Advances in Mathematics · 2012
Finite Groebner bases in infinite dimensional polynomial rings and applications
We prove the Independent Set Conjecture in algebraic statistics.
About
About
My work in pure mathematics focuses on how information is represented — in polynomials, matrices, groups, and the brain. My research spans computational algebraic geometry, matrix analysis, AI, and theoretical neuroscience.
I earned my Ph.D. in Mathematics from U.C. Berkeley under Bernd Sturmfels, following a B.S. in Mathematics and Computer Science from Yale. I was a project scientist at the Redwood Center for Theoretical Neuroscience (U.C. Berkeley) and worked with the Tecott Lab at U.C. San Francisco.
Research
Research
Interests
A single thread runs through my work: finding the structure that makes hard representation problems tractable — from solving structured polynomial systems to theorizing how the brain codes the world.
- Nonlinear algebraStructured polynomial systems.
- Complexity & tensorsHardness of tensor and combinatorial problems.
- Matrix analysis Matrix equations, bounds on traces.
- Neural computation & AIAssociative memory, invariant networks, efficient coding.
- Math & the brainApplying these tools to neuroscience and information theory.
Selected work
Journal of the ACM · 2013
Most tensor problems are NP-hard
The natural generalizations of matrix problems to tensors are, almost without exception, computationally intractable.
arXiv · 2018
Maximum entropy distributions on graphs
Inspired by applications to theories of coding and communication in networks of nervous tissue, we study maximum entropy distributions on weighted graphs with a given expected degree sequence.
Journal of Mathematical Neuroscience · 2018
Robust exponential memory in Hopfield networks
Shows how a recurrent network can store an exponential number of memories robustly, which also solves the hidden clique problem in computer science.
Nature Scientific Reports · 2018
Active state organization of spontaneous behavioral patterns
Mouse genetics are 99% predicted by behavior alone.
IEEE Transactions Signal Processing · 2019
On the uniqueness and stability of dictionaries for sparse representation of noisy signals
We provide very general conditions guaranteeing when dictionaries yielding the sparsest encodings are unique and stable with respect to measurement or modeling error.
ICLR · 2023
Bispectral neural networks
We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups.
COLT · 2024
Harmonics of learning: Universal Fourier features emerge in invariant networks
We formally prove that, under certain conditions, if a neural network is invariant to a finite group then its weights recover the Fourier transform on that group.
Entropy · 2025
Detecting signatures of criticality using divergence rate
Guided by the classical theory of rate–distortion (RD) from information theory, we propose a measure for detecting and characterizing critical phenomena from data.
Preprint · 2026
Implicit bias and invariance: How Hopfield networks efficiently learn graph orbits
Many learning problems involve symmetries, and while invariance can be built into neural architectures, it can also emerge implicitly when training on group-structured data. We study this phenomenon in classical Hopfield networks and illustrate how they can infer the isomorphism class of a graph from a small, random sample. Our results reveal that: (i) graph isomorphism classes can be represented within a three-dimensional invariant subspace, (ii) using gradient descent to minimize energy flow (MEF) has an implicit bias toward norm-efficient solutions, which underpins a polynomial sample complexity bound for learning isomorphism classes, and (iii) across multiple learning rules, parameters converge toward the invariant subspace as sample sizes grow.
Full list
For the complete and current publication record, see my Google Scholar profile.Talks
Talks
Selected talks
Writing
Writing
Articles & expository
Research papers, notes, and expository pieces.
The complete, always-current list lives on Google Scholar.
Code
Code
Software & data
Reference implementations and computational supplements accompanying the papers.
Problems
Problems
Problem of the Month
An open or instructive problem, refreshed periodically — a small standing invitation to think about something hard.
Problem: Characterize those positive integers m,n such that the following holds: For every group of size |G| = m, and any g in the group, we have that g has a unique n-th root.