Research
Research Overview
My research connects explicit mathematics, robust learning, and high-performance scientific computation.
I develop algorithms that turn higher-dimensional existence results into explicit defining equations. I am also exploring how singular learning theory can identify geometric obstructions to robustness in large language and quantitative models, while a complementary direction studies online computational feedback for real-world dynamical systems.
Current doctoral research
From Existence to Equations
An existence theorem establishes that a mathematical object is present, but it does not necessarily tell us how to compute it. Even for a continuous function whose values change sign, the intermediate value theorem guarantees a root without locating it; an algorithm such as Newton's method is needed to approximate the solution under suitable additional hypotheses.
My Ph.D. research develops a higher-dimensional analogue of this passage from existence to computation. I study algebraic surfaces embedded in projective space and create computational methods for finding their explicit defining equations, bridging abstract algebraic geometry with symbolic computation.
Fake Projective Planes
My current focus is on fake projective planes, a class of complex algebraic surfaces whose first example was constructed by David Mumford in 1979. Later work classified further examples, but realizing these abstractly known surfaces in chosen projective coordinates requires an additional layer of explicit computation.
In joint work with Lev Borisov, I use chains of cyclic covers and quotients, together with computational algebra, to obtain equations for a new pair of fake projective planes in the Cartwright–Steger classification.
Developing research direction
Singular Learning Theory and Robust Models
I am developing an exploratory research program that applies singular learning theory to large language models and to overparameterized or latent-variable quantitative models. Such models can be non-identifiable and may exhibit degenerate parameter structures, especially under distribution shifts, tail events, and other extreme regimes.
I aim to use the geometric and asymptotic tools of singular learning theory to characterize obstructions to convergence and robustness, and ultimately to understand how models can maintain reliable behavior across a wider range of data-generating regimes.
Computational systems and applications
Online Feedback and High-Performance Computation
For real-world dynamical systems, computation is part of a feedback loop: observations arrive continuously, the system evolves, and numerical predictions must be updated before they become stale. Algorithms that are accurate offline may still fail to capture live behavior when computational latency is too high. This motivates my interest in online scientific computing, efficient numerical algorithms, and parallel hardware for timely state estimation and decision-making.
My work on G-PASTA demonstrates this principle in an engineering application. By harnessing GPU computing for task-dependency-graph partitioning, G-PASTA achieved up to 41.8× faster partitioning than a state-of-the-art CPU method and improved overall static timing analysis performance by up to 43% on large designs.