• Mathematical Foundations of Deep Learning.
  I study neural networks as objects of pure mathematics: parameterized families of functions whose geometry, topology, symmetries and degeneracies can be analyzed via theorems.  My current research develops the mathematical theory of neural network architectures from the viewpoint of geometry, topology, and dynamical systems.  For a fixed architecture, the space of weights and biases determines a realization map to a corresponding space of functions.  Much of my work investigates the structure of this map: its fibers, its singularities, and the ways in which architectural constraints are reflected in the functions a network can represent.

• Dynamical Systems.
  My work on dynamical systems studies iterated maps as objects whose long-term behavior is organized by topology, combinatorics, and complex geometry.  I am especially interested in one-dimensional real and complex dynamics, where a map gives rise to richly structured parameter spaces, invariant sets, entropy, symbolic dynamics, and orbit portraits.  Much of my work investigates how combinatorial data encodes dynamical behavior: for example, how kneading theory, Thurston sets, and related invariants capture the structure of orbits and determine features of the underlying dynamical system. I have also worked on billiards, where geometric constraints give rise to subtle dynamical behavior through repeated reflection.  Across these projects, the central problem is to understand how simple local rules generate global structure.
  

• Yaoying Fu, Ph.D. student
  

Past:

• Laura Seaberg,  Ph.D., 2025
  
• Ethan Farber, Ph.D., 2023
  
• Henry Bayly, senior thesis, 2022
  
• Alex Benanti, senior thesis, 2022
  
• Jieqi Di, scholar of the college thesis (2nd reader), 2022
  
• Hong Cai, geophysics MS student (2nd reader) 2022

• Regularization Implies Balancedness in the Deep Linear Network (with G. Menon)
  
• Link to arxiv preprint
  
  
• Empirical NTK tracks task complexity (with  E. Grigsby).
  
  
• On Functional dimension and persistent peudodimension (with E. Grigsby)
• Link to arxiv preprint.
  
  
• Hidden symmetries of ReLU neural networks (with E. Grigsby, D. Rolnick)
• Proceedings of the 40th International Conference on Machine Learning, PMLR 202:11734-11760
  
• Link to arxiv preprint and published version.
  

• On the deck groups of iterates of bicritical rational maps (with S. Koch, T. Sharland)
• Link to arxiv preprint.
  

• Bicritical rational maps with a common iterate (with S. Koch, T. Sharland)
• International Math. Research Notices, Mar. 2023.
  
• Link to arxiv preprint.
  
  
• Functional dimension and moduli spaces of ReLU neural networks (with E. Grigsby, R. Meyerhoff, C. Wu)
• Link to arxiv preprint. 
• Advances in Mathematics, Vol 482, Part C, Dec. 2025
  
  
• Local and global topological complexity measures of generic, transversal ReLU neural network functions (with E. Grigsby, M. Masden).
• Topology Proceedings, to appear
  
• Link to arxiv preprint.
  
  
• Existence of maximum likelihood estimates in exponential random graph models (with H. Bayly, A. Khanna).
• Link to arxiv preprint.
  
  
• Master Teapots and entropy algorithms for the Mandelbrot set (with G. Tiozzo, C. Wu).
• Transactions of the American Math. Soc., Mar. 2025
  
• Link to arxiv preprint.
  
  
• On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks (with E. Grigsby)
• SIAM Journal on Applied Algebra and Geometry, vol. 6, issue 2
• Links to published version and arxiv preprint.
  
  
• A characterization of Thurston's Master Teapot (with C. Wu).
• Ergodic Theory & Dynamical Systems, Nov. 2022.
  
• Links to arxiv preprint.
  
  
• Degree-d-invariant laminations (with W. Thurston, H. Baik, Gao Yan, J. Hubbard, Tan Lei, D. Thurston).
• What's Next?: The Mathematical Legacy of Williams Thurston, Princeton Univ. Press., 2021.
• Links to published version and arxiv preprint.
  
  
• The Shape of Thurston's Master Teapot (with H. Bray, D. Davis, C. Wu).
  
• Advances in Mathematics, vol 377, Jan 2021.
  
• Links to arxiv preprint.
  
  
• Fekete polynomials and shapes of Julia sets (with M. Younsi).
• Transactions of the American Math. Soc. v. 371, 2019
• Link to arxiv preprint.
  
  
• Convex shapes and harmonic caps (with L. DeMarco).
• Arnold Math. Journal, (2017) 1-21.
• Link to arxiv preprint.
• Read about this in Quanta Magazine.
  
  
• Horocycle flow orbits and lattice surface characterizations (with J. Chaika).
• Ergodic Theory & Dynamical Systems, vol 39 (6), 2019, p. 1441-1461.
• Links to published version and arxiv preprint.
  
  
• Counting invariant components of hyperelliptic translation surfaces.
• Israel J. Math., 210 (2015), p. 125-146.
• Link to arxiv preprint.
  
  
• Shapes of polynomial Julia sets.
• Ergodic Theory & Dynamical Systems, vol 35, 06, 2015.
• Links to arxiv preprint.
• Read about this result in Scientific American.
  
  
• A Game of Life on Penrose tilings (with D. Bailey).
• Permanent preprint. Link to arxiv preprint.
  
  
• Flat surface models of ergodic systems (with R. Trevino).
• Discrete and Continuous Dynamical Systems - A, vol 36, 10 (2016), 5509-5553.
• Links to published version and arxiv preprint.
  
  
• Measurable Sensitivity (with J. James, T. Koberda, C. Silva, P. Speh).
• Proc. Amer. Math. Soc. 136 (2008), 3549-3559.
• Links to published version and arxiv preprint.
  
  
• On ergodic transformations that are both weakly mixing and uniformly rigid (with J. James, T. Koberda, C. Silva, P. Speh).
• New York Journal of Math. 15 (2009), 393-403.
• Links to published version and arxiv preprint.
  
  
• Families of dynamical systems associated to translation surfaces.
• Ph.D. dissertation, Cornell University, 2014.
  
  
• Descriptive dynamics of Borel endomorphisms and group actions.
• Honors thesis in mathematics, Williams College, 2007.