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Biogeographical regions (geographically distinct assemblages of species and communities) constitute a cornerstone for ecology, biogeography, evolution and conservation biology. Species turnover measures are often used to quantify…
The analysis of the three-dimensional structure of proteins is an important topic in molecular biochemistry. Structure plays a critical role in defining the function of proteins and is more strongly conserved than amino acid sequence over…
Machine learning models of vastly different modalities and architectures are being trained to predict the behavior of molecules, materials, and proteins. However, it remains unclear whether they learn similar internal representations of…
We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from…
In this work, we develop a framework for shape analysis using inconsistent surface mapping. Traditional landmark-based geometric morphometrics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid…
Animals are diverse in shape, but building a deformable shape model for a new species is not always possible due to the lack of 3D data. We present a method to capture new species using an articulated template and images of that species. In…
Understanding the influence of an environment on the evolution of its resident population is a major challenge in evolutionary biology. Great progress has been made in homogeneous population structures while heterogeneous structures have…
In computational anatomy, the statistical analysis of temporal deformations and inter-subject variability relies on shape registration. However, the numerical integration and optimization required in diffeomorphic registration often lead to…
There is no much doubt that biotic interactions shape community assembly and ultimately the spatial co-variations between species. There is a hope that the signal of these biotic interactions can be observed and retrieved by investigating…
The interplay between space and evolution is an important issue in population dynamics, that is in particular crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution…
A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and morphology of structures in the developing…
Statistical and mathematical modeling are crucial to describe, interpret, compare and predict the behavior of complex biological systems including the organization of hematopoietic stem and progenitor cells in the bone marrow environment.…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call "mathematical data science". In this paradigm, one studies mathematical objects collectively rather than…
Despite the remarkable success of large large-scale neural networks, we still lack unified notation for thinking about and describing their representational spaces. We lack methods to reliably describe how their representations are…
In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces $X$, $Y$, $Sh(X,Y)$, defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary…
Statistical models of 3D human shape and pose learned from scan databases have developed into valuable tools to solve a variety of vision and graphics problems. Unfortunately, most publicly available models are of limited expressiveness as…
Urban planning and morphology have relied on analytical cartography and visual communication tools for centuries to illustrate spatial patterns, propose designs, compare alternatives, and engage the public. Classic urban form visualizations…
We introduce the shape module of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves and surfaces across fields of natural sciences and engineering. The shape module first implements widely used shape…
Correspondence-based shape models are key to various medical imaging applications that rely on a statistical analysis of anatomies. Such shape models are expected to represent consistent anatomical features across the population for…