Przebieg SesjiIt was the one-of-a-kind autumn aura in which we were destined to discover connections between Measure and Categories.
Participants arrived in the resort on Friday and helped themselves to freshly prepared salads. After dinner the new participants were used as psychologists (visit us next time if you're curious what does that mean). Bridge and other forms of (Lebesgue) integration took the rest of the evening.
On Saturday we were unpleasantly surprised by our President who woke us up at 8.00 for breakfast, with it needing to be done first. After breakfast, whose main attraction were scrambled eggs made from 90 eggs, a spontaneous group for walking and sightseeing was formed, which went to Karkoszczonka. One does have to add that one of the participants walked out earlier for a longer route, starting at dawn without breakfast.
After traditional spaghetti for lunch, it was time for the talks. Włodek began, talking about the notion of ideal of sets. After him Szymon shared his thoughts about a certain problem (If a certain derivative of a function is constantly equal to zero, then that function is a polynomial. And what if we change the order of the quantifiers?)
Agatka told us about little sets on the real line: measure zero sets and first category sets. Then she proved the (famous) theorem of Baire in this simple case and partitioned the line on two small (in the sense mentioned above) sets. Agnieszka told us about Liouville (pronounce "Ljuwij" - those French!) numbers, constructing a transcendental number and proving its transcendence in the process. Wirginia (first-timer on our session) helped us to master some qualities of sets having the Baire property. The last talk before dinner was given by Ja¶, who presented the Łuzin and Jegorow theorems, which looked much simpler than on the lecture. Talks were interrupted with discussions, funny remarks and munching various munchies. Dr Szostok participated in the Saturday part of our talks.
Traditionally, we had sausages from grill for dinner, prepared in the freezing cold by brave Włodek, with (not only) girls taking care of his warmth, and after it Janek reminded us that, in fact, most of the functions do not have any one-side derivative, ending Saturday talks. The official part of the Saturday evening ended with choosing the topic for the next session. After heated argument we have chosen "Testing statistical hypotheses". It is worth to mention that this topic, given by Włodek, is a "return to the past" - it was one of the two topic of our first outgoing session. After the voting participants washed their teeth and went to sleep ;)
On Sunday we were again woken up by our president and after such a night (full of thinking about measures and categories, of course) waking up is as easy as counting the integral from the exponential of a square. After breakfast other mysteries from the J. Oxtoby book (which served as inspiration and main source of talks) were discovered by Paweł Borówka (he finally presented something, standing ovation for him!), who talked about generalizing the (famous) theorem of Baire on topologically complete (=homeomorphic with complete) spaces. Rafał managed to construct three different (DIFFERENT) non-measurable sets and to prove the theorem of Ulam, which says about the impossibility of constructing a finite measure vanishing on sets of certain cardinalities (with HC on R, if you know what I mean). Finally Michał Lewicki (incredible session debut) presented the Sierpiński-Erdoes theorem. Let us hope that the younger listeners will not use the last one against their tutors and won't take pleasure in exchanging "measure zero" to "first category" and vice versa in various theorems presented to them, knowing that the theorems will remain true, because think of the implications...! And the theorem, well - beautiful, not to quote a more striking comment made on the spot. No one could answer immediately if the function appearing in the theorem may be additive, and after the session it turned out that it can not be. We direct those interested in details to mr Marcin Kysiak webpage.
The session ended with us cleaning up and eating lunch, which was (new!) beetroot soup. In the end we found the keys of one of our colleagues from ŁódĽ.
last update: 13.04.2011
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