Research

My main research interests are Lie theory, representation theory, moduli spaces, and gauge theory—specifically instantons and monopoles.

Published Papers

“Hyperbolic monopoles with continuous symmetries”, J. Geom. Phys. 203, 105258 (2024). DOI, arXiv, Main results

Abstract

We provide a framework to classify hyperbolic monopoles with continuous symmetries and find a Structure Theorem, greatly simplifying the construction of all those with spherically symmetry. In doing so, we reduce the problem of finding spherically symmetric hyperbolic monopoles to a problem in representation theory. Additionally, we determine constraints on the structure groups of such monopoles. Using these results, we construct novel spherically symmetric \(\mathrm{Sp}(n)\) hyperbolic monopoles.

“Construction of Nahm data and BPS monopoles with continuous symmetries”, J. Math. Phys. 63, 013507 (2022). DOI, arXiv, Maple program, Main results

Joint work with Benoit Charbonneau, Anuk Dayaprema, Ákos Nagy, and Haoyang Yu

Abstract

We study solutions to Nahm’s equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data, and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry.

“Scale-dependent anisotropy in forced stratified turbulence”, Phys. Rev. Fluids 4, 044801 (2019). DOI

Joint work with Michael Waite

Abstract

In stratified turbulence, buoyancy forces inhibit vertical motion and lead to anisotropy over a wide range of length scales, which is characterized by layerwise pancake vortices, thin regions of strong shear, and patches of small-scale turbulence. It has long been known that stratified turbulence becomes increasingly isotropic as one moves to smaller length scales, as the eddy timescale decreases towards and below the buoyancy period. This paper investigates the anisotropy of stratified turbulence across scales and the transition towards isotropy at small scales, using a variety of techniques. Direct numerical simulations of strongly stratified turbulence, with buoyancy Reynolds numbers \(\mathrm{Re}_b\) up to 50, are analyzed. We examine the relative contributions of different components of the strain rate tensor to the kinetic energy dissipation, the invariants of the isotropy tensor, directional kinetic energy spectra, and the subfilter energy flux across different length scales. At small scales, the degree of isotropy is determined by \(\mathrm{Re}_b\), while at the Ozmidov and larger scales, the anisotropy also depends on the Froude number. The change in the anisotropy with scale and with the parameters is examined in detail. Interestingly, Ozmidov-scale eddies are found to become increasingly isotropic as \(\mathrm{Re}_b\) increases, as characterized by the isotropy tensor invariants and the subfilter energy flux. At larger scales, the energy spectra for near-vertical wave vectors have a spectral slope around \(−3\), which shallow towards \(−1\) for near-horizontal wave vectors. These spectra converge beyond the Ozmidov scale, increasingly so for large \(\mathrm{Re}_b\). These results suggest that \(\mathrm{Re}_b\gtrsim 500\) would be necessary to obtain the same degree of small-scale isotropy that is found in similarly sized simulations of unstratified turbulence.

Preprints

“Instantons with continuous conformal symmetries: Hyperbolic and singular monopoles and more, oh my!” (2025). arXiv, Main results

Abstract

Throughout this paper, we comprehensively study instantons with every kind of continuous conformal symmetry. Examples of these objects are hard to come by due to non-linear constraints. However, by applying previous work on moduli spaces, we introduce a linear constraint, whose solution greatly simplifies these non-linear constraints. This simplification not only allows us to easily find a plethora of novel instantons with various continuous conformal symmetries and higher rank structure groups, it also provides a framework for classifying such symmetric objects. We also prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we discuss the connections between instantons with continuous symmetries and other gauge-theoretic objects: hyperbolic and singular monopoles as well as hyperbolic analogues to Higgs bundles and Nahm data.

“Fixed points of Lie group actions on moduli spaces: A tale of two actions” (2024). arXiv, Main results

Abstract

In this paper, we examine Lie group actions on moduli spaces (sets themselves built as quotients by group actions) and their fixed points. We show that when the Lie group is compact and connected, we obtain a linear constraint. This constraint makes the problem of finding fixed points one of representation theory, greatly simplifying the search for such points. We obtain a similar result when the Lie group is one-dimensional. For compact and disconnected Lie groups, we show that we need only additionally check a finite number of points. Finally, we show that the subgroup fixing an equivalence class in the moduli space is a compact Lie subgroup.

Ph.D. Thesis

“Solitons with continuous symmetries” (2024). UWSpace, Fixed points on moduli spaces, Symmetric Euclidean monopoles, Symmetric hyperbolic monopoles, Symmetric instantons, Symmetry breaking of monopoles, Representation theory notes, GMoM 2024 talk

Abstract

In this thesis, we develop a framework for classifying symmetric points on moduli spaces using representation theory. We utilize this framework in a few case studies, but it has applications well beyond these cases.

As a demonstration of the power of this framework, we use it to study various symmetric solitons: instantons as well as hyperbolic, singular, and Euclidean monopoles. Examples of these objects are hard to come by due to non-linear constraints. However, by applying this framework, we introduce a linear constraint, whose solution greatly simplifies the non-linear constraints. This simplification not only allows us to easily find a plethora of novel examples of these solitons, it also provides a framework for classifying such symmetric objects. As an example, by applying this method, we prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry.

Additionally, we study the symmetry breaking of monopoles, a part of their topological classification. We prove a straightforward method for determining the symmetry breaking of a monopole and explicitly identify the symmetry breaking for all Lie groups with classical, simply Lie algebras. We also identify methods for doing the same for the exceptional simple Lie groups.