The IMA was a founder member of ICIAM (International Council for Industrial and Applied Mathematics) in 1987 together with GAMM, SIAM, and SMAI. This consortium of mathematical societies now has 54 members from all regions of the world. ICIAM hosts a large congress every four years. The first was held in Paris in 1987 and the IMA hosted one in Edinburgh in 1999. The next congress will be in Tokyo in August 2023.

In recent years, ICIAM has been involved in many other activities than just the quadrennial meeting. These include support for conferences in developing nations, involvement and representation in ISC, production of a quarterly newsletter, and support for international campaigns for diversity and gender balance. A meeting of the Board of ICIAM that comprises representatives of all member societies and a small number of officers is held every year and is hosted by one of the member societies.

The IMA bid successfully to host the 2020 Board meeting. It was cancelled because of the pandemic, but we were asked if we could hold the meeting in 2022. Partly to encourage attendance at the Board meeting, this one-day meeting is preceded by a two-day conference on applied mathematics with many board members participating and giving talks. We partnered with Strathclyde University to hold the workshop and board meeting at their Technology & Innovation Centre (TIC). The local arrangements were spearheaded by Alison Ramage and Phil Knight from the Mathematics Department who are well known for their role in the organization of the Biennial Conference in Numerical Analysis held in the University. Although I was also very involved in the organization, I was doing so at distance and am pleased to acknowledge the central role that they played in ensuring the success of the event with good assistance from the staff at TIC.

We obtained significant funding from the IMA, the LMS, the EMS (Edinburgh Maths Society), QJMAM, and GMJT that enabled us to avoid charging any registration fee for the Workshop. The LMS support assisted participants from Tier5 countries and some of the EMS money was used to support attendees from Scotland. This support was reflected in the speaker list for the Workshop. There were twenty 25-minute talks of which ten were given by people attending the Board meeting, and eight by participants from Scotland, including graduate students from Strathclyde. The programme, abstracts of the talks, and PDF files for most of the talks are available on the website tinyurl.com/3e73bhuu.

For some of the research students it was their first experience of an in-person meeting that benefited greatly from the long breaks for coffee/tea and a finger buffet lunch. It just emphasized to me that early career people and graduate students have suffered greatly because of the pandemic by being unable to network properly with their peers and more senior members of the community.

I must say that I was impressed by the range and generally high standard of the presentations. The programme embraced talks in many areas of applied mathematics so there were obviously some on topics quite foreign to me. Although I will make some very brief comments on all of the talks, I apologize for the non-homogeneity of my commentary. I will group them according to application or mathematical similarities.

Some talks addressed medical issues including work by Avshalom Offner (from Edinburgh) on positron emission particle tracking. The particles are labelled with radio isotopes that decay emitting positrons. These in turn annihilate to emit photons that can be sensed by an array of detectors and lines are drawn between pairs of these detectors that will intersect to identify the position of the particle. The main mathematics is in deciding a priori how many lines should be used and how uncertainty quantification can be used to do this. He discussed the one particle system in detail which was complicated enough and indicated the difficulties in handling the many particle system. The importance of this approach is that flows can be monitored in opaque environments and so can be used in medical applications.

Dmitry Savostyanov (Essex) and Gabriela Gomes (Strathclyde) gave talks that were relevant to the current pandemic with Dmitry overcoming the exponential growth in complexity of the more realistic network based models of the population by using low rank tensor product decompositions. He illustrated this using a transport network in Austria. Gabriela emphasized the high importance of heterogeneity in the modelling and analysis of populations, particularly when characteristics respond to selection forces. She showed that the homogeneous model greatly exaggerated the herd immunity thresholds when considering the COVID-19 pandemic.

Paulo Mancera (Brazil) discussed the modelling of melanoma growth emphasizing the danger of this common form of cancer and its high capacity to cause metastasis. He has developed an ODE mathematical model that incorporates macrophages and immunotherapy with CAR-T cells. The analysis shows radical differences in the presence and absence of tumour-associated macrophages.

The two droplet talks by Stephen Wilson and Sophie McLauchlan from Strathclyde were actually very different. Stephen presented a very detailed description of the evaporation of multiple droplets on a substrate noting that the evaporation of a single droplet is reduced by the presence of another nearby droplet. His numerical results showed excellent agreement with experiment. Sophie is a graduate student and was pleased to be able to present in an in-person meeting her very interesting and well-described work on droplets of nematic liquid crystals, as used for example in LCDs. Her talk focused on radial hedgehog solutions and their instability at low temperatures and large droplet radii. She discussed applications in biosensors.

Two graduate students from Strathclyde, Tasnia Shahid and Stephen Smith, gave a well orchestrated duet on uncertainty quantification of pollutant dispersion models. They use a streamlined upwind Petrov-Galerkin model with the model parameters coming from a distribution and get good results using lower fidelity methods combined with uncertainty quantification as opposed to using more expensive high fidelity computations. In their approach, the pollutant concentration is calculated as a stochastic variable on which they analyse the models sensitivity to different parameters to study the factors that contribute the most to the spread of air pollution.

There were two talks on numerical linear algebra. One by Jim Nagy (Emory University) on Krylov subspace regularization for very large-scale ill-conditioned systems from inverse problems. He discussed combining various regularization techniques, the classical Tikhonov regularization and the use of the approximation to singular vectors from the Golub-Kahan bidiagonalization process. He considered the use of a range of norms in the regularization process. The other NLA talk was by Philip Knight (Strathclyde) indicating the many areas where his matrix balancing to obtain a constant row sum and constant column sum for the scaled matrix had a significant impact. These included page rank computations, work on the human genome, and the apportioning of seats over a range of parameterized proportional representation scenarios.

The talk by Sven Leyffer (Argonne) resulted in a sequence of challenges requiring new or adapted techniques to optimize binary control pulses in quantum systems. He was concerned that linear algebraists did not just exponentiate a matrix entry-by-entry to obtain the matrix exponential and Jim Nagy reminded the audience of a famous paper on this by Moler in 1978 on nineteen dubious ways to compute the exponential of a matrix.

The two software oriented talks were by Carol Woodward (Lawrence Livermore) and Hong Zhang (Argonne). Carol looked at adaptive time step methods in the solution of ODE and DAE problems when solving multi-physics and multi-component systems. Software based on this work has been incorporated in the SUNDIALS package, and she showed its use in scientific applications. Hong gave a detailed description of the current status of the PETSc Library. After a general introduction, he focused on the DMNetwork looking at the solution of large scale multi-physics PDE-based network applications. He applied this to a Mississippi River simulation with over 28 million network variables and showed its performance on a 65,000 core machine at the Argonne Leadership Computing Facility.

The final two talks on the Thursday were related more to outreach with Heather Yorston from Edinburgh describing the preparation and organization of a course in industrial mathematics at MSc level designed for students with a range of mathematical backgrounds. The course was Python-based in terms of programming language and used an agent based modelling approach inter alia. Ivete Sanchez (Mexico) discussed examples of projects coordinated by CIMAT, a multi-campus university with the main centre in Guanajuato. She described some of the projects on long-term energy auctions and a review of the deterioration of bank notes. She discussed the linkage with industry and issues of the interaction between academic institutions and industry.

Kenji Kajiwara from Kyushu University considered curves and surfaces in industrial design as could be generated from CAGD software. Aesthetic shapes are defined in terms of curvature and changes in curvature and are modelled by including equations when extending and generalizing the traditionally used LAC (log-aesthetic curves). The major customers for this were was people working in car design where aesthetics was highly valued.

Wen-shin Lee from the University of Stirling gave a lively and energetic presentation about the use of exponential analysis in CSE emphasizing where this was significantly more powerful than using Fourier-based methods. Methods for regularizing the analysis to overcome issues with ill-conditioning and avoiding combinatorial complexity were also discussed.

There were two talks concerning finite-elements. One was a detailed study by Weizhu Bao from Singapore on parametric finite elements for geometric PDEs. He began reviewing geometric flows including mean curvature flow and discussed different mathematical formulations and numerical methods. An energy-stable semi-implicit parametric finite element method was considered for this case and then extended to more general flow models.

Suzanne Shontz from the University of Kansas motivated her work through a cardiac MPI to assess the functioning of the heart. The idea is to use mesh warping to map meshes from the source domain to the target domain to avoid the need for remeshing. She illustrated this through a high-order mesh warping algorithm for tetrahedral meshes based on a finite element formulation for hyperelastic materials. The ability to generate several 3D dynamic high-order meshes was important when modelling a beating heart.

Peter Benner from MPI Magdeburg discussed dynamic models used both to study transient behaviour and long-term horizon prediction. After commenting on the historical use of neural networks for system identification and its limitations, he discussed the use of machine learning to lift the nonlinear dynamics to a higher dimension and represent it by a quadratic model. The use of NN in this way would have been a natural introduction to the talk by Wil Schilders (Eindhoven and President-elect of ICIAM).

Wil presented a talk based on a document prepared for a workshop in the Netherlands on computational science and machine learning and indicated several aspects of scientific machine learning where mathematics could be used in a machine learning context to advance the solution of problems in scientific computation, often ensuring constraint satisfaction in neural network computations.

(Written for the IMA publication Mathematics Today and reprinted by permission).