*By Hong Qian*

At the Applied Mathematics: The Next 50 Years' conference, a panel discussion of the “Future Research Directions in Applied Mathematics” was put on. The entire discussion was recorded and uploaded on YouTube, the link can be found here.

The panel discussion covered a wide range of issues in connection to the future of applied mathematics. They included research directions and training, both as broad as the field of studies and as narrow as the department at the University of Washington; as well as sociology of applied mathematics and its relation to other fields of learning, outreach, and applied mathematics in the wider world.

From left to right: moderator B. Deconinck, panel members: P. Schmid (Imperial College London), C. K. R. T. Jones (UNC), K. K. Tung, A. Greenbaum, J. N. Kutz, and C. Bretherton.

**Fundamental laws, engineering, and applied mathematics**

Applied mathematics is perhaps the world's oldest ** interdisciplinary** area of research. Since the application areas and focuses shift constantly, it is one of the most dynamic branches of knowledge. It has been a continuous generator of independent academic departments when areas reaching certain maturity ready for the real world: mechanical engineering, electrical engineering, chemical engineering, etc. In contrast, biochemistry at the beginning of 20th century was a highly interdisciplinary subject; but now it has become an established area and no longer considered interdisciplinary.

In addition to developing powerful methods for solving mathematical problems, in the past several decades, there has been a growing awareness of the role ** mathematical modeling**, both in classrooms and as research areas. Many applied mathematicians are now proudly to call themselves biologists, atmospheric scientists, oceanographers, or even social scientists, in addition to fluid mechanists or physicists a half century ago. As academic departments and units, the broader applied mathematics community usually sits in various departments that include mathematics, statistics, in engineering school, and applied mathematics as at University of Washington. It is safe to say that mechanics, or more broadly laws from physics written in dynamical equations, has been a singular most significant intellectual thread of the applied mathematics proper. For majority of the past two hundred years, a genuine knowledge of mechanics was sufficient for an applied mathematical worker to start tackling any scientific problems with confidence.

Facing the rising tide of machine learning and data, scholars have compared and contrasted ** data driven models** with

**In Professor Jones' term, "datum balances modeling''. He reminisced in the 80s and 90s, computers were believed to be able to answer everything in oceanography; and discussed the "law of nature channeled through mathematics.'' Indeed, there is a fundamental difference between pre-Newtonian science including great accomplishments such as Kepler’s laws, but the mechanistic scientific (physical) understanding post Newton, that initiated applied mathematics proper, was a paradigm shift.**

*mechanism based models.***The twin challenges of "scale and complexity''**

When asked to enlist challenges that faced applied mathematics, panel members Professor Bretherton and Professor Kutz talked about “multiscales”, “heterogeneity”, and “complicated” scientific problems. Most can be captured by the ** scale and complexity**. Complexity is actually about systems that are large scale in its “information content”. Looking back, Professor Schmid listed recent past exciting developments in computational mathematics, research topics that had led to mini-paradigm shifts, generated euphoria and certain frenzy with band-wagon: uncertainty quantification, randomized algorithms, compressed sensing, and machine deep learning. Even though these are all highly specialized methodologies, one sees that all deal with

**and with an element of**

*large scale computations***. One origin of the scale and complexity that applied mathematicians confront is that we are still trying to compute things from a very fundamental level where we had the natural laws. However, as lucidly articulated by condensed matter theoretical physicist P. W. Anderson: “At each level of complexity entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other.” In other words, one needs to develop new theories and “laws” rather than starts too remote from "microscopic'' and simply relies on powerful computation, where “the twin difficulties of scale and complexity” appear, exactly the terms Anderson used!**

*randomness and uncertainty*Classical sciences based on Newtonian mathematics are epitomized in the laws, expressed in terms of equation, with certainty. The certainty in mathematics, in return, has also given the world a sense of confidence in the scientific knowledge. A fully embrace of the logic of stochastic dynamics to represent complex systems and processes, and understanding phenomena in terms of probabilities and its limiting determinism, emergence, will constitute a paradigm shift of applied mathematics for the years to come. **E**** mergence,** the statistical reasoning that turns both

**, the two adversarial factors in deterministic mathematics, into a blessing. As pointed out by Professor Greenbaum, in recent years computer graphics did not push for higher precision in computer arithmetic, but actually started half-precision format in order to efficiently capture highlights and shadows.**

*scale and complexity***Applied mathematics education**

Professor Tung briefly recounted applied mathematics as an educational program. In the United States, independent graduate education in applied mathematics started at Brown University in the 1940s. Its core subjects included differential equations; approximation theory which broadly constructed to include representations/variational, asymptotic/perturbation, numerical analysis, and applied probability. An applied mathematical worker is expected to be conversant in all these subjects, as pointed out by Professor Bretherton. On the other hand, for a long time the most natural training in an application area was mechanics. In the past several decades, however, there has been a significant diversification of the application areas for mathematics. It is safe to say now there is almost no area untouched by applied mathematics, *c.f.*, from mathematical linguistics to computational history.

Professor Tung also discussed one of the earlier research focuses of many applied mathematicians, for example C. C. Lin when he was at Brown: *singular perturbation and asymptotic analysis*. Many current programs in applied mathematics have eliminated its teaching, arguing these analytical methods are no longer significant in comparison with numerical computations.* *However, as all great stories that come back with a vengeance, modern theory of phase transitions, and emergence phenomena, convincingly argues that it is the mathematical limit that defines a new "natural law''! The limiting process makes quantitative, gradual changes into a discontinuity with qualitative differences. The limit is usually singular, involving non-uniform convergence.

On the other hand, Professor Kutz wondered how to training our students effectively using all the newly developed technologies in applied mathematics, particularly with all the computational and statistical tools, in parallel to traditional training of computations for solving differential equations. With the rise of the internet, which provides efficient communications, Professor Bretherton pointed out the nature of collaborations have also changed, and that education of individuals could take advantage of the new infrastructure.

**Applied mathematics as a core of liberal education**

Based on mathematics and especially equipped with the new mathematical thinking of probability and statistics, applied mathematics provides an individual the most coherent and penetrating analytic ability to deal with situations and affairs in the world beyond the areas covered by science, technology, engineering, and mathematics (STEM). Indeed, it is urgently needed for people in law, social services, and politics, for example, to understand the analytic foundation of *chance* and appreciate the nature of emergent phenomena. Mathematical thoughts, through its applications, is one of the most time-tested achievements of humanity. Society as a whole cannot afford to let it be only possessed by only a few specialists.

Just as Newtonian, deterministic mathematics has provided a powerful framework and tools to understand the natural world, the stochastic mathematics will provide the needed language, logic, and tools to represent the world of biological species, human societies, and activities. We have seen plenty of evidence for this bold statement: the growing areas such as behavioral finance, agent-based models in sociology and economics, for examples, can all benefit from this new mathematical modeling paradigm.

Professor Jones particularly commented on continuing making the case for applied mathematics as an equal partner of pure mathematics, and taking up more visible leadership role in outreach: articulating the relevance of mathematics in the real world. There is a great need to empower all communities, increasing the participation of minorities to be a part of Applied Mathematics and STEM enterprises.