La Trobe Mathematics and Statistics Colloquium talk
Dr Toen Castle: I will present some different strands of my research which centre on a common theme of applied (low dimensional) geometry, in which (low dimensional) topology turns up in its typical enforcer role. Physics, chemistry, and materials engineering provide the spatially relevant systems and themes of interest to me.
I will first talk about applications of hyperbolic geometry to structural chemistry, both in generating crystal structures and understanding micellar forms (think of detergent in oil/water mixtures) and the shape of materials they template. Spatially embedded graphs are the natural objects of structural chemistry, and exciting new materials which exhibit spatial entanglement have been discovered which show the need for topology to understand physical and chemical properties. This is a subfield in its infancy, in which knot theory, graph theory and crystallography (group theory of symmetries) appear, but have not yet been very productively combined.
Finally I will introduce kirigami – the cool and lazy cousin of origami – where you use cutting and rejoining as well as folding to create a 3D structure from a flat 2D sheet. Kirigami is cooler than origami because the cutting and gluing allows new structures that origami can’t match; and it’s lazy because that extra bit of freedom can save you a lot of intricate folding, especially in creating intrinsic (Gaussian) curvature. I’ll discuss the slight simplification of kirigami on a lattice which allows the usage of ideas and terminology from crystal defects. The resulting structures are relevant to self-assembly and as they are scale independent they can (theoretically) be created on scales from nano- (graphene sheets) to micro- (microfluidics) up to human- (buildings) size.
No specialist knowledge will be assumed.
Host University: La Trobe University
Seminar Convener: Dr Toen Castle
How to participate in this seminar
1. Book your University’s Access Grid Room, or a university or APAC etc. Access Grid Room that you are able to use, and
2. Send an email to the seminar convener at La Trobe University (Toen Castle) to advise that you will be attending the seminar.