Hard Problems 
Why It’s Worth Tackling Difficult Problems: Perspectives from Mathematics and Computer Science

It is not your responsibility to complete the work, but you are not free to shirk it. 

- Sayings of the Fathers [Pirkei Avot] 2:21 
 

You must not attempt to prove the parallel lines by that method; I know that path to its very end—I implore you by God! Leave the theory of parallel lines alone (…) it can rob you of all your leisure, your health, your peace of mind, and all your happiness in life.

- Farkas Bolyai (co-developer of non-Euclidean geometry) to his son Janos (1820) 

Rombuch 2022 as a PDF
 

Even if some believe that all of mathematics—and even areas of computer science—consist exclusively and indiscriminately of difficult problems that are, miraculously, quite easy for experts, even a cursory glance at their respective histories shows that the “difficulty scale” of a problem or a set of problems is of the utmost importance for the development of mathematics and computer science. A significant portion of mathematical research is, in fact, dedicated precisely to the endeavor of formulating problems that are sufficiently difficult yet solvable—that is, interesting but not hopeless. 
If one looks beyond the day-to-day routine of research, even the “unsolvable”—that is, extremely difficult—problems often prove to be extremely fruitful for the further development of mathematics. Here are just a few examples:

1. The classical construction problems in geometry—namely, squaring the circle, doubling a cube, or trisecting an angle, each using only a compass and straightedge—are all unsolvable, yet they have inspired a wealth of mathematical work over millennia.
2. Questions that are easy to formulate, such as the parallel lines problem or the search for a general formula for solving higher-order polynomials, have—precisely because of their unsolvability—led to entirely new fields of research, namely non-Euclidean geometry and Galois theory.
3. The undecidability of the continuum hypothesis—the first of David Hilbert’s famous open problems around 1900—has led to a deeper understanding of the structure of the real numbers and set theory.

In the field of computer science, it has also become possible to classify the computational effort required to solve a problem using an algorithm; thus, the “difficulty” of a problem can in turn be measured mathematically. A famous open problem in this context is characterized by the equation P = NP. This still-unsolved problem tops the list of the seven “Millennium Problems” established by the Clay Institute in 2000, for the solution of each of which one million dollars is offered. So cracking tough nuts can even be profitable! 
However, the Rome seminar is also intended to broaden our perspective beyond the narrower fields of mathematics and computer science. In a societal context, we are currently facing challenges that can certainly be classified as “hard problems”: climate change and sustainability, migration, issues of (social or international) justice, the search for a global legal or peace order, and, most recently, dealing with pandemics. Here, public discourse in particular—ranging from populist slogans to conspiracy theories—shows that seemingly very simple solutions to tough problems are repeatedly offered. And it, in turn, proves to be a difficult problem to deal with these fundamentalist positions in a wise manner. In these and other complex situations, we are often faced with the question of how to reach a legitimate—and perhaps even wise—decision based on insufficient information and under time pressure. 

After all, a look at history shows how a wide variety of cultures were able, in astonishing ways, to solve the most difficult problems. Reconstructing how they did this as accurately as possible is a fascinating field of archaeology, history, and sociology. To this day, it remains a challenge to analyze how, for example, “prehistoric” societies were able to erect a monument like Stonehenge, how ancient peoples were able to build irrigation systems and lay out gardens in what are now deserts, or how various advanced civilizations developed calendars that replicate subtle celestial movements with astonishing accuracy. 

The 2021 Rome Seminar will address the topic of “hard problems” on several levels. On the one hand, it aims to explore what makes a problem “hard,” whether this classification remains constant, and what implications arise when a “hard” problem has a long tradition. On the other hand, the seminar will address the question of what “unsolvability” means both within and outside of mathematics and computer science. Can a deeper understanding of this concept also be helpful in addressing other problems, and does it help strengthen critical judgment, even when one’s own understanding is only partial? Finally, it addresses the debate over whether one should tackle hard problems or rather avoid them, since, as we know, life is short and failure cannot be ruled out. Overall, we hope that by engaging with “hard problems,” we will discover solutions that are at times surprisingly and fascinatingly simple, develop critical judgment regarding solutions that are “too simple,” and learn from the masterful handling of unsolvable problems. 

Overall, the seminar thus offers a unique opportunity to look beyond the boundaries of one’s own field of study and to get to know students from three other universities (Tübingen, Kiel, and Dresden). Of course, no one is expected to solve any of the open problems discussed in a presentation—but one goal of the preparation is to learn how to communicate the essence of a field generally considered difficult in an effective and engaging way. To this end, techniques for successful presentations will be explained and practiced. 

Over the course of the winter semester, we will first explore and familiarize ourselves with the topic through shared reading, short introductory presentations, and discussions. By the end of December, each participant should have identified a topic for a presentation and tested it in dialogue with local students by the end of the semester. This presentation will ultimately be delivered and discussed during the field trip in Rome (March 1–6, 2021) before a group comprising students from all four universities. Through a diverse accompanying program, we will also draw inspiration from places in the “Eternal City” that are not otherwise accessible.

We must imagine Sisyphus as a happy man. 

- Albert Camus, *The Myth of Sisyphus* (1942)