ABOUT

RESEARCH

TEACHING

CONFERENCES

GALLERY

OTHER

---

• Mathematical theory of neural networks.
• My research investigates the geometric foundations of deep learning theory, the study of neural networks as function classes for supervised learning.  My  focus is on feedforward ReLU neural networks -- a family that coincides with the class of finitely piecewise linear functions.   
  
  
• Dynamical systems with connections to geometry/topology.
• The focus of much of my recent and ongoing work is topological entropy and Thurston sets.  The overarching question my work investigates is ``which numbers are topological entropy of which dynamical systems, and what does that tell us about the dynamical systems?"
  

Research advisees:

Current:

• Laura Seaberg, Ph.D. student
• Yaoying Fu, Ph.D. student
  

Past:

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

Publications:

• On Functional Dimension and Persistent Pseudodimension (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.
  
  
• 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).
• To appear in Transactions of the American Math. Soc. 
  
• 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 published version and 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 published version and arxiv preprint.
  
  
• Fekete polynomials and shapes of Julia sets (with M. Younsi).
• Transactions of the American Math. Soc. v. 371, 2019
• Links to published version and arxiv preprint.
  
  
• Convex shapes and harmonic caps (with L. DeMarco).
• Arnold Math. Journal, (2017) 1-21.
• Links to published version and 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.
• Links to published version and arxiv preprint.
  
  
• Shapes of polynomial Julia sets.
• Ergodic Theory & Dynamical Systems, vol 35, 06, 2015.
• Links to published version  and arxiv preprint.
• Read about this result in Scientific American.
  
  
• A Game of Life on Penrose tilings (with D. Bailey).
• 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.