(This time we are dead serious, because this went to documentation :P)

I Summer School of Mathematics of the Cracow University of Technology took place on 4-8th July 2009r in Krynica Górska. There were 22 participants, including four doctors from the University (dr Magdalena Grzech, dr Katarzyna Pałasińska, dr Beata Strychacz-Szemberg and dr Margareta Wiciak), 5 students who received their bachelor's degree in the academic year 2008/2009 and three guests from the Silesian Mathematical Society.

During the session part of the participants gave short talks. They were grouped in four sessions. During the first, after an official welcome from the president of the Student Mathematical Society of the Cracow University of Technology, Piotr Pokora, and the Society's tutor, dr Katarzyna Pałasińska, the talks presented were from various branches of mathematics. The session began with a talk entitled „Ciała uporządkowane" (Ordered fields), gave by Jola Marzec from the Silesian Mathematical Society of the Silesian University, during which definitions and some qualities of ordered and formally real fields were presented, along with I and II Artin-Schreier theorems with proofs. During the next talk, "O skończenie wymiarowych algebrach z dzieleniem i algebrach Liego" (Finite-dimensional algebras with division and Lie algebras), Piotr Pokora defined a few kinds of algebras, for example an associative algebra, commutative algebra and algebra with division, and described some examples of the last ones, like quaternions, octaves, sedenions or Clifford algebras. He also presented the Frobenius theorem and gave the definition and some qualities of the Lie algebras. Our next reader, mgr Tomasz Wiewiórka, gave an overview of his master's thesis with a talk „Teoria informacji i twierdzenie Shannona o kodowaniu” (Information theory and the Shannon coding theorem). He described the way the decoded information are sent and introduced the notions of a channel, coder, decoder and noise, along with a scetch of Shannon's coding theorem. He also told us about the influence of entropy on the efficiency of sending the coded information. Our talk session was ended with dr Katarzyna Pałasińska giving a talk „BCK-algebry jako przykład struktur z rezyduacją” (BCK-algebras as an example of structures with residuation). The SMS tutor presented the definition of residuation and described examples of algebraic structures with it, like Boolean algebra with alternative and partial ordering. Dr Pałasińska also introduced the definition of a BCK-algebra and showed its application in Łukasiewicz logic.

The next talk session was entitled „Financial mathematics". Dr Margareta Wiciak opened it with a talk „If I knew how to make money", which introduced first- and second year students to the world of financial mathematics, defining notions like arbitration, arbitration strategy, derivatives or a replicating strategy. She also mentioned the Brown motion - an experiment explained ultimately by Marian Smoluchowski, the Wiener process, filtration, the Black-Scholes process and Ito integral. Our next talk, „Mathematical transaction systems + the 1% Risk rule", was given by Adam Marszałek. He explained the basic division of transaction systems, the structure of investment strategies, how does the OTC (Over The Counter) market works, the 1% Risk rule, the Stop-Loss level. He described the working of the Forex (Foreign Exchange) market, the scalping stratey and midterm trading. Our next talker was msc. Mateusz Sosnowski, who presented an overview of some concepts from his master's thesis in his talk „Optimal stopping moment". He described the issue in the discrete case, introducing the notion of the Snell envelope and giving an example of an optimal moment, and in the continuous case. The talker gave the definition of a stochastic process, a stochastic process adapted to an F filtration, along with the notions of martingale, supermartingale and submartingale with respect to an F filtration. This session was ended with msc. Kinga Wójcik giving a talk „Pricing of financial instruments based on value at risk", in which she referenced her master's thesis. The talker presented i.a. the financial market with discrete time model and the optimal strategy in the sense of value at risk of a payment H for a given starting capital x.

The third talk session was entitled „Theoretical mathematics and more". It was opened by our guest from the Silesian Mathematical Society of Silesian University, Piotr Idzik, who presented a talk „Liouville numbers in short". The next talk, „A little something from game theory", was given by msc. Katarzyna Zacłona, who defined i.a. the notions of game, strategy, result, payments, dominating, mixed and pure strategies and paretooptimal result. The talker also presented how a two-player game works on many examples, along with the ways of optimal strategy selection, establishing domination in a game and applying the saddle point test and Pareto test, along with the Nash arbitral diagram and cooperative solutions. Our third talk, „Applied mathematics", was given by dr Beata-Strychacz-Szemberg, who, in a pleasant and clear way, explained their listeners, what can a mathematician specialised in applied mathematics do. Examples provided by dr Strychacz-Szemberg were: computer tomography, impendant tomography and acoustic tomography. The talker also decribed i.a. the parametrisation rules of a system, simple and inverse problems, Hadamard conditions and an application of simple algebraic problems to solving technical problems.

Our last talk session was wholly dedicated to theoretical mathematics and such was its title. There were two talks presented: „Manifolds with and without a border" and „The multitude of mathematical worlds". Our first talker was Piotr pokora. He introduced basic notions concerning manifolds and presented some basic examples. Our last talker during the session was dr Magdalena Grzech. The second tutor of SMS talked about model (mathematical world) theory and about certain set theory problems.

There were several discussion panels during the camp. Two of them were scientific in nature: first of them was a discussion about financial options and market (including news about this year financial mathematics conference), second - about Millennium Prize Problems, in particular the six unsolved to this day and the seventh, solved recently by Perelman. Some problems of modern algebra and topology (the jacobian and Poincare hipothesis) were also discussed. Next two panels were dedicated to SMS activities: a summary of its previous ones and about the tasks which the Society was to undertake in the following academic year, with a particular interest of an October integration party for new students.

Humankind cannot live by mathematics alone, however, and the School was also to be integrating and recreational in nature. The participants took an excursion to the Góra Parkowa, which is one of the attractions of Krynica Górska, and went sightseeing during several walks. Moreover, some evenings for gaming and free discussions were organised, along with grills. I Summer School of the Cracow University of Technology was a great success and we hope it began another successful tradition of Students' Mathematical Society of Cracow University of Technology.