Math courses at UCLA are difficult

Mathematics: Always between joy and pain

Before I start, I would like to give you the chance to refrain from reading this post: I am a mathematician. With this sentence I've already brought many a burgeoning conversation to an abrupt end, so why not read a blog post? The fact is that my science triggers a reaction in many people that for many has to do with forced contact with a school subject of the same name. So if you'd rather run away, I'm sure not to stop you - but I would like to point out right away that the story I am about to tell you will be about something completely different from the kind of mathematics that you use in the Had to get to know school.

Still, one more quick word of warning: I am indeed a mathematician through and through. I always find it very difficult to explain what my scientific work actually consists of. In fact, a conversation like the one above, in which the other person did not flee, often leads to the question of what a mathematician actually does all day long (in our case, when he is not writing a blog post). If I want to answer this question in terms of content and not just list the external signs of my work (sitting, thinking, writing, talking about mathematics), that's simply difficult for me.

The language of mathematics

In order to explain the main reason for this difficulty, I have to go back a little. Objects of mathematical research can, depending on the viewer's philosophical convictions, have very different shapes. The views of what this figure actually looks like have an astonishing range: from ghostly appearances that have no real (material) reality to the only objects that have real reality, you can find something for every taste. This also has to do with the fact that for some, mathematics is a language with which we want to understand natural phenomena, while for others it has its own reality.

All of these approaches (almost) always have the fact that mathematics is not an experimental science: We do not formulate theories that can be tested (i.e. falsified) with the help of experiments. We formulate theorems (sentences) which make general statements about (mathematical) objects. Accuracy is crucial: A sentence is valid through the flawless logical chain of its proof.

This means that the mathematical language is a really fine blade: the power of mathematics rests on the clarity with which it is expressed, and this means that its language has strayed very far from the usual view. When I try to explain math problems in everyday language, it is almost impossible to do it correctly in terms of content, and I have to give up the accustomed sharpness. This step is very difficult (for me). I try anyway; and I will try to show a little of the bigger picture and perhaps make my (still great) enthusiasm for mathematics a little understandable.

Research area in the border area

My own research area is located in the border area between geometry and analysis. Geometry is no longer what you may have got to know in school - in modern mathematics and in large areas of the natural sciences, geometry is now almost a universal approach that allows us to look at many different phenomena in a relatively uniform way and way of thinking. In mathematics, geometry helps not only to think about systems of algebraic equations, but also to understand important inequalities. Geometry is also essential to describe the behavior of solutions to partial differential equations that we use to understand natural phenomena with the help of modern physics.

Analysis, on the other hand, is one of the most powerful tools mathematics has developed over the centuries. It establishes connections between the behavior of dependencies between different quantities: from behavior on a small scale to behavior on a large scale, from a short time to a long time, over structural problems (singularities), analysis creates an arc that encompasses many phenomena not only in mathematics, but also makes nature understandable or even analyzable.

Differential equations and natural phenomena

One of the most important methods that nature makes accessible to us with the help of analysis is the description of natural phenomena by (partial) differential equations: With these, the dependencies of the change in (observable) quantities are related to one another. The first examples that one usually hears about are the heat conduction equation and the wave equation. These equations describe the development of a quantity with the passage of time. The temperature distribution in an object changes over time, the faster the greater the change in temperature in the room at that point in time (just ask your refrigerator about it). The heat conduction equation describes this process. For example, the wave equation is used to describe how a piano string vibrates after it has been struck.

Partial differential equations other than their solutions describe quantities that are in a specially balanced state: When you create a soap bubble on a wire ring, the soap film tries to minimize its surface tension and thus creates an area of ​​minimal size; this characteristic of the soap film is described by the Laplace equation.

The partial differential equations that I deal with in my work are the Cauchy-Riemann differential equations. Not only do they bear the name of two famous mathematicians (Augustin-Louis Cauchy, 1789–1857; Bernhard Riemann, 1826–1866), but also two famous mathematicians from competing nations at the time (France and Germany). We call the solutions to these differential equations holomorphic functions, and the branch of mathematics that deals with them is called complex analysis. This has to do with the fact that these types of functions are best understood in the context of complex numbers.

The "bounty" of the Clay Foundation

Holomorphic functions are relatively special and Cauchy-Riemann differential equations are of the second type discussed above (no time evolution, but rather a balanced state; holomorphic functions are closely related to solutions of the Laplace equation). Nevertheless, they appear not only across all of mathematics, but also in many important areas of application. All "elementary" functions are holomorphic: exponential functions, trigonometric functions, power functions, logarithms and many others; Complex analysis allows us, among other things, to effectively calculate these important quantities.

Another example of a holomorphic function that has achieved a certain fame is the Riemann zeta function: The position of its zeros is the content of the Riemann conjecture, which has been unsolved for more than 150 years, and the solution of which is a "bounty" of the clay -Foundation of one million US dollars is suspended. Although this question may sound very theoretical, a solution would also have a direct impact on the common encryption techniques on the Internet, as they are used every day in business transactions.

No shortage of open questions

But we are not only concerned with centuries-old problems. One of the questions we are often confronted with is whether there is anything left to research in mathematics (it is a relatively old science after all). In my opinion, the answer to this is a resounding yes: the longer I work in mathematics, the more I understand how little we actually know. There is no shortage of open, unresolved questions. As I said, mathematics differs here from other sciences - Euclid's theorems are just as valid today as they were 2,000 years ago. But that does not mean that one should expect that all mathematics can ever be fully described; one of Kurt Gödel's results can be interpreted in such a way that this is in principle impossible.

How do we actually find the problems that we are working on? Often the questions that need to be worked on arise from previous work, of course. Sometimes the aim is to extend the validity of known results to more general situations, sometimes to refine it. On the other hand (and this is often more interesting) one comes across questions that can no longer be answered using the existing methods. One of my colleagues likes to compare the work of a mathematician with always having a suitable tool with you for many different tasks. In the case where our methods reach their limits, one simply has to tinker with a suitable tool in this comparison; this is one of the most demanding parts of our job, but not necessarily the most pleasant.

Thinking ahead - like playing chess

There are much nicer ways to describe the contents of math work than the toolbox mentioned above. The comparison to a chess game, for example: Just as you have to think several moves ahead when playing chess, you also have to think a few steps ahead with a chain of arguments. And while it can be a lot of fun to play a good game of chess, it is often necessary to work out new variations or come up with answers to new game variations. And hopefully, in the end, these new things can actually be used successfully in a game.

The joy of math

A similarly beautiful analogy, which probably comes closest to my understanding of and my experience with mathematics, is that of music, which is often summarized in J. J. Sylvester's quote "mathematics is the music of reason". The comparison with music manages to illustrate well an aspect of mathematical work that has not yet been mentioned - and yet one of the most important: the incredibly exciting creative process inherent in mathematical work. Similar to a good piece of music, there is math that develops by itself; in which you initially have the feeling that you can barely grasp it, and as the piece progresses, it develops from a fleeting idea into its own melody, sustains life and, freed from all constraints, finds new expressions forever new shape.

For me, mathematics is primarily characterized by the creative challenges, where you first encounter unclear phenomena, try to put them into a tangible form, and then help them to express themselves. This is what makes mathematics so much fun for me: that I often don't know where it will lead me; that I am often surprised by it; that I almost never get bored in my work. (Bernhard Lamel, April 25, 2018)

Bernhard Lamel is Professor of Complex Analysis at the Faculty of Mathematics at the University of Vienna. Stations of the academic career before Vienna were the University of California, San Diego (Ph.D. 2000), KTH Stockholm and University of Illinois at Urbana Champaign. The former START prize winner was elected a member of the Young Academy (then still Young Curia) of the Austrian Academy of Sciences in 2013. He is currently leading several research projects, including international cooperation projects between the FWF, the ANR (France) and the RSF (Russia), and is vice dean of the Faculty of Mathematics.

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Event tip

On June 6th, the Austrian mathematician and Fields Medalist Martin Hairer will speak at 6.30 p.m. at the Austrian Academy of Sciences in Vienna. In his lecture entitled "Bridging Scales" he deals with probability theory.