In Fall 2018, the CMSA will host a Program on Mathematical Biology, which aims to describe recent mathematical advances in using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The plethora of natural shapes that surround us at every scale is both bewildering and astounding – from the electron micrograph of a polyhedral virus, to the branching pattern of a gnarled tree to the convolutions in the brain. Even at the human scale, the shapes seen in a garden at the scale of a pollen grain, a seed, a sapling, a root, a flower or leaf are so numerous that “it is enough to drive the sanest man mad,” wrote Darwin. Can we classify these shapes and understand their origins quantitatively?
In biology, there is growing interest in and ability to quantify growth and form in the context of the size and shape of bacteria and other protists, to understand how polymeric assemblies grow and shrink (in the cytoskeleton), and how cells divide, change size and shape, and move to organize tissues, change their topology and geometry, and link multiple scales and connect biochemical to mechanical aspects of these problems, all in a self-regulated setting.
To understand these questions, we need to describe shape (biomathematics), predict shape (biophysics), and design shape (bioengineering).
For example, in mathematics there are some beautiful links to Nash’s embedding theorem, connections to quasi-conformal geometry, Ricci flows and geometric PDE, to Gromov’s h principle, to geometrical singularities and singular geometries, discrete and computational differential geometry, to stochastic geometry and shape characterization (a la Grenander, Mumford etc.). A nice question here is to use the large datasets (in 4D) and analyze them using ideas from statistical geometry (a la Taylor, Adler) to look for similarities and differences across species during development, and across evolution.
In physics, there are questions of generalizing classical theories to include activity, break the usual Galilean invariance, as well as isotropy, frame indifference, homogeneity, and create both agent (cell)-based and continuum theories for ordered, active machines, linking statistical to continuum mechanics, and understanding the instabilities and patterns that arise. Active generalizations of liquid crystals, polar materials, polymers etc. are only just beginning to be explored and there are some nice physical analogs of biological growth/form that are yet to be studied.
The CMSA will be hosting a Workshop on Morphometrics, Morphogenesis and Mathematics from October 22-24 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).
For a list of lodging options convenient to the Center, please visit our recommended lodgings page.
Time | Speaker | Title/Abstract |
8:30 – 9:00am | Breakfast | |
9:00 – 9:15am | Welcome and Introduction | |
Tutorial on Morphometrics | ||
9:15 – 10:30 | Anuj Srivastava | Geometric methods for functional data analysis |
10:30 – 10:45 | Coffee Break | |
10:45 – 12:15 | Sayan Mukherjee | Title: Integral Geometry for Modeling Shapes
Abstract: We study transformations of shapes into representations that allow for analysis using standard statistical tools. The transformations are based on Euler integration and are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. By using an inversion theorem, we show that both transforms are injective on the space of shapes—each shape has a unique transform. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. The main theoretical result provides the first (to our knowledge) finite bound required to specify any shape in certain uncountable families of shapes, bounded below by curvature. This result is perhaps best appreciated in terms of shattering number or the perspective that any point in these particular moduli spaces of shapes is indexed using a tree of finite depth. We also show how these transformations can be used in practice for medical imaging applications as well as for evolutionary morphology questions. |
12:15 – 1:30 pm | Lunch | |
1:30 – 2:15pm | Jukka Jernvall | |
2:15 – 3:00pm | Arkhat Abzhanov | |
3:00 – 3:30pm | Coffee Break | |
3:30 – 4:15pm | Siobhan Braybrook | |
4:15 – 5:00pm | Karen Sears |
Time | Speaker | Title/Abstract |
8:30 – 9:00am | Breakfast | |
Tutorial on Morphometrics (methods and applications) | ||
9:00 – 10:30 am | David Gu | Geometric flows for shape matching |
10:30 – 11:00am | Coffee Break | |
11:00 – 12:30pm | L. Mahadevan/Serra | Maps and flows for shape comparison |
12:30 – 2:00pm | Lunch | |
2:00 – 2:45pm | Eric Klassen | |
2:45 – 3:30pm | Laurent Younes | Title: Estimation of Laminar Coordinate Systems between Two Surfaces.Abstract: Given two disjoint open surfaces, we discuss the problem of estimating a parametrization (x,y,z) -> F(x,y,z) of an open subset of the three-dimensional space such that the set [z=0] corresponds to one of the surfaces, [z=1] to the other one, and for each t, (x,y) -> F(x,y,t) is an embedding onto the surface [z=t], with the additional constraint that the derivative of F with respect to t is always normal to this surface.
The resulting construction is a natural way to build a foliation between the two surfaces for which the length of the transverse lines provides a good definition of thickness of the considered volume. It is designed with the objective of analyzing cortical volumes, for which a robust definition of thickness is essential for the characterization of degenerative diseases. Our estimation algorithm uses a version of the large deformation diffeomorphic metric mapping between surfaces in which normality constraints of the velocity field are enforced. The estimated coordinate system can also be used to estimate “cortical layers’’ as laminar segmentations of the inter-surface region that satisfy a local equi-volumetric condition called Bok’s hypothesis. Experimental results on human and feline brains will be presented as illustrations. This is joint work with Tilak Ratnanather, Sylvain Arguillere and Kwame Kutten. |
3:30 – 4:00pm | Coffee Break | |
4:00 – 4:45pm | Cassandra Extavour | |
4:45 – 5:30pm | Nipam Patel |
Time | Speaker | Title/Abstract |
8:30 – 9:00am | Breakfast | |
9:00 – 9:45am | David Mumford | |
9:45 – 10:30am | Peter Olver | |
10:30 – 11:00am | Coffee Break | |
11:00 – 11:45am | Stephanie Pierce | |
11:45 – 12:30pm | Alain Trouve | |
12:30 – 2:00pm | Lunch | |
2:00 – 2:45pm | Shing-Tung Yau | |
2:45 – 3:30pm | Olivier Pourquie | Summary of Conference |