Fixed points (mathematics) | Fixed-point theorems | Topology
Nielsen theory is a branch of mathematical research with its origins in topological . Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name. The theory developed in the study of the so-called minimal number of a map f from a compact space to itself, denoted MF[f]. This is defined as: where ~ indicates homotopy of mappings, and #Fix(g) indicates the number of fixed points of g. The minimal number was very difficult to compute in Nielsen's time, and remains so today. Nielsen's approach is to group the fixed-point set into classes, which are judged "essential" or "nonessential" according to whether or not they can be "removed" by a homotopy. Nielsen's original formulation is equivalent to the following:We define an equivalence relation on the set of fixed points of a self-map f on a space X. We say that x is equivalent to y if and only if there exists a path c from x to y with f(c) homotopic to c as paths. The equivalence classes with respect to this relation are called the Nielsen classes of f, and the Nielsen number N(f) is defined as the number of Nielsen classes having non-zero fixed-point index sum. Nielsen proved that making his invariant a good tool for estimating the much more difficult MF[f]. This leads immediately to what is now known as the Nielsen fixed-point theorem: Any map f has at least N(f) fixed points. Because of its definition in terms of the fixed-point index, the Nielsen number is closely related to the Lefschetz number. Indeed, shortly after Nielsen's initial work, the two invariants were combined into a single "generalized Lefschetz number" (more recently called the ) by and Reidemeister. (Wikipedia).
85 Years of Nielsen Theory: Periodic Points
Part 2 of a 3 part series of expository talks on Nielsen theory I gave at the conference on Nielsen Theory and Related Topics in Daejeon Korea, June 25, 2013. Part 1- Fixed Points: http://youtu.be/1Ls8mTkRtX0 Part 3- Coincidence Points: http://youtu.be/Wu2Cr3v_I44 Chris Staecker's intern
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Part 1 of a 3 part series of expository talks on Nielsen theory I gave at the conference on Nielsen Theory and Related Topics in Daejeon Korea, June 24, 2013. Part 2- Periodic Points: http://youtu.be/Ic26_F8UUBE Part 3- Coincidence Points: http://youtu.be/Wu2Cr3v_I44 Chris Staecker's int
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A talk about Nielsen theory at the 2016 conference on Nielsen Theory and related topics at UNESP Rio Claro, SP Brazil. Given July 5, 2016. Conference website: http://igce.rc.unesp.br/#!/departamentos/matematica/nielsen-theory/ Link to Chris Staecker webarea: http://cstaecker.fairfield.ed
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From playlist Research & conference talks
85 Years of Nielsen Theory: Coincidence Points
Part 3 of a 3 part series of expository talks on Nielsen theory I gave at the conference on Nielsen Theory and Related Topics in Daejeon Korea, June 27, 2013. Part 1- Fixed Points: http://youtu.be/1Ls8mTkRtX0 Part 3- Coincidence Points: http://youtu.be/Wu2Cr3v_I44 Chris Staecker's intern
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