A year is too small a measuring unit for mathematics in general. Mathematics has existed for over 5,000 years, and approaches to it that are ideologically comparable to modern perspectives have existed for over 2,500 years. Therefore, we can’t reliably say that mathematics will change drastically within a given year. The key principles of this field and its core methods won’t change. In this sense, mathematics is one of the most conservative fields around.
Going beyond key principles, and considering specific results in this field, we can still see fast-moving trends. Every year, specific and long-existing problems are solved, while important and long-set hypotheses are either proved or refuted. For instance, a mathematician, who is able to predict in 2016 that a certain important problem (e.g., the Riemann hypotheses on the location of zeros in a zeta function) will be solved in 2017, is likely to be the person who will eventually solve this problem. At the same time, such people don’t usually make predictions like this.
It probably makes more sense to try to make predictions not for a year, but for a century ahead. Such attempts are usually made at the eve of a century. Two such attempts are well-known. In 1900, David Hilbert presented a list of 23 mathematical problems at the 2nd International Congress of Mathematicians in Paris, and almost 16 of them have been solved to date (some of the open-ended problems can’t be solved completely, while some of them haven’t been sufficiently and clearly formulated). Only one of these problems — the Riemann hypotheses — was included in the Millennium Problems list. This list was compiled by the Clay Mathematics Institute in 2000, and features seven problems. A prize of USD 1,000,000 is offered for the solution to each of them. As of today, only one of them has been solved, the Poincaré conjecture, which was solved by Grigori Perelman.
Furthermore, what can we say about general trends in mathematics (not limiting ourselves to just one year)? There are two processes underway, and they are essentially opposite. On one hand, interdisciplinary areas are developing, such as those based on methods of geometry (topology), algebra, and analysis. In this sense, mathematics is now in an integration process due to the growing number of links between its respective areas. In addition, mathematics is interacting with physics more and more actively.
On the other hand, and this is a reason for concern, on average, researchers are becoming more specialized. In the 19th century, there were universal mathematicians, and in the 20th century, there were universal specialists in areas such as analysis, geometry, and topology. However, even at that time, there were only a few of them. In the 21st century, this universality has been lost, at least for specific researchers. This may, in turn, result in the next crisis in mathematics over the long term.
However, today it’s too early to talk about a crisis. Mathematics is living its life to the fullest, and very interesting things are happening. We are looking forward to next year with great optimism.
On the eve of New Year’s, it is customary to take a look into the near future. We asked HSE experts in various fields to share their forecasts on which areas of research might be the most interesting and promising in 2017. They tell us about what discoveries and breakthroughs await us in 2017, as well as how this could even change our lives.