Not long ago, I was attending a birthday party for a friend of my then six-year-old son. The father of the birthday girl was a friend and colleague of mine in the Department of Mathematics at Drexel University. Several newer department members whom I did not know were also in attendance, so we struck up a conversation around the natural question, “So, what do you work on?” One worked in Lie theory, another in the area of differential geometry, yet another in real analysis, etc. When it came to my turn I said, “Well, my most recent project involved creating decision aides for signal corps officers, a role the Army calls the S6, to make better network management decisions.” Suddenly, conversation ceased and the looks on their faces changed. Something was clearly wrong, wrong in a third-arm-growing-out-of-my-head kind of way. Was it something I said?
Realizing my faux pas was the human users and their scruffy human problems, I quickly backpedaled and blurted out, “Optimization. I work in network optimization algorithms.” All was well. Conversation resumed. Songs were sung, cake was consumed, and network management on the battlefield was reduced to a mathematical optimization problem.
This was, on my part, a terrific over-simplification of my project, which had demanded structured interviews of several S6 officers about their current cognitive workflows and development of a task model for their activities across different network and mission software systems. All of this involved novel elements of mathematics, computing, software engineering, human-machine interfaces, and, yes, optimization. Sure, there were some interesting algorithmics, but this was about much more than an optimization problem. The real excitement in the project lay in our effort to take a fuzzy, human-driven process and accelerate it with cognitive models and algorithmics. Why couldn’t I better convey the intellectual excitement of trying to use formal techniques to make this scruffy problem more tractable?
My birthday party conversation was a reminder of how natural it is for us to self-identify with our academic sub-disciplines – our tribe. But describing problem-driven and use-inspired research is still a hard task, and sometimes results in a combination of curiosity and derision. I’ve been wondering how to change this perception.
Figure 1. Grumman F6F-3 Hellcats in tricolor camouflage. The Grumman F6F Hellcat was developed during World War II. Photo credit: Wikimedia Commons.
As we look around the science and engineering landscape, there are many such opportunities for mathematics. In most of these cases, the nature of the opportunity is the same; people are desperate for new tools and techniques that can transform a scruffy, ill-posed problem into one that is more precise, and can ideally then be fully captured analytically or attacked with computational techniques.
In many ways, we are back at the beginning. The origin story of mathematics often centers on how the ancients developed primitive accounting techniques and tools for everything from documenting recipes for beer and handling agricultural inventories to performing land management. Mathematics emerged from the need to solve these then-muddled, real-world problems. What new vistas can we open for exploration today?
In the Defense Sciences Office (DSO) at DARPA, we try to break assumptions about use-inspired research, and mathematics is one of our main enablers. The approach followed by several of our program managers is to make the mathematics codependent with a problem-specific question. Here are just a few examples.
Mathematicians in our Enabling Quantification of Uncertainty in Physical Systems (EQUiPS) program are working on new ways to manage the curse of dimensionality in the context of physical systems, such as marine vehicle design. The use of physical phenomena to focus and constrain the fundamental mathematical inquiry yields new insights for both members of the mathematical community and the applied sciences domains they study. Members of SIAM’s Activity Group on Uncertainty Quantification are actively involved in this multi-year program.
Graph-theoretic Research in Algorithms and the PHenomenology of Social Networks (GRAPHS) is a program in theoretical computer science and combinatorial algorithms intended to create new techniques for processing graph-based data. In this project, each performer brought a domain context within which to test and evaluate the algorithms they were developing. The program, which is nearing completion, has shown how to improve social media analysis, reduce hospital readmissions, and optimize resource allocation in adversarial games.
The Complex Adaptive System Composition and Design Environment (CASCADE) program addresses the mathematics of an emerging class of design problems: systems of systems. If systems design (i.e., design of a new aircraft or satellite) was not hard enough, designing a “systems of systems” in which the functions are disaggregated across many simpler elements is positively baffling. Such design requires formal means that consider how to best break down the collective objectives into more primitive behaviors and then coordinate the entities producing these behaviors to achieve shared objectives, all the while adapting to changing states. Most existing approaches to this problem space (e.g., the design of communications networks) are empirical or based on simulations and Monte Carlo methods. In seeking a more analytic framework, the CASCADE program looks toward emerging areas like category and sheaf theory to provide breakthrough insights and become the foundations to design tools.
Our TRAnsformative DESign (TRADES) program aims to leverage the huge advances in materials science and manufacturing technologies over the past decade. In spite of these advances, the mathematical structures that underlie current computer-aided design systems are based on work by mathematicians like Bezier, Braid, Riesenfeld/Cohen/Lyche, and Voelcker/Requicha in the 1960s and 1970. Members of SIAM’s Activity Group on Geometric Design have made many contributions to this body of work as well. However, we have reached a crisis point where the materials structures we can conceivably produce are far beyond the mathematical modeling tools and software systems we use to design things. The TRADES program attempts to create new mathematical foundations for these possibilities, but will require mathematicians to embed themselves with both materials scientists and traditional engineers.
Figure 2. Information technologies now make it possible to design configurations of matter across 106 to 109 orders of magnitude. Various computational techniques can describe and predict the mechanics and physics of materials on many different length and time scales. Figure credit: Dennis Kochmann.
In each of these cases, the mathematics and computation do not exist in a vacuum. Rather, they are wholly situated within the context of the larger problem. This is hardly a new idea; these programs are a continuation of the use-inspired scientific and research culture in the United States that emerged from the problem-driven needs of our nation during World War II. It is worth remembering that Bezier’s curves are based on the shape of the hoods of Renault cars, Monte Carlo techniques were the Manhattan Project’s means of simulating the movement of neutrons during atomic detonation, and graph theory originated in response to the need to plan a nice Sunday stroll across the bridges of a Prussian town on the Baltic Sea.
Embedded in the car parts, neutrons, and bridges are mathematical realms that we have yet to fully explore. We are wrestling with issues such as the implications for machine learning and deep learning technologies, the role of computation and data as a potential accelerator for scientific discovery, the appearance of the human-machine innovation team, and our understanding of what makes us human – from the workings of our neurons and our individual behaviors to our social systems. In this current world, with its data-rich and increasingly complex problems, what other mathematical frontiers are waiting to be discovered?