I have done several presentations in high schools aimed at introducing scientific concepts to students unaware of them, or presenting materials they may have heard about in a thought-provoking manner. Here are (in French) the slides of these talks. Please contact me if you wish to use these documents or if you find any mistakes.
The first talk is a presentation of Hilbert's third problem. The idea was to present the ways in which research in mathematics is done, applied to a problem in geometry. There are very few computations and the ones that remain can easily be skipped if the audience is not receptive. The talk includes an optional part at the end about the continuum hypothesis which is too ambitious, and which probably should be skipped as well.
The second talk is a relatively informal discussion on cognitive biases. It presents different ways in which our brains tend to take demonstratively suboptimal decisions or make logical mistakes. The slides are voluntarily light because the main interest of the talk lies in oral examples of different biases, hopefully sparking discussions with students.