Applied Differential Geometry seeks opportunities for applying differential geometry to science
and technology, with a focus on healthcare technology driven by complex imaging modalities and
on extensions of general relativity theory.
Applying differential geometry to solve 21st century grand challenges
Societal challenges and scientific curiosity are important actuators for mathematics, and vice
versa. This is especially true for differential geometry, a generic framework straddling many areas
of science and technology, as it essentially deals with `stuff’ that furnishes space and time.
Our strategy is to engender and exploit synergies among said `X-geometries’ by investigating their
common ground. This argues for an approach beyond axiomatics in terms of operational
concepts (`algorithms’). Our stratagem is to pursue our mission by combining inductive and
analytical reasoning, establishing differential geometry (i) as an overarching framework for distinct
application areas to fruitfully exchange ideas, and (ii) as a firm basis for incorporating artificial
intelligence techniques to learn from annotated data while respecting a priori geometric
constraints (geometric deep learning).
We are currently working on the following programmes:
- Neurogeometry: Interpreting diffusion weighted magnetic resonance images in terms of
underlying neural pathways for neurosurgical applications (`tractography’). This programme
focuses on personalised care for brain tumor patients and is executed in collaboration with the
Department of Neurosurgery of ETZ, with support from NWO TTW (OTP programme), ZonMw
(TZO programme), and various academic and industrial partners.
Details: Diffusion weighted imaging (DWI) is a magnetic resonance imaging modality capable of
producing scalar images on a six-dimensional ‘(x,q)’-domain. Three of the independent variables refer
to position (‘x’) and three to a machine-controllable diffusion gradient magnetic field in the scanner (‘q’). The signal S(x,q) acquired with the help of a suitable diffusion sensitisation protocol can be related to a probability density function P(x,y) for the y-displacement of diffusing (water-bound) hydrogen spins at each position x in the brain, aka the ‘ensemble average propagator’. These high-dimensional images provide a rich source of information about the brain, which can be seen as a porous medium affecting the local diffusivity of water molecules. However, we must then solve the notoriously hard inverse problem to extract quantitative and/or visually meaningful information about the brain’s neural
architecture from these complex diffusion data. Our point of departure is to try and relate the physical
signals S(x,q) and P(x,y) to mutually conjugate representations of scalar fields that naturally live on the
so-called slit-tangent bundle in a Finsler geometric framework, and to exploit the machinery of Finsler
geometry for solving the inverse problem. This endeavour is part of the 21st century grand challenge to unravel the human brain, coined ‘connectomics’, which has drawn massive attention worldwide,
boosted by the advent of DWI. We are particularly interested in turning scientific results into instruments for improving the neurosurgical workflow.
- Cardiovascular geometry: Analysing vascular images from various medical imaging modalities,
based on the generic prior that vessels are spatially elongated structures. Complicating factors,
such as the occurrence of bifurcations and inevitable imaging imperfections, are effectively
handled by a geometrical paradigm that (i) `lifts’ spatial evidence to an abstract manifold in
which space and orientation are pried apart, and (ii) exploits geometric deep learning.
Details: Having a healthy cardiovascular system is vital to our health and well-being. A proper
assessment of the circulatory system from a medical scan is not merely a matter of depicting arteries
and veins but requires a topologically correct mapping of full vessel trees. Data noise and vessel
crossings and bifurcations often lead to confusion due to ambiguities or missing pieces of information.
To properly deal with these issues we consider ‘liftings’ of spatial data into higher-dimensional
homogeneous spaces, a trick of the trade from differential geometry that allows one to pry apart
position and orientation. After lifting, empirical data are more sparsely distributed over a higher dimensional domain but, in a precise sense, still carry exactly the same information. Subject to
geometric partial (linear, morphological) differential equations of parabolic type, the higher-dimensional images can be processed so as to enhance, complete, and delineate the line-like structures of interest, with much less chance of confusion as compared to traditional ‘direct’ routes. After processing, the lifted data can be back-projected into 3-space for human inspection. We furthermore propose ‘geometric deep learning’ as a natural augmentation to learn from extrinsic expert annotations, thus complementing the data intrinsic analysis sketched above.
- Spacetime geometry: Pushing the boundaries of Einstein’s relativity theory. This `classical’
theory is written in the language of pseudo-Riemannian geometry. In spite of its tremendous
success, it fails to explain striking phenomena at galactic and cosmological scales, and cannot
be reconciled with quantum theory. This programme, supported by a personal grant from NWO
(ENW domain, Physics discipline, formerly known as FOM), seeks remedies within the more
powerful framework of pseudo-Finsler geometry.
Details: In order to accommodate unexplained spacetime and gravitational phenomena observed at
galactic and cosmological scales, we need to extend our purview of Einstein’s relativity theory beyond
its classical reach. Finsler geometry is a canonical extension of Riemannian geometry, aptly
paraphrased as ‘Riemannian geometry without the quadratic restriction’. Since the classical theory is
formulated in the language of (pseudo-)Riemannian geometry, a promising avenue is to view it as some (as yet poorly understood) limiting case of a more comprehensive (pseudo-)Finslerian framework. But,
without any guiding principles, the latter brings in infinitely many new degrees of freedom, begging the question of how to apply Ockham’s razor in order to ‘make things as simple as possible, but not
simpler’ in the light of new physical phenomena to be accounted for. Our approach to figure this out is
to venture upon a heuristic program with a modest amount of foresight, stipulating a priori geometric
constraints, and then investigating the pros and cons implied by these. Intriguing results have been obtained.