ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζῴων τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

Mathematical Colloquim




                       OPŠTI MATEMATIČKI SEMINAR




NAPOMENA: Predavanja ce se odrzavati u Sali 301f na trecem spratu Matematickog instituta SANU, Knez-Mihailova 36 (zgrada preko puta SANU).


Petak, 26.09.2014. u 14:00h, sala 301f, MI SANU

Ulrich Koschorke (University of Siegen)

Abstract: In topological fixed point theory we are mainly interested in the following question. Can a given selfmap $f$ of a manifold $M$ be deformed continuously until it has no fixed points? And we try to measure to what extent $f$ fails to be 'homotopically fixed point free' in this sense. The classical Lefschetz number yields a necessary condition. But a much better measure is the Nielsen number of $f$. It vanishes precisely if $f$ is homotopic to a fixed point free map EXCEPT when $M$ is a surface with strictly negative Euler characteristic (in which case this statement can be dramatically wrong).

In coincidence theory we do not just compare a selfmap $f$ with the identity map but we study the coincidence set $C$ of an arbitrary pair $f,g$ of maps from the domain $M$ to a possibly different target manifold $N$ (i. e. $C$ is the set of points $x$ in $M$ where $f(x)=g(x)$). Can the maps $f,g$ be deformed away from one another? In other words: can $C$ be made empty by suitable homotopies?

Since the dimensions $m$ and $n$ of the domain and the target need not agree, generically the coincidence set $C$ will be an ($m$-$n$)-dimensional manifold (and not just consist of isolated points as in the fixed point setting). Thus the geometric methods of differential topology come into play, and deep notions such as bordism, Kervaire invariants, Hopf invariants etc. enter the picture. In particular, Nielsen numbers get a new, deeper meaning and answer some, but not all central questions (and may allow us to measure what can go wrong).

REFERENCE: U. Koschorke, Minimum numbers and Wecken theorems in topological coincidence theory. I ,J. Fixed Point Theory Appl. 10,1 (2011), 3-36.

Rukovodioci Odeljenja za matematiku Matematickog instituta SANU i Opsteg matematickog seminara na Matematickom fakultetu u Beogradu, Stevan Pilipovic i Sinisa Vrecica predlazu zajednicki program rada naucnih sastanaka.

Predavanja ce se odrzavati na Matematickom Institutu (sala 2), petkom sa pocetkom u 14 casova. Odeljenje za matematiku je opsti seminar sa najduzom tradicijom u Institutu.

Svakog meseca, jedno predavanje ce biti odrzano na Matematickom Fakultetu u terminu koji ce biti posebno odredjen.

Molimo sve zainteresovane ucesnike u radu naucnih sastanaka da posebno obrate paznju na vreme odrzavanja svakog sastanka. Na Matematickom fakultetu su moguce izmene termina.

Obavestenje o programu naucnih sastanaka ce biti objavljeno na oglasnim tablama MI (Beograd), MF (Beograd), PMF (Novi Sad), PMF (Nis) i PMF (Kragujevac).

Odeljenje za matematiku Matematickog instituta SANU

Stevan Pilipovic

Opsti matematicki seminar na Matematickom fakultetu u Beogradu,

Sinisa Vrecica

Ako zelite da se obavestenja o Vasim naucnim skupovima pojave u Newsletter of EMS (European Mathematical Society) i na Internetu na lokaciji EMS, onda se obratite na emsvesti@mi.sanu.ac.rs gde cete dobiti format obavestenja.