My doctoral research explored the geometry of optimization and how optimization landscapes change as their underlying parameters vary. I used problems in machine learning and signal processing as concrete settings in which to investigate a broader mathematical question: how does the structure of a model or data distribution shape the geometry of its optimization landscape?
Discriminants and parameter space. I studied the discriminant—the set of parameter values at which the critical-point structure of an optimization problem becomes degenerate. These hypersurfaces partition parameter space into regions in which the qualitative behavior of the problem is stable.
Neural networks and data geometry. I studied shallow polynomial neural networks through their connection to low-rank tensor approximation, using this framework to understand how the geometry of an optimization problem is shaped by the higher moments of the data distribution.
Algebraic complexity of optimization. I used tools from algebraic geometry and tensor geometry to study critical points, Euclidean Distance Degree, normal bundles, and the geometry of low-rank approximation.
Fourier optimization and perturbation. I studied SO(2) alignment and the effect of low-pass filtering on trigonometric optimization landscapes, using discriminant-based ideas to understand when perturbations preserve—or change—the structure of the solution.
More broadly, my mathematical interests extend beyond the specific problems of my doctoral research, while remaining closely connected to its themes. I enjoy thinking about connections between different mathematical viewpoints—especially when an algebraic, geometric, probabilistic, or computational perspective reveals something unexpected about a problem.
Geometry & optimization. Critical points, optimization landscapes, perturbation theory, stability, tensor geometry, and the geometry of learning.
Algebra & symmetry. Algebraic geometry, algebraic combinatorics, complex reflection groups, group actions, and the role of symmetry in mathematical problems.
Machine learning & statistics. Mathematical aspects of machine learning, particularly optimization, regularization, statistical modeling, generalization, and the geometry induced by data.
Connections across fields. I am especially interested in how ideas from different areas illuminate one another—for example, when a computational procedure admits a geometric interpretation, an algebraic formulation reveals an optimization problem, or a statistical method acquires a deeper interpretation through a probabilistic perspective.