Research Overview

My research lies in dynamical systems and the mathematical foundations of deep learning. Across both areas, I study how geometry, topology, combinatorics, algebra, and symmetry organize parameterized families of mathematical objects. A recurring theme is the passage from a parameter space to a space of objects: how singularities and symmetries arise, how redundancy is encoded in the fibers of this map, and which invariants capture the structure of the resulting family.

In my work on neural networks, I study the geometry of neural-network parameterization. For a fixed architecture, the weights and biases parameterize a family of functions, giving a realization map from parameter space to function space. This map is redundant and singular: distinct parameters can represent the same function, its effective dimension can vary across parameter space, and its fibers carry nontrivial geometric structure. My work studies this geometry—its generic structure, fibers, symmetries, singularities, and dimension—and connects it to the topology and expressivity of neural-network functions and to optimization and regularization.

My work in dynamical systems includes real and complex dynamics, entropy, Thurston theory, symbolic dynamics, translation surfaces, and billiards. Although the objects are different, the two research programs share a common structural viewpoint: understanding how families of mathematical objects are organized by their parameter spaces, singularities, symmetries, and invariants.

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Mathematical Foundations of Deep Learning

Across a series of papers, my collaborators and I have developed complementary parts of this framework: a generic and polyhedral geometry for ReLU networks; functional dimension as an invariant of the represented function family; a theory of hidden symmetries and parameter redundancy; topological invariants of represented functions; and, since 2024, a geometric approach to optimization and regularization along realization fibers. These questions are organized by a common object—the realization map from parameters to functions—and by the geometry of its image and fibers.

Geometry of neural-network parameterizations

A central theme of this program is the discrepancy between parameter space and function space: different parameter settings may realize the same function, and the amount and structure of this redundancy varies throughout parameter space. I study the realization map from complementary directions: the dimension of its image, the geometry of its fibers, and the symmetries responsible for parameter redundancy.

• The Geometry of Limits in ReLU Function Spaces
• Preprint (ArXiv, 2026)
• Studies the geometry and closure of spaces of functions realized by ReLU neural networks of fixed architecture. Constructs an explicit sequence of networks with architecture (2,3,2|1) whose limit is not realizable by that architecture.  Establishes that, in contrast, for continuous piecewise-linear functions, the minimum number of convex sets on which a function is affine is lower semicontinuous under pointwise convergence.
• Link to arxiv preprint.
  

• Functional dimension of feedforward ReLU neural networks (joint with J. Elisenda Grigsby, Robert Meyerhoff, Chenxi Wu)
  • Journal article: Advances in Mathematics, vol. 482 (2025).
  • Defines functional dimension as a local invariant of the ReLU realization map and shows that it is inhomogeneous across parameter space. Relates it to the ranks of batch neural tangent kernels, proves an upper bound that accounts for known symmetries and is attained for narrowing architectures, and gives examples of realization fibers that are disconnected, have non-constant functional dimension, or carry non-transitive symmetry-group actions.
  • Links to published version and arxiv preprint.
  
  
• On Functional Dimension and Persistent Pseudodimension (joint with J. Elisenda Grigsby)
  • Preprint (arXiv, 2024).
  • Introduces persistent pseudodimension, a local VC-type complexity measure for ReLU network classes. Proves that it lies between functional dimension and the supremum of the ranks of the evaluation Jacobians at almost every parameter, conjectures that it equals functional dimension, and proves a local product structure for parameter space near almost every parameter, separating effective directions from directions tangent to batch realization fibers.
  • Link to arxiv preprint.
  
  
• Hidden symmetries of ReLU networks (joint with J. Elisenda Grigsby, David Rolnick)
  • Conference proceedings: ICML 2023 (Proceedings of the 40th International Conference on Machine Learning), PMLR 202:11734–11760.
  • Studies how faithfully the parameter space of feedforward ReLU networks represents the functions they compute. Proves that for any architecture in which no layer is narrower than the input, there are parameters with no hidden symmetries; identifies mechanisms that generate hidden symmetries; and measures functional dimension empirically at initialization.
  • Links to published version and arxiv preprint.

Geometry, topology, and expressivity of ReLU functions

This line of work treats ReLU networks as structured families of piecewise-linear functions and asks how architecture constrains the geometry and topology of the functions they can realize. The approach draws on differential topology, polyhedral geometry, and Morse-theoretic ideas.

• On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks (joint with J. Elisenda Grigsby)
  • Journal article: SIAM Journal on Applied Algebra and Geometry, vol. 6, issue 2 (2022).
  • Defines generic, transversal ReLU networks and proves that almost all ReLU networks are generic and transversal. Introduces a partially oriented linear 1-complex that detects obstructions to bounded connected components of decision regions, and uses it to restrict the topology of the decision regions that a fixed architecture can realize; for example, the decision region of a generic, transversal single-hidden-layer network of width n+1 on inputs in Rⁿ has at most one bounded connected component.
  • Links to published version and arxiv preprint.
  
  
• Local and global topological complexity measures of ReLU neural network functions (joint with J. Elisenda Grigsby, Marissa Masden)
  • Journal article (forthcoming): Topology Proceedings (arXiv preprint, 2022).
  • Applies a piecewise-linear version of Morse theory to define local and global notions of topological complexity for ReLU network functions. Constructs a canonical polytopal complex and a deformation retraction that give a compact model for computation, and shows that local topological complexity can be arbitrarily high.
  • Link to arxiv preprint.

Geometry of optimization and training

Recent work (2024–2025) asks how the same parameter-space geometry interacts with optimization and regularization. In particular, realization fibers separate changes of representation from changes of the represented function, providing a geometric framework for distinguishing motion along a fiber from learning in function space.

• Regularization Implies Balancedness in the Deep Linear Network (joint with Govind Menon)
  • Preprint (arXiv, 2025).
  • Uses geometric invariant theory, via the Kempf–Ness theorem, to prove that the L² regularizer is minimized on the balanced manifold. Introduces balancing flows along realization fibers that converge exponentially, and decomposes training dynamics into a regularizing flow on fibers and a learning flow on the balanced manifold.
  • Link to arxiv preprint.
  
  
• Empirical NTK tracks task complexity (joint with J. Elisenda Grigsby)
  • Manuscript in preparation.
  • Identifies empirical-NTK rank with batch functional dimension and uses it to measure the effective number of parameter directions visible on a finite dataset, connecting training-time NTK rank to the geometry of the realization map.

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Dynamical Systems

My work in dynamical systems studies how topology, combinatorics, algebra, and complex geometry organize the long-term behavior of iterated maps. Recurring themes include entropy and Thurston invariants in one-dimensional dynamics, geometric questions concerning Julia sets and rational maps, and the dynamics of translation surfaces. As in my work on neural networks, a central concern is to understand how global behavior is constrained by the geometry and combinatorics of an underlying parameterized family.

Entropy, kneading theory, and Thurston sets

• Master Teapots and entropy algorithms for the Mandelbrot set (joint with Giulio Tiozzo, Chenxi Wu)
  • Journal article: Transactions of the American Mathematical Society, vol. 378 (2025).
  • Constructs an analogue of Thurston's Master Teapot for each principal vein of the Mandelbrot set. Proves that eigenvalues outside the unit circle vary continuously and that a persistence property holds inside the unit disk, develops kneading theory for principal veins, and proves that several algorithms for computing core entropy are equivalent.
  • Links to published version and arxiv preprint.
  
  
• A characterization of Thurston's Master Teapot (joint with Chenxi Wu)
  • Journal article: Ergodic Theory & Dynamical Systems, vol. 43 (2023), 3354–3382.
  • Proves an explicit characterization of the points of Thurston's Master Teapot, which can be implemented algorithmically to test whether a point lies in its complement. As an application, shows that the intersection of the Master Teapot with the unit cylinder is not symmetric under reflection through the product of the imaginary axis and R.
  • Links to published version and arxiv preprint.
  
  
• The Shape of Thurston's Master Teapot (joint with Harrison Bray, Diana Davis, Chenxi Wu)
  • Journal article: Advances in Mathematics, vol. 377 (2021).
  • Establishes basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal continuous self-maps of intervals: the Master Teapot is connected, contains the unit cylinder, and its intersection with a horizontal slice grows monotonically with the height of the slice.
  • Links to published version and arxiv preprint.
  
  
• Degree-d-invariant laminations (joint with William Thurston, Hyungryul Baik, Gao Yan, John Hubbard, Tan Lei, Dylan Thurston)
  • Book chapter: What's Next? The Mathematical Legacy of William Thurston, Princeton University Press (2021).
  • Develops degree-d invariant laminations as combinatorial models of polynomial dynamics and introduces primitive majors, whose space serves as a spine for monic degree-d polynomials with distinct roots. Shows that the core entropy of postcritically finite polynomials can be computed from this combinatorial structure without constructing Hubbard trees.
  • Links to published version and arxiv preprint.

Complex dynamics

• On the deck groups of iterates of bicritical rational maps (joint with Sarah Koch, Thomas Sharland)
  • Preprint (arXiv, 2022).
  • Classifies the groups that arise as deck groups of iterates of bicritical rational maps. Each such group is cyclic or dihedral, and for maps that are not power maps its order is bounded in terms of the degree.
  • Link to arxiv preprint.
  
  
• Bicritical rational maps with a common iterate (joint with Sarah Koch, Thomas Sharland)
  • Journal article: International Mathematics Research Notices, vol. 2024, issue 2 (2024).
  • Uses deck groups to prove that bicritical rational maps sharing an iterate have the same critical points and the same critical values, and that two bicritical maps of even degree sharing an iterate also share a second iterate and lie in the symmetry locus of bicritical maps of that degree.
  • Links to published version and arxiv preprint.
  
  
• Fekete polynomials and shapes of Julia sets (joint with Malik Younsi)
  • Journal article: Transactions of the American Mathematical Society, vol. 371 (2019).
  • Proves that a nonempty proper subset of the plane can be approximated by polynomial filled Julia sets if and only if it is bounded and the complement of its interior is connected. The proof is constructive, uses Fekete polynomials, and gives estimates for the rate of approximation.
  • Links to published version and arxiv preprint.
  
  
• Shapes of polynomial Julia sets
  • Journal article: Ergodic Theory & Dynamical Systems, vol. 35 (2015).
  • Proves that any Jordan curve in the plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials, and that finite collections of disjoint Jordan domains can be approximated by basins of attraction of rational maps.
  • Links to published version and arxiv preprint.
  • Read about this result in Scientific American.
  
  
• Convex shapes and harmonic caps (joint with Laura DeMarco)
  • Journal article: Arnold Mathematical Journal (2017), 1–21.
  • Shows that any planar shape embeds isometrically as part of the boundary of a convex subset of R³ so that the boundary of the shape supports the positive curvature, and studies the resulting cap when the curvature is harmonic measure. Of particular interest is the case where the shape is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.
  • Links to published version and arxiv preprint.
  • Read about this work in Quanta Magazine.

Ergodic theory and translation surfaces

• Horocycle flow orbits and lattice surface characterizations (joint with Jon Chaika)
  • Journal article: Ergodic Theory & Dynamical Systems, vol. 39 (2019), 1441–1461.
  • Proves that the orbit closure of a translation surface under the horocycle flow in almost every direction equals its SL(2,R)-orbit closure, and derives new characterizations of lattice surfaces in terms of the horocycle flow.
  • Links to published version and arxiv preprint.
  
  
• Counting invariant components of hyperelliptic translation surfaces
  • Journal article: Israel Journal of Mathematics, 210 (2015), 125–146.
  • Characterizes the pairs (p,m) for which a translation surface in the hyperelliptic components of the strata H(2g−2) and H(g−1,g−1) has a directional flow whose decomposition into closed invariant sets has exactly p periodic components and m minimal components.
  • Links to published version and arxiv preprint.
  
  
• Infinite type flat surface models of ergodic systems (joint with Rodrigo Treviño)
  • Journal article: Discrete and Continuous Dynamical Systems - A, vol. 36 (2016), 5509–5553.
  • Develops a general framework, based on Bratteli diagrams and cutting-and-stacking constructions, for building infinite-type flat surfaces of finite area and describing their translation flows. Shows that such surfaces can carry flows that are mixing, have positive topological entropy, or are minimal with uncountably many ergodic invariant probability measures, which are phenomena that do not occur for finite-type surfaces.
  • Links to published version and arxiv preprint.
  
  
• On ergodic transformations that are both weakly mixing and uniformly rigid (joint with Jennifer James, Thomas Koberda, Cesar Silva, Peter Speh)
  • Journal article: New York Journal of Mathematics, 15 (2009), 393–403.
  • Examines uniformly rigid transformations and their compatibility with measurable weak mixing. Shows that on Cantor space there is no finite measure-preserving, totally ergodic, uniformly rigid transformation, and that weak mixing and uniform rigidity can coexist for more general group actions.
  • Links to published version and arxiv preprint.
  
  
• Measurable Sensitivity (joint with Jennifer James, Thomas Koberda, Cesar Silva, Peter Speh)
  • Journal article: Proceedings of the American Mathematical Society, 136 (2008), 3549–3559.
  • Introduces measurable sensitivity, a measure-theoretic version of sensitive dependence on initial conditions. Shows that it is a consequence of light mixing, that it implies only finitely many eigenvalues, and that it does not occur for ergodic infinite measure-preserving transformations.
  • Links to published version and arxiv preprint.

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Research Grants and Fellowships

• NSF Award #2133822: Collaborative Research, Probabilistic, Geometric and Topological Analysis of Neural Networks, from Theory to Applications, 2022
  
  
• NSF Award #1901247: Shapes of Julia sets, Thurston sets, and Neural Networks, 2019
  
  
• Major Grant, Boston College Institute for Liberal Arts, “Women in STEM,” 2018
  
  
• Research Incentive Grant, Boston College, 2018
  
  
• NSF Award #1401133: Mathematical Sciences Postdoctoral Research Fellowship, 2014
  
  
• NSF Graduate Research Fellowship, 2009
  
  
• DoD National Defense Science and Engineering Graduate Fellowship, 2009
  
  
• U.S. State Department Critical Languages Scholarship (Chinese), 2008
  
  
• Graduate Fellowship, Cornell University, 2007

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Research Supervision

I supervise graduate and undergraduate research in mathematical foundations of machine learning, dynamical systems, and related areas. Research advisees and thesis students include:

• Yaoying Fu, Ph.D., 2026
• 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 M.S. student (2nd reader), 2022

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Other Work

• Existence of maximum likelihood estimates in exponential random graph models (joint with Henry Bayly, Aditya Khanna)
  • Preprint (arXiv, 2022).
  • Gives a streamlined proof that the maximum likelihood estimate exists if and only if the target statistic lies in the relative interior of the convex hull of the realizable statistics.
  • Link to arxiv preprint.
  
  
• A Game of Life on Penrose tilings (joint with David Bailey)
  • Preprint (arXiv, 2017).
  • Defines cellular automata on quasiperiodic tilings of the plane, including Penrose tilings, that are isomorphic to Conway's Game of Life. Although the tilings are nonperiodic, the next state of each tile depends only on the local structure of the tiling and the states of finitely many nearby tiles.
  • Link to arxiv preprint.

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Theses

• 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.