Research 

                                                       Completed Papers

1. Sublinear Morse Geodesics and First Passage Percolation

Joint w/ Yulan Qing | arXiv | Submitted

Abstract: Given an infinite connected graph, one way to randomly perturb its metric is to assign independent and identically distributed (i.i.d.) random lengths to its edges, a process known as first-passage percolation. Assuming that the graph has uniformly bounded degree, that the edge-length distribution is strictly positive with finite expectation, and that the graph contains a sublinearly Morse bi-infinite geodesic, we prove that the randomly weighted graph almost surely admits a bi-infinite geodesic. This generalizes a previous result concerning Morse geodesics.

2. First-Passage Percolation, Non-Positive Curvature, and Radial Maps

Joint w/ Dominic Bair and Yulan Qing | arXiv | Submitted

Abstract: We investigate the effects of first-passage percolation on the large-scale geometry of infinite, bounded-degree graphs. Under suitable assumptions on the edge-length distribution, we show that non-positive curvature properties, including Gromov hyperbolicity and the coarse CAT(0) property, are almost surely not preserved. We also establish that Morse geodesic rays need not remain Morse under the randomly perturbed metric. Finally, we study the radial behavior of the identity map between the original and randomly weighted graphs.

3. First-Passage Percolation Preserves Sublinearly Morse Boundaries

Joint w/ Yulan Qing | arXiv | Updated Version | Submitted

Abstract: Sublinearly Morse directions in proper geodesic spaces are characterized by a stability property involving sublinear neighborhoods. We establish an alternative characterization of sublinearly Morse geodesics using a middle-recurrence property. Applying this characterization to first-passage percolation on bounded-degree graphs, we prove that, under suitable assumptions on the edge-length distribution, the sublinearly Morse boundary is almost surely preserved by the random perturbation of the metric.

                                                               In Preparation

(Listed in order of anticipated completion)

1. Horofunction Compactification of Horospherical Products of Trees (Joint w/ Behrang Forghani, Ilya Gekhtman, and Keivan Mallahi-Karai)

We investigate the horofunction compactification of horospherical products of trees, focusing on the relationship between coordinatewise convergence and convergence in the horofunction boundary. 

2. Metrizability of the Sublinearly Morse Boundary (Joint w/ Manisha Garg and Yulan Qing)

We study visual-type metrics on the stratum of sublinearly Morse boundaries of groups and spaces, including right-angled Artin groups and CAT(0) spaces. Our goal is to understand their topological and metric structures and the behavior of these metrics under quasi-isometries.

3. Actions of Big Mapping Class Groups (Joint w/ Yulan Qing)

We investigate the dynamics of mapping class groups of infinite-type surfaces on non peripheral curve graph. In particular, we study constructions of parabolic actions and their connections with the topology and geometry of infinite-type surfaces.

4. On Martin Boundaries of HHS

Miscellaneous:

Fractal Dimensional Approach to Seizure Detection in EEG Signals ( Joint w/ Milap Rajgor,  Aakash Mishra, Eva Kaushik, and Shipra Baranwal)

We are developing a framework for detecting epileptic seizures in EEG signals using tools from topological data analysis and fractal geometry. In particular, we investigate the Higuchi fractal dimension of cumulative sum (CUSUM) curves constructed from the total persistence of EEG data, aiming to improve the accuracy and reliability of seizure detection.