Research
I am an Assistant Professor of Mathematics at Florida State University. I graduated in Spring 2016 with a Phd in mathematics from the University of Georgia. My advisor was Jason Cantarella. As a postdoc at The Ohio State University, I worked with Facundo Mémoli and the Topology, Geometry and Data Group at OSU. My research is in applications of geometry and topology to applied problems in geometrical and topological data analysis and mathematical signal processing; more specifically:
Metric Geometry and Topological Data Analysis: I use tools from optimal transport and algebraic topology to develop new methods for the analysis of data with complex structure. I'm interested in algorithmic advances, concrete applications and mathematical theory.
Statistical Shape Analysis via Riemannian Geometry: Some of my work models shapes (e.g., closed plane curves) as points in an infinite-dimensional Riemannian manifolds. The Riemannian structure provides a rich set of tools for principled analysis of shape data.
Mathematical Signal Processing: I work in compressed sensing and frame theory. My current work in frame theory is based on applying tools from the field of symplectic geometry to study spaces of frames by exploiting their symmetries.
My research is partially supported by NSF Grant DMS--2107808.
Papers
Preprints
Geometric Approaches to Matrix Normalization and Graph Balancing. With C. Shonkwiler, 2024. Preprint.
Information on Oral Temperature is More Robustly Encoded Than Taste in Neurons of the Mouse Gustatory Cortex. A. N. Nash, C. G. Bouaichi, K. E. Odegaard, T. Needham, M. Bauer, R. Bertram, R. Vincis, 2024. Preprint.
Topological optimal transport for geometric cycle matching. With S. Zhang, M. P. H. Stumpf and A. Barbensi, 2024. Preprint.
Path constrained unbalanced optimal transport. With M. Bauer, N. Charon and M. Nishino, 2024. Preprint.
On the Existence of Parseval Frames for Vector Bundles. With S. A. Ballas and C. Shonkwiler, 2023. Preprint.
Interleaving Distances, Monoidal Actions and 2-Categories. With P. K. McFaddin, 2023. Preprint.
Convergence of Leray Cosheaves for Decorated Mapper Graphs. With J. Curry, W. Mio, B. Okutan, F. Russold, 2023. Preprint.
Flexible and Probabilistic Topology Tracking with Partial Optimal Transport. M. Li, X. Yan, L. Yan, T. Needham, B. Wang, 2023. Preprint.
Comparison Results for Gromov-Wasserstein and Gromov-Monge Distances. With F. Mémoli, 2022. Preprint.
Statistical Analysis of Shape Graphs with Applications to Retinal Blood Vessel Networks. A. Basu Bal, X. Guo, T. Needham, A. Srivastava, 2022. Preprint.
Grassmannian geometry of framed curve spaces. Phd Dissertation, 2016. Preprint
Published/To Appear
A Wasserstein-type Distance for Gaussian Mixtures on Vector Bundles with Applications to Shape Analysis. M. Wilson, T. Needham, C. Park, S. Kundu, A. Srivastava. Accepted to SIAM Journal on Imaging Sciences, 2024. Preprint.
Elastic Metrics on Spaces of Euclidean Curves: Theory and Algorithms. With M. Bauer, N. Charon, E. Klassen, S. Kurtek, T. Pierron. Journal of Nonlinear Science, 2024. Article. Preprint.
Stability and Approximations for Decorated Reeb Spaces. With J. Curry, W. Mio, O. B. Okutan and F. Russold. Accepted to Symposium on Computational Geometry, 2024. Preprint.
Comparing Morse Complexes Using Optimal Transport: An Experimental Study. With M. Li, C. Storm, A. Y. Li and B. Wang. Accepted to IEEE Visualization Conference, Short Paper, 2023. Preprint.
Topologically Attributed Graphs for Shape Discrimination. With J. Curry, W. Mio, B. Okutan, F. Russold, 2023. Accepted to ICML Workshop Topology, Algebra and Geometry in Machine Learing, 2023. Preprint.
Hypergraph Co-Optimal Transport: Metric and Categorical Properties. With S. Chowdhury, E. Semrad, B. Wang, Y. Zhou. To Appear: Journal of Applied and Computational Topology, 2023. Preprint.
Gromov-Wasserstein Distances, Distance Distributions and Inverse Problems for Metric Measure Spaces. With F. Mémoli. Oberwolfach Reports, 2023. Article.
Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent. With C. Shonkwiler. Preprint. Accepted to Journal of Fourier Analysis and Applications, 2023.
Three proofs of the Benedetto-Fickus theorem. With D. Mixon, C. Shonkwiler, S. Villar. Chapter in: Sampling, Approximation, and Signal Analysis, 2024. Article. Preprint.
Geometric averages of partitioned datasets. With T. Weighill. SIAM Journal on Applied Algebra and Geometry, 2023. Article. Preprint.
Active licking shapes cortical taste coding. C. Neese, C. G. Bouaichi, T. Needham, M. Bauer, R. Bertram and R. Vincis. The Journal of Neuroscience. Article. Preprint.
Distance distributions and inverse problems for metric measure spaces. With F. Mémoli. Studies in Applied Mathematics, 2022. Preprint. Article.
Toric Symplectic Geometry and Full Spark Frames. With C. Shonkwiler, 2021. Applied and Computational Harmonic Analysis, 2022. Preprint. Article.
Admissibility and Frame Homotopy for Quaternionic Frames. With C. Shonkwiler. Linear Algebra and Its Applications, 2022. Preprint. Article.
Decorated Merge Trees for Persistent Topology. With J. Curry, H. Hang, W. Mio and O. Okutan. Journal of Applied and Comptutational Topology, 2022. Preprint. Article.
Statistical shape analysis of brain arterial networks. With X. Guo, A. Basu Bal and A. Srivastava. Annals of Applied Statistics, 2021. Preprint.
Quantized Gromov-Wasserstein. With S. Chowdhury and D. Miller, 2021. Proceedings of ECML 2021. Preprint.
Generalized Spectral Clustering via Gromov-Wasserstein Learning. With S. Chowdhury. Proceedings of AISTATS 2021. Article. Preprint.
The (homological) persistence of gerrymandering. With M. Duchin and T. Weighill. Foundations of Data Science, 2021. Article. Preprint.
Symplectic geometry and connectivity of spaces of frames. With C. Shonkwiler. Advances in Computational Mathematics, 2021. Article. Preprint.
Gromov-Wasserstein Averaging in a Riemannian Framework. With S. Chowdhury. CVPR workshop DIFF-CVML proceedings, 2020. Preprint.
The Weighted Euler Curve Transform for Shape and Image Analysis. With Q. Jiang and S. Kurtek. CVPR workshop DIFF-CVML proceedings, 2020. Preprint.
Simplifying transforms for general elastic metrics on the space of plane curves. With S. Kurtek. SIAM Journal on Imaging Sciences, 2020. Article. Preprint.
Knot types of generalized Kirchhoff rods. Journal of Knot Theory and its Ramifications, 2019. Preprint. Article.
Random triangles and polygons in the plane. With J. Cantarella, C. Shonkwiler and G. Stewart. American Mathematical Monthly, 2019. Article. Preprint.
Shape analysis of framed space curves. To appear in Journal of Mathematical Imaging and Vision, 2019. Preprint.
The Paulsen problem made symplectic. With M. Ehler, M. Hering, C. Manon and C. Shonkwiler. Oberwolfach Reports, 2018. Proceedings.
Symplectic Geometry and Frame Theory. With C. Shonkwiler. Oberwolfach Reports, 2018. Proceedings.
Kahler structures on spaces of framed curves. Annals of Global Analysis and Geometry, 2018. Article. Preprint.
Sparse recovery from saturated measurements. With S. Foucart. Information and Inference, 2016. Article.
Sparse disjointed recovery from noninflating measurements. With S. Foucart and M. Minner. Applied and Computational Harmonic Analysis, 2015. Article.
Dictionary-sparse and disjointed recovery. In: Proceedings of Sampling Theory and Applications 11th Annual Conference, 2015. Article.
Book
Introduction to Applied Algebraic Topology. Preprint. (Updated 4/10/19) These lecture notes serve as the textbook for a course that I developed at OSU with Facundo Mémoli and Crichton Ogle. The class is intended for undergraduate math and data analytics majors and aims to keep the prerequisites low. In particular, it begins with advanced linear algebra and metric topology and works through the foundational results of persistent homology.
The notes are constantly being updated. Please contact me to report any errors or to make suggestions!