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## Rotation of a sword

Lecturer:

Prof. Dr. Stefanie Petermichl

Summary:

Motivated by the art of sword fighting, the Japanese mathematician S. Kakeya asked in 1917: Among all surfaces in the plane that allow for a complete rotation by 360 degrees of a straight line of length 1 (the sword) within, what is the smallest area we can hope for? We quickly see that a disk of diameter 1 fulfills the condition by placing the center of the straight line onto the center of the circle and thus simply rotating the sword around its center. We will construct a much smaller area by sliding and overlapping certain triangular surfaces. We will get the amazing answer to S. Kakeya's question.

## Mathematics of paper folding - from handmade fascination to tangible proof!

Lecturer:

Dmitri Nedrenco, Wissenschaftlicher Mitarbeiter

Summary:

What does origami have to do with math? Is it possible to look at paper folds with a mathematical prospective? In the workshop some interesting models will be presented, as well as we will analyze in groups their patterns. To this end, the following questions will be addressed: When does a given folding pattern convert to a flat object? Are there any fold patterns that cannot be reduced to a flat object? Are there any theorems that can clarify these questions? How do you prove these theorems?

Folding paper is a valuable asset for creating mathematics with your own hands; the path from playing with paper to guesswork and even proofing is interesting and entertaining.

## Evidence can be so easy!

Lecturer:

Prof. Dr. Jürgen Appell

Summary:

"Math always makes me confused!" is the usual complaining of many students, who don’t understand neither the most innocent proofs at school nor their application. This situation is very common, independently on having a good teacher.  What is reason for that? Can we change this behavior? If it is possible to change it, what is the best procedure to go through? We want to address these questions in the lecture with several examples.

The picture shows the Würzburg mathematician Prof. Dr. med. Hans-Georg Weigand discussing a particularly straight proof.

## How does a ball go quickly from A to B?

Supervisor:

Dr. Barbora Benesova

Summary:

We deal with mathematical modeling in physics. We will explain what a mathematical model is and the main tasks of mathematicians in this setting. We proceed with several examples that were addressed by Riemann and Newton. More precisely, in which way a ball comes from a point in space A to a point B (which is slightly lower).

This problem is known as the problem of brachistochrons, and we have to minimize some size: the trick is that you have to work in an infinitely dimensional space and use sine methods of modern mathematics. We learn two ways to solve the problem, an analytical one and a numerical one. We will see how certain mechanics can be solved today by the scientific community and how mathematicians can contribute here.

## How do you connect two railway lines so that you can drive over them as pleasantly as possible?

Supervisor:

Prof. Dr. Sergey Dashkovskiy

Summary:

Let us take a look at two straight railroads that lie on two different lines (as in the picture).

How can these two routes be connected so that a modern train can travel over it?

The shortest link connecting A and B through a straight line does not make sense because the train cannot drive over it. Obviously, a smooth connection is required. One can find a circular connection that provides a smooth transition. However, a ride over it will be quite uncomfortable as the centrifugal forces change abruptly. We consider how to design a suitable connection curve (clothoid) by means of a mathematical model. Starting from the concept of tangential straight lines, we learn about the curvature of a curve and how it can be quantified. This helps us to answer this specific question.

## How stars are made, why does an airplane fly? Mathematics helps to understand.

Lecturer:

Prof. Dr. Christian Klingenberg

Summary

The central importance of mathematics for our view of the world is illustrated by examples from technology and nature. We will show a suitable computer-simulated description of the evolution of the universe, showing how the initial stages of the universe have transformed stars and galaxies over time. This description is possible only through mathematical theories.

## Paul Erdös and the drawer principle

Supervisor:

Dr. Jens Jordan

Summary:

Paul Erdös was one of the most important but also the most curious mathematicians of the 20th century. He was always looking for "beautiful evidences”. We will take a closer look about a nice proof which uses the drawer principle.

## What shape could the universe have?

Supervisor:

Prof. Dr. Jürgen Appell

Summary:

The question of what form and structure our universe as a whole possesses has often moved physicists, astronomers, and philosophers. But even mathematicians can make interesting contributions here, because the answer can be formulated in the language of mathematics: is it finite or infinite? Bounded or unbounded? Curved or flat? What geometry could it have? Does it perhaps look like the funny bottle in the picture, from which one should rather not drink?

None of these questions can be answered (yet), but one can approach the problem through theoretical considerations: this is the goal of the lecture.

## Complex numbers: from the midnight formula to black holes

Supervisor:

Dr. Daniela Kraus/Prof. Dr. Oliver Roth

Summary:

We embark on a journey through the world of complex numbers and complex functions. The starting point is the midnight formula for quadratic polynomials. We explain why complex numbers help us to understand even more complicated functions and learn how some famous mathematicians such as C.F. Gauss and E. Galois know each other. Our foray ends with an outlook on current mathematical research showing how complex numbers are used in conjunction with Einstein's Theory of Relativity to compute the deflection of light rays through massive bodies. This is illustrated by numerous pictures taken with the Hubble Space Telescope.

## Develop, explore, prove and refute

Supervisor:

Dr. Jens Jordan

Summary:

What do mathematicians do? Is not everything already known? Why research? What does mathematics mean? And can we already calculate everything with computers?

## Rabbit plague in Pisa

Supervisor:

Dr. Jens Jordan

Summary

The Fibonacci sequence is an infinite sequence of numbers in which the sum of two adjacent numbers gives the immediately following number. It is named after Leonardo de Pisa, who described in 1202 the growth of a rabbit population. In the lecture, this sequence will be presented and connected to nature and art. Furthermore, a few mathematical properties of the Fibonacci numbers will be discussed.

## Find the right way with math

Supervisor:

Prof. Dr. Jörn Steuding

Summary:

How do I cheaply get from Würzburg to Berlin? How to organize efficient garbage collection? How do you behave in a labyrinth? Such everyday questions about orientation in complex environments can be treated with an interesting mix of mathematics and computer science. The concept of the graph allows a simplified representation of these problems (by reducing them to essential information) and allows clever algorithms to find solutions or at least strategies for dealing with them. The lecture should provide a first insight into mathematics at universities (especially graph theory) and in particular show that there are many interesting and often unresolved questions that are nevertheless easy to understand.

## Sausage guess and sausage disaster

Vortragender:

Dr. Richard Greiner

Summary

How to pack 5, 50, 5000, 5,000,000 balls as mush efficiently as possible? Atoms (and globular molecules) regularly assemble into crystals. In fact, there are surprises that we come across in the lecture. This requires basic geometric knowledge about simple surface calculations (circle, triangle, parallelogram) and some spatial imagination. Considerations that could be made in a W seminar stand next to hard nuts in current mathematical research. It is not only visible how mathematicians work at the university, but also why seemingly abstract mathematics is responsible for technical innovation.

Target group: from Jgst 11.

## A_15 - a breeze

Supervisor:

Dr. Gunther Dirr

Summary

The lecture deals with some mathematical aspects of so-called "sliding puzzles". Based on the concrete question of whether each initial configuration can be transferred to each final configuration, simple terms about permutation groups are illustrated. At the end of the lecture, a surprisingly short proof provides the solution to the above question.

The lecture is suitable for interested students and it does not depend on their mathematical education, since it manages almost without school mathematics. This "unconventional" kind of mathematics amazes most listeners.

## What is mathematics for, why are mathematicians used?

Supervisor:

PD Dr. Christian Zillober

Summary:

"Mathematics is interesting, but what can I do with it later?"