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Research 

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I am working on questions at the intersection of algebraic geometry and machine learning. More concretely, I am interested in how to apply algebro-geometric tools to understand the geometry of function spaces of algebraic neural networks. For a fixed neural network architecture, we have a finite-dimensional embedding of the network parameters into its ambient space. In the case of polynomial or rational activation functions, we can choose this ambient space to be finite-dimensional. Therefore, the corresponding embedding is a polynomial map, and its image is a semi-algebraic set. We call this set a neuromanifold. When we update the parameters of the network, we traverse the neuromanifold. The geometry and topology of the neuromanifold affect the training dynamics of neural networks.

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My goal is, on one hand, to explore how different neural network setups are connected to various classical algebraic geometry objects, such as Chow varieties, secant varieties of Veronese and Grassmann varieties, and Weyl’s conjectures. On the other hand, I am interested in how what we learn in terms of topology, geometry, and invariants can improve our understanding of black-box neural network training and potentially bring practical advantages.

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I am currently focused on the following projects.

  • Monomial neural networks and tensor decompositions. The goal of this project is to study the geometry of neuromanifolds from the perspective of tensor decomposition. For example, a family of shallow polynomial neural networks with a monomial activation function (sigma(x) = x^r) corresponds to simultaneous Waring rank decomposition.

  • Rational neural networks. One motivation for this project is to learn the location and type of singularities from data. A beautiful fact is that a family of shallow rational neural networks with a 1/x activation function corresponds to a projection of a Chow variety.

  • Neural networks over finite fields. Finally, the goal of this project is to explore the expressivity and geometry of networks with low-precision weights.

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Here is my CV.

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Research Articles

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  • Algebraic geometry of rational neural networks

with Alexandros Grosdos and Elina Robeva
Submitted, 2026. [arXiv]

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  • Expressivity of Shallow Neural Networks Over Finite Fields

with Carol Wu, Shiwei Yang, Param Mody, and Yifei Chen
Submitted, 2026. [OpenReview]

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  • Geometry of Rank Constraints in Shallow Polynomial Neural Networks

with Param Mody
Published at MOSS @ ICML, 2025. [OpenReview]

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  • Sign patterns of principal minors of real symmetric matrices

with Tobias Boege and Jesse Selover
Published in Linear Algebra and its Applications, 738 (2026), 161–188. [arXiv] (ScienceDirect)

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  • Likelihood Geometry of Determinantal Point Processes

with Hannah Friedman and Bernd Sturmfels
Published in Algebraic Statistics, 15 (2024), no. 1, 15–25. [arXiv] (MSP)

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  • Chromatic Graph Homology for Brace Algebras

with Vladimir Baranovsky
Published in New York Journal of Mathematics, 23 (2017), 1307–1319.

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