# Summer reads 3: mathematical insights

For this third and final part, we have collected articles about interesting mathematical insights, and how they can yield ideas about real-life problems.

#### Mathematical Insights

It is undeniable that natural, integer, rational, and real numbers are necessary to explain what happens in our daily lives, but in high school, some of us also heard about some weird objects called imaginary numbers, also known as complex numbers. In his article, Francisco Escudero Gutierrez wrote about complex numbers and a recent endeavor to prove their necessity in understanding the laws of nature! In order to see them in nature we need to look deep into quantum mechanics and non-local games!

Common sense tells us that objects of comparable size should be equally hard to find. Yet, when searching inside a random network, surprises are awaiting... In her article, Noela Müller wrote about finding cliques in networks. It may be a very specific topic to write about, but, in reading the article you will learn about the overlap-gap property, a pioneering idea described by Massachusetts Institute of Technology professor David Gamarnik.

Often when mathematicians popularise their work they focus on the importance of mathematics in various applications. The mathematics community should focus instead on creating new ways and techniques to communicate what makes them really enthousiastic in mathematics, like the deep ideas, beauty and structure that you can find in it. In his article, Jop Briët wrote about two beautiful and incredibly interesting mathematical results. He discusses the Erdős conjecture on arithmetic progressions and Szemerédi's regularity lemma! This regularity lemma shows that tools from graph theory can be used in studying the natural numbers!

In the following animation, we show how Szemerédi's result works on a certain graph.

#### Some help with daily problems?

Have you ever experienced long waiting times at the security checks of an airport? A year ago for example the situation at Dutch airports, as well as many other airports in Europe, was untenable for many months. As we all have heard in the media, this issue was a consequence of the shortage of trained personnel at the airports. But, what if we tell you that is not the only issue? What if we tell you that long queues are also caused by the impatient passengers who arrive too far in advance at the airport? In her article, Elene Anton analyzed how the organization of the security check queue affects the waiting time of passengers!

Have you ever participated in a tournament and had the feeling that there was some unfairness in how the schedules of the matches were made? In his article, Roel Lambers explained how some teams during the Volleyball Nations Leagues 2018 may have been (dis)favored by the schedule of the matches. And how mathematical techniques yield a schedule that would have been fairer!

Finally, Federico Capannoli wrote about his research on the Voter Model. Would you like to learn how the development of your opinion during the last political issue, the spread of a virus among your acquaintances during the current pandemic, and the alignment of some particles lying inside the device from which you are reading this article are extremely comparable phenomena? Then have a look at his article. Mathematicians love to find similarities between phenomena that at first sight seem unrelated!

From next week, that almost everyone will be back from their vacations, we will start with new articles!

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