• Mathematical Foundations of Deep Learning. I develop a rigorous, theorem-driven understanding of neural networks using tools from geometry, topology and dynamical systems.  My work analyzes how structural features of neural network parameterizations shape the behavior, expressivity, and training dynamics of the functions they represent. Three complementary themes motivate this research:

• ReLU neural networks and piecewise-linear geometry.   Fully-connected multilayer perceptrons with ReLU activations coincide with the class of piecewise linear functions and are the most "vanilla" type of neural network. I study these networks through the geometry of their associated polyhedral complexes, analyzing how the arrangement of linear regions is determined by the network's architecture and parameters.
  
  
• Parameter space symmetries and functional redundancy. Modern deep learning models are highly redundant, meaning that many distinct parameter settings yield the same realized function. I analyze the fibers, symmetries, and quotient geometry of the non-injective realization map from parameter space to function space, and study how these structures influence training dynamics, implicit bias, generalization, and model selection.
  
  
• Topological expressivity.  We do not have a good understanding of which functions can be realized by which neural network architectures.  I investigate how the architecture constrains the topological properties of the functions a network can represent. 

Prior to transitioning to the mathematics of deep learning, I worked in dynamical systems.  Although the mathematical objects I study have shifted, my approach remains the same.  In both areas, I combine geometry and dynamical perspectives with precise mathematical reasoning to illuminate deep structural phenomena about parameterized families of functions.

• Dynamical Systems.
  My work on dynamical systems explores topological entropy and Thurston sets in real and complex one-dimensional dynamics, with an emphasis on how combinatorial models of interval maps and polynomials reflect and encode dynamical complexity (kneading theory). I have also worked in holomorphic dynamics and billiards.
  
  

• 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).
• 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.