Partial Order in Materials: at the Triple Point of Mathematics, Physics and Applications (17w5059)

Arriving in Banff, Alberta Sunday, November 26 and departing Friday December 1, 2017

Organizers

(University of Bath)

(Purdue University)

(Peking University)

(University of Waterloo)

Objectives

The theory, analysis, simulations and applications of partially ordered materials constitute novel, thriving and somewhat disconnected research fields in their own right. The lack of effective connections stems from the fact that different research communities in mathematics, physics and computational chemistry have different approaches to similar problems and the approach strongly relies on the scale and scope of the problem. For example, researchers use quantum mechanics to study atomistic-level details, molecular-level models to study spatially homogeneous nanoscale systems and phase transitions, mean-field models to study mesoscale phenomena and phenomenological continuum models to study macroscopic phenomena. An underpinning problem is the absence of a rigorous microscopic-to-macroscopic derivation of theories or even rigorous error bounds for coarse macroscopic theories. Partially ordered materials have both solid and fluid characteristics and hence, this workshops aims to discuss analytical, numerical and applications-oriented challenges in this field by bringing together a stellar and diverse group of researchers from solid mechanics, fluid dynamics, PDE experts, mutiscale analysis of materials and numerical analysis. The theoretical researchers will be complemented by at least four leading experimentalists from the USA, Canada and India who can motivate new mathematical approaches and suggest new applications for theory simultaneously.



The workshop aims resonate with the aims of Banff as an excellent venue to host groups of researchers who do not ordinarily meet at workshops and meetings. The organizers will ask the experimentalists to give an overview of some modern experiments on liquid crystals, colloidal suspensions and new materials, including details on the design and fabrication, the experimental predictions and experimental limitations and the big future challenges. The mathematicians and theoretical researchers will similarly be asked to give an overview of their fields, their techniques, the strengths and limitations of their approaches, the success stories and the mathematical challenges for the future. The organizers will encourage the delegates to identify common collaborative projects at the interface of rigorous analysis, computations and applications, which are well-aligned with the future challenges in the field. The overarching workshop goal is to create a new network of young and emerging researchers in this field across the globe, who can collaborate on cutting-edge problems that can have a profound and lasting impact in the mathematical and applied sciences alike.



The scientific agenda is based on (not exclusively) four specific themes listed below.



[1] PDEs for partially ordered materials with emphasis on the Landau-de Gennes theory for nematic liquid crystals and the Ginzburg-Landau theory for superconductivity, and their generalizations to new classes of intermediate materials. As discussed above, this is an active area of research in the applied analysis and applied mathematics communities around the globe. There has been substantial work on energy formulations for such materials, the corresponding energy minimizers and spatio-temporal pattern formation in these systems, as solutions of systems of nonlinear coupled partial differential equations, with emphasis on singularities and their characterization. The organizers have invited new and emerging researchers in this area, who have made recent contributions but have not had many opportunities to present them at international forums. This includes a cohort of researchers from the Far East .



[2] Partial order on curved surfaces. This is a relatively new and under-developed area of research compared to theme [1] above. The organizers have invited delegates who work on curvature-dependent energies for partially ordered materials, the corresponding partial differential equations and their analysis, numerical methods for PDEs on surfaces and experimentalists who can talk about the curved geometries in an interdisciplinary context. Some interesting questions include the evolution of defects on curved surfaces and how curved surfaces coated with mesogenic molecules can be used as binding sites for the manufacture of macromolecules. There are new analytical and numerical challenges and the organizers are optimistic that the workshop will provide an ideal platform for new collaborations in this direction.



[3] Multiscale analysis of partially ordered materials. There is a substantial body of work on domain decomposition methods and HMM (Heterogeneous Multiscale Methods) for coupling atomistic and macroscopic theories in solid mechanics and a portion of the invited delegates represent that research community. The organizers believe that these analytical and numerical methods can be applied to some extent, to partially ordered materials although the inherently fluidity and absence of systematic microscopic-to-macroscopic derivations will present difficulties. The aim is to formulate some model problems that are amenable to adaptations of the multiscale methods for solids and discuss the relevance or usefulness of such approaches with the experimentalists and decide on future strategies.



[4] Modelling of new partially ordered systems. There are a plethora of new experiments on colloidal suspensions of liquid crystals, viruses in confined chambers, polymers, complex fluids for microfluidics and mesogenic molecules for new applications. The theoreticians can design toy models for these new systems and can also predict, to some extent, the strength and limitations of such toy models. The experimentalists can shed light on what mathematics can actually do for the applications. The invited delegates include modellers, computational scientists, theoretical physicists and experimentalists who can understand each other and as such, the team should be in a strong position to formulate and collaborate on scientific questions of mutual interest.



The invited delegates are divided into the following categories: (i) general PDE experts with expertise in fluid dynamics, harmonic maps and variational theories in materials science, (ii) experts in elasticity, (iii) numerical analysts some of whom have expertise in multiscale methods for solids, (iv) theoretical physicists and computational scientists with expertise in complex fluids and (v) experimentalists. The workshop will largely consist of research talks by all the invited delegates and several interactive networking sessions, concluded by a collaborative session wherein the delegates will report on the scientific exchanges and new collaborations facilitated by the workshop. Additionally, the organizers will invite Professors Lia Bronsard, Gui-Qiang Chen, Fanghua Lin, and Oleg Lavrentovich to give overview-type plenary talks on the state-of-art in their research fields and international networks in their respective countries. The nominated plenary speakers are internationally known for the breadth and depth of their scientific work and for their service to the scientific/mathematical communities in their respective countries. Professor Lia Bronsard is currently the head of the Canadian Mathematical Society.



The organizers have paid special attention to under-represented communities of mathematicians. The principal applicant is a young female associate professor, within 10 years of her Ph.D. The invited delegates include six mathematicians from the Far East (China and Hong Kong) and two professors in polymer science from India, about a fifth of the invited delegates. The six mathematicians from the Far East have not spoken at many European or North American meetings, to the best of our knowledge. The organizers are also keen to pan out the Canadian landscape in this field and have tried to invite relevant Canadian researchers from University of British Columbia, University of Waterloo, University of Toronto and McMaster University. The organizers are in an excellent position to organize this workshop. The lead applicant, Dr Apala Majumdar, is Associate Professor in applied mathematics and EPSRC Career Acceleration fellow at the Department of Mathematical Sciences, University of Bath, United Kingdom and has considerable experience of organizing international workshops, for example at the IPAM (USA) and CRM (Montreal) in 2016. She was recently awarded the London Mathematical Society Anne Bennett Prize in 2015 for her outstanding contributions to the mathematics of liquid crystals and for her service to mathematics. The co-organizers are Professor Jeff Chen (Department Chair and Professor of Physics and Astronomy, University of Waterloo, Canada), Professor Changyou Wang (Professor of Mathematical Analysis, University of Purdue, USA) and Professor Pingwen Zhang (Vice Dean and Changjiang Professor , Department of Scientific & Engineering Computing , Peking University, China). All three co-organizers are internationally leading researchers in their respective fields and collectively, the organizers have expertise in mathematical modelling, rigorous analysis, scientific computation and theoretical physics, all of which are key ingredients for the scientific theme of the proposed workshop.