Geometrical and Topological Data Analysis

In geometric data analysis, we have adapted ideas from quasi-conformal and Teichmuller maps (from complex analysis) and statistics to study wing shapes in insects, recognizing that this framework reflects aspects of the biophysical development of shape in 2-dimensions. We have also applied ideas from projective geometry to study avian egg shape across species, and then deployed a mechanical perspective to quantify the observed variation – thus linking the low-dimensionality of descriptive approaches to the low-dimensionality of predictive approaches. And we have used conformal maps to understand the variability in folding patterns in developing brains, along with physical approaches to explain the patterns mechanistically.

In topological data analysis, we played with the idea of Generalized Erdos Numbers to consider the closeness of nodes in a network, and based on this, thought of a simple clustering algorithm to detect communities of friends. And we are in the process of adapting ideas from homology theory to study the pigmentation patterns in avian eggs with the aim of understanding the psychophysical aspects of pattern recognition among brood parasites.

Related Publications

Avian egg shape: Form, function, and evolution M.C. Stoddard, E.H. Yong, D. Akkaynak, C. Sheard, J.A. Tobias, L. Mahadevan,  Science  356, 1249-1254, 2017. [View PDF] [Download PDF]
Planar morphometrics using Teichmüller maps Gary P. T. Choi and L. Mahadevan,  Proceedings of the Royal Society  A 474:20170905, 2018. [View PDF] [Download PDF]
Planar morphometry, shear and optimal quasi-conformal mappings G.W. Jones and L. Mahadevan,  Proceedings of the Royal Society (A) , 469, 20120653, 2013. [View PDF] [Download PDF]
Generalized Erdös numbers for network analysis Greg Morrison, Levi H. Dudte and L. Mahadevan,  Royal Society Open Science , 172281, 2018. [View PDF] [Download PDF]
Discovering communities through friendship G. Morrison, L. Mahadevan.  PLoS ONE  7(7): e38704. 2012. [View PDF] [Download PDF]
Computational analysis of size, shape and structure of insect wings. M.K. Salcedo, J. Hoffmann, S. Donoughe and L. MahadevanBiology Open (2019) 8, bio040774. 2019. [DOI] [View PDF] [Download PDF]
Evolution of avian egg shape: underlying mechanisms and the importance of taxonomic scale C. Sheard, D. Akkaynak, EH Yong, L. Mahadevan & J. A. Tobias, IBIS 161 (4) 15 July 2019. [DOI] [View PDF] [Download PDF]
On the growth and form of cortical convolutions T. Tallinen, J.Y. Chung, F. Rousseau, N. Girard, J. Lefèvre, and L. Mahadevan,  Nature Physics , 12, 588-93, 2016. [DOI] [View PDF] [Download PDF]