truong, “Local fields, iterated extensions, and Julia sets,” submitted. (arXiv version)
  • J. Anderson, M. Manes, and B. Tobin, “Some applications of dynamical Belyi polynomials,” La Matematica 3 (2024), no. 3, 891–906. (arXiv version)
  • A. Carter, M. Lalín, M. Manes, and A. B. Miller, “Dynamical Mahler measure: a survey and some recent results,” in Research Directions in Number Theory, 219–252. (arXiv version)
  • A. Carter, M. Lalín, M. Manes, A. B. Miller, and L. Mocz, “Two-variable polynomials with dynamical Mahler measure zero,” Res. Number Theory 8 (2022), 22 pp. (arXiv version)
  • J. Anderson, M. Manes, and B. Tobin, “Cubic post-critically finite polynomials defined over ℚ,” ANTS XIV — Proceedings of the Fourteenth Algorithmic Number Theory Symposium, Open Book Series 4 (2020), 23–38. (arXiv version)
  • X. Faber, M. Manes, and L. Walton, “An alternate proof of idempotent relations among periodic points and quotients,” Journal of Number Theory 205 (2019), 44–49. (arXiv version)
  • R. Benedetto, P. Ingram, R. Jones, M. Manes, J. H. Silverman, and T. Tucker, “Current trends and open problems in arithmetic dynamics,” Bulletin of the AMS 56 (2019), no. 4, 611–685. (arXiv version)
  • J. Juul, H. Krieger, N. Looper, M. Manes, B. Thompson, and L. Walton, “Arboreal representations for rational maps with few critical points,” in Research Directions in Number Theory: Women in Numbers IV, Springer (2019), 133–151. (arXiv version)
  • M. Manes, G. Melamed, and B. Tobin, “Dessins d’enfants for single-cycle Belyi maps,” in Research Directions in Number Theory: Women in Numbers IV, Springer (2019), 153–159. (arXiv version)
  • M. Manes and B. Thompson, “Periodic points in towers of finite fields for polynomials associated to algebraic groups,” Rocky Mountain Journal of Mathematics 49 (2019), no. 1, 171–197. (arXiv version)
  • M. Manes and J. H. Silverman, “A classification of degree 2 semi-stable rational maps ℙ² → ℙ² with large finite dynamical automorphism group,” Annales de la Faculté des Sciences de Toulouse (6) 28 (2019), no. 4, 733–811. (arXiv version)
  • J. Anderson, I. Bouw, O. Edjer, N. Girgin, V. Karemaker, and M. Manes, “Dynamical Belyi maps,” in Women in Numbers Europe 2, Springer (2018), 57–82. (arXiv version)
  • A. Bridy, P. Ingram, R. Jones, J. Juul, A. Levy, M. Manes, S. Rubinstein-Salzedo, and J. H. Silverman, “Finite ramification for preimage fields of postcritically finite morphisms,” Mathematical Research Letters 24 (2017), no. 6, 1633–1647. (arXiv version)
  • F. Balestrieri, J. Berg, M. Manes, J. Park, and B. Viray, “Insufficiency of the Brauer-Manin obstruction for Enriques surfaces,” in Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop, Springer (2016), 1–31. (arXiv version)
  • I. Bouw, J. Cooley, K. Lauter, E. Lorenzo Garcia, M. Manes, R. Newton, and E. Ozman, “Bad reduction of genus-3 curves with complex multiplication,” in Women in Numbers Europe, Springer (2015), 109–151. (arXiv version)
  • X. Faber, M. Manes, and B. Viray, “Computing conjugating sets and automorphism groups of rational functions,” Journal of Algebra 423 (2015), 1161–1190. (arXiv version)
  • B. Hutz and M. Manes, “The field of definition for dynamical systems on ℙᴺ,” Bulletin of the Institute of Mathematics Academia Sinica 9 (2014), 585–601. (arXiv version)
  • D. Lukas, M. Manes, and D. Yap, “A census of quadratic post-critically finite rational functions defined over ℚ,” LMS J. Comput. Math. 17 (2014), 314–329. (arXiv version)
  • A. Levy, M. Manes, and B. Thompson, “Uniform bounds for pre-periodic points in families of twists,” Proceedings of the AMS 142 (2014), 3075–3088. (arXiv version)
  • R. Jones and M. Manes, “Galois theory of rational functions with nontrivial automorphisms,” Comment. Math. Helv. 89 (2014), 173–213. (arXiv version)
  • M.-J. Bertin, A. Feaver, J. Fuselier, M. Lalín, and M. Manes, “Mahler measure of some singular K3-surfaces,” in Women in Numbers 2, Contemp. Math. 606 (2013), 237–247. (arXiv version)
  • M. Manes and Y. Yasufuku, “Explicit descriptions of quadratic maps on ℙ¹ defined over a field K,” Acta Arithmetica 148 (2011), 257–267. (arXiv version)
  • B. Malmskog and M. Manes, “Ramified covers of graphs and the Ihara zeta functions of certain ramified covers,” in WIN — Women in Numbers, Fields Institute Communications 60 (2011), 237–247.
  • B. Malmskog and M. Manes, “Almost divisibility in the Ihara zeta functions of certain ramified covers of q+1-regular graphs,” Linear Algebra and Its Applications 432 (2010), 2486–2506.
  • X. Faber, B. Hutz, P. Ingram, R. Jones, M. Manes, T. Tucker, and M. Zieve, “Uniform bounds on pre-images under quadratic dynamical systems,” Mathematical Research Letters 16 (2009), 87–101. (arXiv version)
  • M. Manes, “Moduli spaces for families of rational maps on ℙ¹,” Journal of Number Theory 129 (2009), 1623–1663. (arXiv version)
  • M. Manes, “ℚ-rational cycles for degree-2 rational maps having an automorphism,” Proceedings of the London Mathematical Society 96 (2008), 669–696.

Books and edited volumes

  • A. Cuoco, K. Waterman, B. Kerins, E. Kaczorowski, and M. Manes, Linear Algebra and Geometry, AMS/MAA Textbooks, vol. 46, American Mathematical Society (2019).
  • J. Balakrishnan, A. Folsom, M. Lalín, and M. Manes (eds.), Research Directions in Number Theory: Women in Numbers IV, Springer (2019).
  • C. David, M. Lalín, and M. Manes (eds.), Women in Numbers 2: Research Directions in Number Theory, Contemporary Mathematics 606, AMS (2013).