1. Mathematics—a broad perspective

 

Mathematics is a remarkable invention of human thought, a purely logical construction independent of experience and yet the basis for our understanding of the laws of nature. Much of modern mathematics, beginning with the invention of calculus in the seven-teenth century, was developed in a close interplay with discoveries in physics and other natural sciences. It is now generally recognized that the need for mathematics is increasing, as mathematics provides the appropriate tool for modeling and understanding complex phenomena in nature, technology, and society.

 

While the interaction with science has persisted to this day, a distinct feature of mathematics in the twentieth century is its division into “pure” mathematics as opposed to “applied” mathematics. The extensive development of abstract mathematics in the beginning of the twentieth century, leading to new fields such as topology and functional analysis, was fundamental for the blossoming of mathematics since that time. This development could be viewed as an important intellectual liberation, but the downside was that parts of mathematics lost touch with its origins. During recent years, however, we have witnessed a notable reversal; nowadays, we see ever more examples of advanced mathematics, previously considered “pure” and useless, penetrating applied areas. The lesson is that the notions of  “pure” and  “applied” mathematics at best are time dependent, and serious mathematics should never be viewed as “useless”.

 

Mathematics is a vast area, and its literature grows exponentially. There are currently about 600 research journals covered by the authoritative Mathematical  Reviews, and some 75000 reviews produced every year. The total volume of the current literature is estimated to some 50 million pages.  Mathematics is today such a formidable construction that no one is able to overlook all of it, and most mathematicians can only hope to master a tiny piece of it. Making bold predictions about what parts of mathematics will come to use is therefore virtually impossible.

 

A safe prediction, however, is that mathematics will be applied in new areas, and more advanced mathematics will be used in areas where it is currently not in use. This development will continue at a rapid speed.  No area is ”safe” from the influence of mathematics and no area of mathematics is ”safe” from being potentially applicable.

 

An interesting example is algebraic geometry, a mathematical discipline that until quite recently was considered to be of purely academic interest. The prestigious Institute for Mathematics and Applications (IMA) at the University of Minnesota has devoted the academic year 2006/2007 to applications of algebraic geometry. The background for this initiative is that effective use of algebraic geometry is now found in applications such as optimization, control problems, bioinformatics, data communication, and computational geometry. This example illustrates the time dependency of the notion of “applied mathematics” and the potential utility of advanced mathematics of any kind.

 

Teaching and research in mathematics should keep the long-time perspective that serious mathematics is important and potentially useful. We should maintain and develop our excellence in the field[1].

 

2. Mathematics and computer technology

 

Modern computer technology has increased the need for mathematics, as well as the range of scientific problems for which mathematics is relevant. Computers have in many ways revolutionized the research in science and technology. Mathematical modeling and numerical and statistical computations are becoming essential components in an increasing number of fields, in the analysis of computer-generated data, in predictions, simulations, and verifications of theoretical models. Research in fields such as biology, chemistry, geology, and physiology depends more and more on computer simulations, and as a result, these fields are getting “mathematized”. Also, the mathematics needed in these fields becomes more advanced and sophisticated.

 

The 21^st century will show tremendous progress in our understanding of biological systems and the human body, and ICT along with mathematics and statistics will play a significant role in this development.

 

Many computations that were previously unimaginable, have now become routine exercises. This is not only due to better and faster computers, but also to more efficient algorithms. This point is illustrated by the following statement from the NSF report “Mathematics and Science” [4]: “For more than 40 years, gains in problem solving power from mathematical algorithms have been comparable to the growth of raw computing speed, and this pattern is likely to continue. In many situations, especially for multiscale and chaotic problems, fast hardware alone will never be sufficient; methods and theories must be developed that can extract the best possible numerical solutions whatever computers are available.” Thus the use and development of  “supercomputers” must be accompanied by appropriate mathematical competence.   

  

Another important consequence of the development of computer technology is that numerical simulations are relevant and pertinent alternatives in situations in which controlled physical experiments are either impossible or too expensive. An illustrative example is that of astrophysicists studying solar activities, a field in which the amounts of data are increasing exponentially. The use of modeling and computations, as a supplement or alternative to physical experiments, will be become increasingly important, both in scientific and industrial research.

    

The development of computer technology faces mathematics with new challenges that lead to new fields. An interesting example of high current interest is that of multiscale modeling. Strictly speaking, multiscale phenomena have been studied for a long time in mathematics, but the current development is driven by applied sciences and computational capabilities. The need for multiscale modeling can be seen in physics, where processes at different scales are governed by physical laws of different character, say, by quantum mechanics at one scale and classical mechanics at another. To quote [1]:

“Our computational capability has reached the stage when serious multiscale problems can be contemplated, and there is an urgent need for science and technology - nanoscience  being a good example – for multiscale techniques”.

 

Mathematicians and statisticians at NTNU should contribute to the “mathematization” of scientific and industrial research and be relevant and active participants in this development. Mathematicians at NTNU should be proactive in identifying those parts of the mathematical sciences that are important for modern applications, and this should be reflected in teaching and research collaboration.     

 

3. Predictions concerning boundary conditions for teaching and research at NTNU in 2020

 

• No revolution will take place concerning ways of learning. In recent years we have seen various novel ways of presenting new material: Using various electronically based methods we can display animations, illustrations, etc that make presentations more elegant and comprehensible than ever before. Furthermore, there has been considerable increase in the understanding of the human learning process. However, when it comes to the ”last step”, the individual understanding of new concepts, ideas, and techniques, it still requires hard and focused work by each of us. Our claim is that this will persist. In particular, there will be no effortless way of learning.


• Printed text will still be important. TV, radio, Internet, etc. represent competitors to the printed word, which, however, has proved to be surprisingly resistant to the fierce competition.  New handheld gadgets to represent text will appear, but traditional books and newspapers will continue to be important.


• There will be full convergence of all means of communication. We have seen an unprecedented development of our means of communication, e.g.,  mobile phones and internet.  There will be full convergence of Internet, TV, radio, phones, etc


• Proper and rapid treatment of information will become even more vital. We see that as a result of the tremendous development of the Internet we have an incredible amount of information available at our fingertips.  The understanding of this information, our ability to find, check, understand, and store this information will become increasingly important.


• Norwegian students will still mainly go to their local universities. We now see that most students in Norway go to the nearest university. NTNU has been an exception, and it is absolutely vital to keep it that way. The claim is that Norwegian students will mostly study in Norway, and the sincere hope is that NTNU will recruit from all of Norway. The competition between Norwegian institutions for the best students will escalate. The vast majority of students at NTNU will be Norwegian.


• All teaching of science and engineering at universities will take place in English. More and more students will come from abroad, and will demand lectures in English.


• At least one of the above claims will turn out to be blatantly wrong. Looking back at bold predictions about the future show that one should approach the present task of forecasting with caution.


• We know the potential number of students in 2020. One of the few sure things we can say about the future is the number of people in each age group. Our students in 2020 are already born!  We have to check carefully the demographic numbers.

 

 

Below are some additional references not referred to in the above text, but of relevance to our discussion of the future role of mathematics in general [2], [3], and at NTNU and IME in particular [7].

 

References

 

[1] Weinan E and B. Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc. 50 (2003), 1062-1070.

[2] G. Høst, Hvorfor satser USA på matematikken? Kronikk, Aftenposten, October 30, 2003
http://www.aftenposten.no/meninger/kronikker/

[3] Mathematics: Giving Industry the Edge, The Smith Institute for Industrial Mathematics and Systems Engineering, UK, April 2002. http://www.smithinst.ac.uk

[4] Mathematics and Science, National Science Foundation, USA, April 1999.
http://www.nsf.gov

[5] Preliminary Program; IMA Special Thematic Year on Applications of Algebraic Geometry,
http://www.ima.umn.edu/AlgGeom/

[6] Research in Mathematics in Norwegian Universities and Colleges, Research Council of Norway, August 2002,
http://www.forskningsradet.no

[7] Basic Research in the Interface between Mathematics and Information and Communication Technologies, Research Council of Norway, August 2002,
http://www.forskningsradet.no