ИСТИНА

Комментарий к докладу. Это пленарный доклад-лекция - для полного состава участников конференции. Вот ссылка на сайт конференции: https://www.kuleuven-kulak.be/nielsen/ Там в разделе программа указана эта моя лекция на 60 минут. Таких лекций на конференции было только 5. Далее Тезисы этого доклада. NIELSEN THEORY AND RELATED TOPICS IN RECENT PAPERS BY RUSSIAN AUTHORS Tatiana Fomenko M.V.Lomonosov Moscow State University, Moscow, Russia tn-fomenko@yandex.ru In 2018, Bo Ju Jiang and Xue Zhi Zhao presented a remarkable survey of developments inNielsen Fixed Point Theory [1] covering its main directions, methods and problem statements.This survey contains also a large list of references, showing the significance and the extensive effortin the development of the Nielsen theory during many years.In this lecture, we would like to give a short survey of more or less recent results by Russianauthors concerning the Nielsen theory and some related problems. We hope that the results tobe presented will somewhat supplement the mentioned survey. We will mainly consider problemsconnected with minimizing the preimage of a given subset under a mapping acting between topo-logical spaces (see [2–7]). Particularly, in the case of a self-mapping of a (not necessary oriented)surface, we describe some results on the minimization of the number of roots, that is preimagesof a given point (see [2–4]). In addition, we give some results concerning the minimization of thecoincidence set of mappings between manifolds in positive codimension (see [8,9]). In the final partof the lecture, we give some results and remarks, in the field of equivariant Nielsen theory (see[10,11]). Without detailed proofs, we give the description of the listed problems, formulate theresults, and outline the ideas of the proofs and some connections with constructions of the classical Nielsen theory. References [1] Jiang B. J., Zhao X. Z.,Some developments in Nielsen fixed point theory,Acta. Math. Sin.-English Ser.,34(2018), 1, 91–102. [2] Bogatyi S. A., Goncalves D. L., Kudryavtseva E. A., Zieschang H.,Minimal Number of Preim-ages Under Maps of Surfaces,Mathematical Notes75(2004), 1–2, 13–18. [3] Bogatyi S., Goncalves D. L., Zieschang H.,The minimal number of roots of surface mappingsand quadratic equations in free groups, Math. Z.,236(2001), 419––452. [4] Goncalves D. L., Kudryavtseva E., and Zieschang H.,Roots of mappings on nonorientablesurfaces and equations in free groups, Manuscripta Math.,107(2002), 311––341. [5] Frolkina O. D.,Minimizing the number of Nielsen preimages classes, Geom. Topol. Monogr.,14(2008), 193–217. [6] Frolkina, O. D.,Relative preimage problem, Math. Notes,80(2006), 1–2, pp. 272–283. [7] Frolkina, O. D.,Estimation of the number of points of a preimage on the complement, Mosc.Univ. Math. Bull.,61(2006), 1, 18–26. [8] Fomenko T. N.,Nielsen type invariants and the location of coincidence set in positive codimen-sion, Topology Appl.,155(2008), 2001–2008. [9] Fomenko T. N.,Minimizing Coincidence in Positive Codimension, Math. Notes,84(2008), 3,407––416. [10] Fomenko T. N.,On the Least Number of Fixed Points of an Equivariant Map, MathematicalNotes69(2001), 1, 88––98. [11] Fomenko T. N., Zhu J.,Nielsen type invariant for equivariant mappings preserving the or-bit structure, Topological methods in nonlinear analysis, (2000), Voronezh State University,Voronezh, 125–131.