Here are my research publications and preprints.
7. An inverse problem on eigenfunction triple products
Preprint ()
Abstract. On a connected closed smooth Riemannian manifold, the algebraic structure of the Laplace eigenfunctions, as described by eigenfunction triple products, uniquely determines the geometry. We refine this correspondence by introducing the notion of an \(N\)-product eigenbasis, which consists of eigenfunctions whose pairwise products may be written as linear combinations of at most \(N\) basis elements. We prove that a manifold admits a \(2\)-product eigenbasis if and only if it is a flat torus. We also prove an analogous result for Laplace eigenvectors of bounded-degree graphs.
6. Steklov rigidity of Euclidean balls
Preprint ()
Abstract. Let \(\Omega\subset \mathbb{R}^n\), \(n\geq 3\), be a bounded domain with smooth boundary. We show that if the Steklov spectrum of \(\Omega\) tends to that of a ball at a sufficiently fast rate, then \(\Omega\) must itself be a ball. In particular, in any dimension and among all bounded domains with smooth and possibly disconnected boundary, Euclidean balls are uniquely determined by their Steklov spectrum.
5. Stability for commutativity properties of the Dirichlet-to-Neumann map
International Mathematics Research Notices ()
Abstract. Viewing \(\mathbb{S}^{n-1}=\partial \mathbb{B}^{n}\subset \mathbb{R}^{n}\), the boundary Laplacian \(\Delta_{\mathbb{S}^{n-1}}\) can be explicitly expressed in terms of \(\Lambda\), the Dirichlet-to-Neumann map, as \(\Delta_{\mathbb{S}^{n-1}}=\Lambda^{2}+(n-2)\Lambda\). In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally to measure the failure of such a relationship in terms of geometric data. To this end, we obtain a stability estimate which shows that a smoothly bounded domain \(\Omega\) in \(\mathbb{R}^{3}\) must be close to the ball if the commutator \([\Lambda,\Delta_{\partial \Omega}]\) is small. We then study the case of manifolds conformal to the ball, show that such a relationship implies a radial metric structure, and discuss stability in this setting. Finally, we provide a modern exposition of Gohberg’s lemma, a foundational result in microlocal analysis which we employ as a starting step for our reasoning.
4. Surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map
Abstract. For \(M\subset \mathbb{R}^{d\geq 3}\) a smooth, connected, compact \(d\)-dimensional submanifold with boundary, equipped with the standard metric, the Laplacian on \(\partial M\) is known to commute with the corresponding Dirichlet-to-Neumann map if and only if \(M\) is a ball. In this paper, we investigate the \(d=2\) case and show that, surprisingly, there exists a one-parameter family of submanifolds of \(\mathbb{R}^2\) as above for which the boundary Laplacian and the Dirichlet-to-Neumann map commute, thus answering an open problem posed by Girouard, Karpukhin, Levitin, and Polterovich. We then classify all such Riemannian surfaces of genus \(0\) or whose boundary has \(k\geq 3\) connected components.
3. A deformation approach to the BFK formula
Canadian Mathematical Bulletin ()
Abstract. Understanding how spectral quantities localize on manifolds is a central theme in geometric spectral theory and index theory. Within this framework, the BFK formula, obtained by Burghelea, Friedlander, and Kappeler in 1992, describes how the zeta-regularized determinant of an elliptic operator decomposes as the underlying manifold is cut into pieces. In this article, we present a novel proof of this result.
2. Continuous limits of generalized pentagram maps
Journal of Geometry and Physics ()
Abstract. We provide a rigorous treatment of continuous limits for various generalizations of the pentagram map on polygons in \(\mathbb{RP}^d\) by means of quantum calculus. Describing this limit in detail for the case of the short-diagonal pentagram map, we verify that this construction yields the \((2,d+1)\)-KdV equation, and moreover, the Lax form of the pentagram map in the limit is proved to become the Lax representation of the corresponding KdV system. More generally, we introduce the \(\chi\)-pentagram map, a geometric construction defining curve evolutions by directly taking intersections of subspaces through specified points. We show that its different configurations yield certain other KdV equations and provide an argument towards disproving the conjecture that any KdV-type equation can be discretized through pentagram-type maps.
1. Evaluating ensemble robustness against adversarial attacks
arXiv ()
Abstract. Adversarial examples, which are slightly perturbed inputs generated with the aim of fooling a neural network, are known to transfer between models; adversaries which are effective on one model will often fool another. This concept of transferability poses grave security concerns as it leads to the possibility of attacking models in a black box setting, during which the internal parameters of the target model are unknown. In this paper, we seek to analyze and minimize the transferability of adversaries between models within an ensemble. To this end, we introduce a gradient based measure of how effectively an ensemble's constituent models collaborate to reduce the space of adversarial examples targeting the ensemble itself. Furthermore, we demonstrate that this measure can be utilized during training as to increase an ensemble's robustness to adversarial examples.