About
I am a Ph.D. candidate in Statistical Science at Indiana University Bloomington. My research program develops rigorous statistical methodology for data in non-Euclidean spaces. Many modern datasets violate classical Euclidean assumptions: brain connectivity networks live on manifolds of positive definite matrices, and spatial processes evolve on curved surfaces. My work addresses how to perform valid statistical inference when the geometry of the data space itself is non-trivial.
I approach this challenge through two complementary directions. First, I develop statistical inference methods for manifold-valued data, with applications to brain connectomics and multi-site neuroimaging studies. Second, I construct and analyze Gaussian processes on compact manifolds for spatial statistics on curved domains. I also have a background in mathematical physics, having worked on braid group representations arising from topological quantum field theories.
Research Interests
Statistical Inference for Manifold-Valued Data
Developing Riemannian ANOVA methods for comparing samples of symmetric positive definite matrices, with applications to brain connectomics and multi-site neuroimaging harmonization.
Stochastic Processes on Non-Euclidean Spaces
Establishing mathematical foundations for Gaussian random fields on tori using spectral analysis and pseudo-differential operator theory, including sample path regularity of Matérn processes.
Statistical Software for Riemannian Data
Developing R packages (riemtan, riemstats) that provide scalable, numerically stable tools for Riemannian geometry in connectome analysis, including Fréchet ANOVA and multi-site harmonization.
Topological Quantum Computing
Investigating braid group representations from braiding gapped boundaries of Dijkgraaf-Witten theories and their implications for universal gate sets in topological quantum computing.
Software
riemtan
Streamlined statistical analysis of connectomes
An R package implementing Riemannian geometry tools for symmetric positive definite matrices. Features include:
- Five metrics: AIRM, Log-Euclidean, Euclidean, Log-Cholesky, Bures-Wasserstein
- Logarithmic and exponential maps
- Frechet mean computation and covariance estimation
- Apache Arrow/Parquet backend for memory efficiency
- Parallel processing via futureverse (3-8x speedup)
riemstats
ANOVA on Riemannian manifolds
An R package for statistical analysis on Riemannian manifolds, designed for ANOVA testing on manifold-valued data. Features include:
- Permutation-based ANOVA for exact inference
- Multiple test statistics (Log-Wilks lambda, Pillai's trace)
- Batch effect correction (ComBat, rigid alignment)
- Preprocessing tools for manifold data
Publications
For a complete list of publications, please visit my Google Scholar profile.
Peer-Reviewed Articles
- Patel, R., Stall, M., Hohman, R. and Escobar-Velasquez, N. "Investigating the Effects of Vocal Frequency and Intensity on Cepstral Peak Prominence and Laryngeal Kinematics via Electroglottography in Vocally Healthy Children". Journal of Voice. 2025.
- Martinez-Romero, J., Fernandez, M. E., Bernier, M., Price, N. L., Mueller, W., Candia, J., Camandola, S., Meirelles, O., Hu, Y.-H., Li, Z., Asefa, N., Deighan, A., Vieira Ligo Teixeira, C., Palliyaguru, D. L., Serrano, C., Escobar-Velasquez, N., Dickinson, S., Shiroma, E. J., Ferrucci, L., Churchill, G. A., Allison, D. B., Launer, L. J., de Cabo, R. and The Study of Longitudinal Aging in Mice (SLAM) Investigators. "A Hematology-Based Clock Derived from the Study of Longitudinal Aging in Mice to Estimate Biological Age." Nature Aging, 4(12):1882–1896, 2024.
- Escobar-Velásquez, N., Galindo, C., and Wang, Z. "Braid Group Representations from Braiding Gapped Boundaries of Dijkgraaf–Witten Theories." Pacific Journal of Mathematics, 300(1):1–16, 2019.
Preprints
- Escobar-Velasquez, N. "riemtan, riemstats: R packages for Riemannian Geometry Techniques in the Analysis of Multiple Samples of Connectomes." arXiv:2508.01342, 2025.
- Escobar-Velasquez, N. and Harezlak, J. "A Novel Testing Approach for Differences Among Brain Connectomes." arXiv:2511.09431, 2025.
- Escobar-Velasquez, N. "Pseudo-Differential Operators and Generalized Random Fields over Tori." arXiv:2511.09423, 2025.
- Ji, H., Beyaztas, U., Escobar-Velasquez, N., Luan, Y., Chen, X., Zhang, M., Zoh, R., Xue, L., and Tekwe, C. "MECfda: An R Package for Bias Correction Due to Measurement Error in Functional and Scalar Covariates in Scalar-on-Function Regression Models." arXiv:2510.21661, 2024.
Other Contributions
- Jamshidi-Naeini, Y., Escobar Velasquez, N., Golzarri-Arroyo, L., Ali, S., Howard, L. R., Dickinson, S., and Allison, D. B. "Promoting Trustworthiness of Science: Reproducing and Verifying Agarwal et al.'s (2022) Findings Through Collaborative Endeavors." Journal of Alzheimer's Disease Reports, 8(1):677–679, 2024.
Teaching
Instructor of Record
- Introduction to Business Statistics Fall 2025
-
Introduction to College Mathematics I
Spring 2022
Led a section of 45 first-generation students through a course designed to strengthen the skills needed to succeed in the Calculus series.
- Introduction to College Mathematics II Fall 2021
-
Calculus I
Spring 2018
Universidad de los Andes, Bogotá, Colombia
Teaching Assistant
- Exploratory Data Analysis Fall 2022
- Brief Survey of Calculus I Summer 2020, Fall 2021, Spring 2022
-
Mathematics of Decision and Beauty
Summer 2021
Directed a discussion session with 30 students in a virtual setting. The course introduced students to elementary game theory and voting systems.
- Introduction to Calculus with Applications Fall 2020
Presentations & Media
Contact
Office
Swain Hall East 215
Indiana University Bloomington