Koło Naukowe Matematyków U¦ PL SMS

XXX SMS' Session
Pathologies and paradoxes in mathematics

Szczyrk, 29th Apr - 3rd May 2011

Our leading topic is "Patologies and paradoxes in mathematics". The talk does not need to be long nor mathematically complicated - it may refer to other branches of science. Giving a talk is especially encouraged among guests from outside the Silesian University - of course it is not a necessary condition of participating in the session.

The talk's topic may be chosen, for example, from the following propositions:

- An algebra talk: Neighbor Rings (articles about them (in Polish) may be found at: [1] [2] [3] [4] [5]);
- A longer talk for, for example, two people, from logics and set theory. [6]
- Various topics from elementary up to advanced: [7]
- Properties of discontinuous solutions of the Cauchy equation;
- Empty set;

Something more advanced:

- Strange Functions in Real Analysis [13]

Harder propositions:

- An example of a non-topologizable algebra;
- Non-reflexive Banach space isomorphic with its second dual;
- An example of a non-measurable in product set without axiom of choice;
- A non-regular Borel measure on a compact space;
- Goedl's solution of Einstein's equation;
- Undecidability of Naimark's problem about some C*-algebras representation;
- Irreducible Banach spaces;
- Banach algebras with infinitely many maximal ideals;
- ?N mappings;
- Non-standard analysis and ultrapowers;
- The l1 space as a dual space to uncountably many non-isomorphic Banach spaces;
- Niemycki plane;
- One may also take inspiration from a list of (usually simple) counterexamples in Banach algebras: http://profsite.um.ac.ir/~moslehian/cfa/BA.HTM.

Of course, these topics are only propositions - you may come up with your own.

Downloadable literature:
[1] Konstrukcja pier¶cieni S±siady, Daniel Simson
[6] Yanofsky, Noson S. A universal approach to self-referential paradoxes, incompleteness and fixed points. Bull. Symbolic Logic 9 (2003), no. 3, 362â??386.
[14] Counterexamples in Banach Algebras

To be found in our mathematical library:
[2] Riddles in mathematics : a book of paradoxes / Eugene P. Northrop. - Repr. - Harmondsworth : Penguin Books Ltd., 1975.
[3] Paradoxes of measures and dimensions originating in Felix Hausdorff's ideas / Janusz Czyż. - Singapore : World Scientific, cop. 1994.
[4] The Banach-Tarski paradox / Stan Wagon.

In main library:
[5] Nonplussed! : mathematical proof of implausible ideas / Julian Havil. - 4th print., and 1st paperback print. - Princeton ; Oxford : Princeton University Press, cop. 2010.

Other literature:
[7] Paradoxes in probability theory and mathematical statistics / Gabor J. Szekely - Reidel, 1986
[8] Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology, Springer, New York 1978, ISBN 0-486-68735-X.
[9] Joseph P. Romano and Andrew F. Siegel: Counterexamples in Probability and Statistics, Chapman & Hall, New York, London 1986, ISBN 0-412-98901-8.
[10] Gary L. Wise and Eric B. Hall: Counterexamples in Probability and Real Analysis. Oxford University Press, New York 1993. ISBN 0-19-507068-2.
[11] Bernard R. Gelbaum, John M. H. Olmsted: Counterexamples in Analysis. (1965), Dover Publications, Mineola, NY 2003, ISBN 0-486-42875-3.
[12] Jordan M. Stoyanov: Counterexamples in Probability. 2nd edition, Wiley, Chichester 1997, ISBN 0-471-96538-3.
[13] Strange Functions in Real Analysis, A. B. Kharazishvili, Chapman and Hall/CRC, 2006
last update: 06.07.2011


Students' Mathematical Society of the University of Silesia
(Koło Naukowe Matematyków Uniwersytetu ¦l±skiego)
40-007 Katowice, ul. Bankowa 14 (room 524)
tel. (032) 359-20-96, email: knm@knm.katowice.pl