# All good things must come to an end

It has been a fantastic couple of weeks, but all good things must come to an end. The time has come for the preparation of the final report and the video presentation.

My project has reached a satisfactory position and leaves space for future research. Semidefinite programming is, in general, a strong tool to approximately solve very hard problems from optimization: mixed integer linear and quadratic programming problems, a wide range of non-linear programming problems, and also to detect positivity of commutative and non-commutative polynomials, an inspiring problem from real algebraic geometry. We can find a nice application of semidefinite programming in graph theory. For example, given a graph we can ask ourselves what is the smallest numbers of colours needed to color the vertices of the graph such that no two adjacent vertices share the same color. This quantity is called the chromatic number and its computation is an NP-hard problem. Instead, we can bound this number with other quantities obtained by solving semidefinite programs in polynomial time. Are you wondering what the terms NP, polynomial time, NP-hard mean? If the video below doesn’t look interesting to you, I don’t know what does.

I took the chance to travel some more during the last weekends of my stay in Slovenia. Me and Ben visited Vienna over the weekend and we found out it would be a really nice city to live in. Some of the places we’ve visited include Schönbrunn palace and the Natural Museum of History.

Last weekend my mentor and I went hiking. Our goal was Mt. Stol, a massive ridge that forms a natural boundary (not to be confused with a boundary point method 🙂 ) between Slovenia and Austria. We started from a mountain shelter located at 1180 m of altitude and the trail towards the peak was marked as easy. I still haven’t figured out what they mean by ‘easy’, since we conquered Mt. Stol (2236 m) after almost 2 hours and a half of hiking climbing. Anyway, once on top you understand that the journey is the reward. Below is a photo of us on Slovenian territory with the Austrian landscape just behind us. You can tell we are happy for the achievement.