Left-right reversal illusions (show description) Goal:It is almost unbelievable, but true: there exist 3d objects that, when seen in a mirror, are exactly reversed. The aim of the project is to understand the underlying mathematics (elementary geometry), types of billiards to design such objects and to 3d print some of them. Ergodicity of the polygonal billiard flow follows from bounds on the degeneration of the geometry along a certain deformation of the phase space. Cubic surfaces over finite fields (show description) Goal:Cubic surfaces have been at the center of algebraic geometry ever since Cayley and Salmon showed all the way back in 1849 that (over an algebraically closed field) they always contain exactly 27 lines (if the surface is smooth). Over a finite field we have to modify this count, as the maximum of 27 is not always attained. Maybe we can even have a small collection in the end? In this edition of the project, we would like to focus more on new interesting features: 1. "Zoom" into some of the interesting parts of the picture of zeros, by, for instance, going higher and higher in the degree (maybe only even degrees) and adding points to the picture while taking more and more polynomials into account.

At that point, the rest of the balls are racked while the one left standing remains in position. While the results wont be as quick as using the audio recordings, slowly but surely this will bring about improvements, so be patient. You will experiment with visualising group elements using certain generating sets. An action of a countable group is called totally nonfree if, generically, all points have distinct stabilizers. In 1968 it became apparent that all known classes off groups have either polynomial or exponential growth and John Milnor formally asked whether groups of intermediate growth exist. Transitive subgroup art (show description) Goal:It is possible to visualise elements of a symmetric group as explicit permutations, similar to how braid groups are represented. There are also interesting subgroups of the symmetric group that one can consider, with other interesting visualisations. There are various choices of generating sets one can use, and they will give different visualisations of the elements of the symmetric group. It can be a nice team work because there are a lot of tasks, both from computer science and mathematical.

The project will be a computer experimentation with integral polynomials and their roots. Most of these questions can be intuited with the help of the computer tool and can give rise to nice visuals of chessboard filling. Whereas the real points (in 2- or 3-dimensional projective space) lead to nice pictures, we will be interested in rational points, i.e. points with rational coordinates. The connected question of the proportion of irreducible (i.e. one cannot write them as a product of two integral polynomials in a non-trivial way, as you will learn (resp. This question can be recast as a question on rational points on the intersection of quadrics. This is actually quite general because a theorem of Mumford asserts that every projective variety can be written as an intersection of quadrics (if a good embedding into a projective space is chosen). They were also underlying many of the incredible discoveries of Ramanujan in the beginning of the 20th century, they played a crucial role in the solution of Fermat's Last Theorem and they are still a very active field of research. From this criterion we derive an ergodicity theorem for non-rational billiards under a full measure Diophantine condition on the angles.

Our work generalizes this approach to the locally solvable 3-dimensional phase space of a non-rational polygonal billiard flow. Rational billiards are pseudo-integrable, that is, their phase space is foliated by invariant surfaces. The only known results on the ergodicity of non-rational billiards have been obtained by fast approximation methods based on the rational case (Kerckhoff, Masur and Smillie, Vorobets) and apply to a zero measure G-delta dense set of polygons. Billiards in polygons are of two fundamentally different types: rational and non-rational. In this talk we will formulate a new ergodicity criterion which applies to both the rational and non-rational case. In the rational case this idea goes back to Masur in 1982 and has been revisited more recently by Cheung-Eskin and Trevino. In this talk, I will discuss the above results which are analogous to the case of circle diffeomorphisms. 3. Last of all, the most powerful and targeted method for accelerating results is with customized mental training recordings, personalized to your exact needs, training, schedule, mindset and situation.