My research is in differential geometry, more specifically Riemannian geometry. My early work began the program of studying ambient obstruction solitons, which applies the study of geometric flows to tensors with conformal properties. While I still work on pushing forward this program, my work has broadened to include both the study of Einstein metrics on homogeneous manifolds and length bounds on geodesics. This past year I also began working in mathematical formalization using Lean and co-founded the Northwestern Undergraduate Lean Lab (NULL) to incorporate undergraduate students into this work.
Broadly stated, research in differential geometry aims to answer the question “what is the best metric?” with the hope that the answer to this question will provide valuable insights to the universe we inhabit. Riemannian geometry focuses on answering this question for Riemannian manifolds (M, g). A Riemannian manifold is a smooth manifold M paired with a Riemannian metric, g. Riemannian metrics are positive-definite inner products defined on the tangent space of M , TpM , that vary smoothly across M .
The study of geometric flows evolved as a way to use curvature to identify best metrics. A geometric flow is a differential equation in which the metric is considered as a function of time and is changed over time in accordance with the curvature of the manifold. Solitons are solutions to the geometric flow that, over time, change only by diffeomorphism and/or rescaling. Heuristically: if the flow is supposed to make metrics “nicer” over time, solitons isolate metrics that are not improving as they are pushed through the flow. In quantitative geometry, similar techniques such as curve shortening flow arise to produce length bounds on geodesics chords, as I discuss in subsection 2.
The study of homogeneous manifolds emerges as a way to identify potential best metrics by focusing on Riemannian manifolds with maximal symmetry. A Riemannian manifold is said to be homogeneous if for each pair of points p, q P M there is an isometry (a map preserving angle and distance) mapping p to q. While I discuss how this is characterized algebraically in subsection 3, we can intuitively think of these as manifold looks the same at every point and, thus, have constant scalar curvature. Some familiar examples of homogeneous manifolds are spheres and Rn.
Summarizing my main results:
1. Classified steady and shrinking homogeneous gradient Bach solitons. I also show that the study of these solitons is distinct from that of Ricci solitons by showing the existence of an expanding homogeneous gradient Bach soliton on SU(2).
2. Proved that compact homogeneous ambient obstruction solitons are ambient obstruction flat. My approach using general q-solitons establishes a more refined system of results for general q.
3. Established that, with additional regularity or curvature conditions, non-compact q-solitons are also q-flat. Applied these results to Cotton solitons, homogeneous ambient obstruction solitons, and Bach solitons with harmonic Weyl curvature. (Joint with A.W. Cunha)
4. Proved that any closed ambient obstruction soliton is ambient obstruction flat. We defined extended ambient obstruction solitons to extend result to nonhomogeneous manifolds. Notably, solutions to the geometric flow that stay within their conformal class are extended solitons. (Joint work with R. Poddar, R. Sharma, W. Wylie)
5. Establish natural geometric conditions that yield length estimates for the shortest possible orthogonal geodesic chords. (Joint with I. Beach, H. Contreras-Peruyero, R. Rottmann, and C. Searle)
6. Examine the existence of Einstein metrics on a class of homogeneous manifolds with contractible nerve complex and 3 or 4 irreducible isotropy summands. (Joint with I. Beach, H.Contreras Peruyero, M. Kerr, and C. Searle)
7. Begin work to formalize the higher-dimensional second derivative test in mathlib. (Joint with A. Kapota, B. Kjos-Hanssen, J. Lakshmanan)
Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds with I. Beach, H. Contreras-Peruyero, R. Rotman, C. Searle (September 2025)
Extended Solitons of the Ambient Obstruction Flow with R. Poddar, R. Sharma, and W. Wylie (May 2024)
On non-compact gradient solitons with A.W. Cunha (July 2021)
Dissertation: Gradient Ambient Obstruction Solitons on Homogeneous Manifolds (August 2020)
Einstein Metrics on Homogeneous Manifolds (Anticipated Winter 2026)