The Istituto per le Applicazioni del Calcolo "Mauro Picone" (IAC), one of the eldest CNR-Institutes, is a research institute of applied mathematics. It was founded by Mauro Picone in 1927, and since 2002 it has locations in Rome, Bari, Florence and Naples with about 50 researchers. Amazingly enough, its current research policy still reflects many of Picone's original ideas. Indeed, long before the introduction of digital computers, Picone had the intuition of the potential impact on real-life problems of the combination of computational methods with mathematical abstraction.
Modern society is becoming increasingly complex, and new technologies are being developed more and more rapidly. In this context mathematics is not merely a language to describe scientific processes and technologies. Its real value lies much more in depth. The high level of abstraction of mathematics does not only increase theoretical insight, it also provides, at low costs, flexible methods, efficient algorithms and more accurate solutions. In other words, a correct and intelligent combination of mathematics and computing power opens up entirely new possibilities to deal with complexity and to design efficient solution techniques which meet with the flexibility requirements which are typical for today's applications.
It is in this context that the Institute is working, in the firm belief that the CNR offers the best conditions in Italy for promoting this kind of activities. In fact the specific mission of the Institute is "to develop highly advanced mathematical, statistical and computational methods in order to solve, in a mostly interdisciplinary context, problems with strong relevance to society and industry". Applications can be found in many fields having direct impact on the society such as engineering (material science, turbulence, Bose-Einstein condensation, microflows), medical sciences and biology (medical image processing, genomics, the human immune system, blood flow), environment (analysis of satellite data for earth observation, subglacial lakes at the south pole), transportation (urban traffic modeling), finance and economics (optimization of the management of the Public Debt in Italy, micro-dynamics of financial markets), cultural heritage (degradation of ancient monuments), manufacturing (robotics, computer vision, scheduling problems), computer science (networks, security).
For two different reasons interdisciplinarity is a key concept for the research at the Institute. The first - and most obvious - one concerns the intensive interaction between mathematics and applications. Mathematics does not only provide solutions for all sort of problems, but viceversa, complex real-world problems directly stimulate the development of new mathematical techniques and approaches. For example, practical problems in computer vision require the development of new theoretical and numerical methods in the field of calculus of variations; medical applications of magnetic resonance require the development of new results in both statistics and computational linear algebra; the simulation of the human immune system leads to new types of computational modeling. Even classical fields, such as fluid dynamics, when looked from a more general mathematical perspective, yield new methodologies of wide interdisciplinary applicability in science and engineering. The choice of the examples is arbitrary and many other examples can easily be added to the list.
A second aspect of interdisciplinary research is less obvious but equally important. It refers to the interaction between different mathematical fields, often necessary to obtain the optimal solution of a problem. Indeed complex real-life problems usually require the combination of more than one single technique in order to obtain satisfactory solutions. This sort of interaction is at a relatively early stage of development, but this is not so surprising: combining highly sophisticated methods from different fields requires the formation of teams of first rank researchers, and can be considered as one of the great challenges of applied mathematics for the near future. Its strategic importance should lead to a future key action the European Commission. At the Instutute several pioneering experiments have led to most promising and convincing results. For example, in a collaboration with the Italian Ministery for Economic Affairs concerning the optimal management of the Italian Public Debt, only a highly nontrivial combination of methods coming from stochastic control theory, discrete mathematics, statistics, mathematical analysis and computational mathematics leads to optimal results. Similarly, image processing requires the combination of methods in the fields of real and complex analysis, statistics, probability theory, discrete and continuous optimization, linear algebra and numerical analysis. And realistic traffic modeling requires the use of nonlinear partial differential equations at one side and methods of discrete optimization at the other side. It should be emphasized that in particular combining methods in discrete mathematics and continuuum mathematics often adds considerable value to the solution of complex and/or multiscale problems. Having a team of researchers working in this direction, all stimulated by real-life problems, is one of the major hallmarks of the Institute. The particular blend of knowledge and experience of the IAC research staff members allows an effective employment of common techniques in disciplines that look very different like finance and biology. In fact most of the staff members are involved in more than one of the applications mentioned below.
The overall know-how of the Institute can be summerized by the following list of keywords: mathematical modeling, partial differential equations, probability theory, statistics, operations research, inverse problems, signal and image processing, mathematical physics, numerical analysis, computational linear algebra, fluid mechanics, kinetic theory, nonlinear dynamical systems, complex systems, relativity theory, mathematical biology, mathematical finance, data analysis, calculus of variations, optimization, control theory, computer science, computer graphics, scientific visualization, computing theory, traffic modeling.