XI Północne Spotkania Geometryczne
4 Lipiec 2017
XI Północne Spotkania Geometryczne odbędą się w Gdańsku, w dniach 21-22 czerwca 2017 r.
Gościem głównym konferencji będzie Hans Havlicek (Wiedeń).
Środa, 21.06 (Audytorium 2, Wydzialu Mat. Fiz. i Inform.)
|09.00-10.30||H. Havlicek (Wien), 170 Years of Harmonicity Preserves PDF|
|10.50-11.30||J. Jakobowski (UWM Olsztyn), Some noncommutative rings constructed on the base of polynomials x^2+tx+1 and their zero divisors PDF|
|11.35-12.15||B. Hajduk (UWM Olsztyn), Classical methods revisited: cobordism, thickenings, group actions and symplectic topology
I will discuss the following problem: under what assumptions a given manifold F can be the fixed point set of a symplectic action? For smooth actions on disks it is known (due to Oliver and Pawałowski) that the necessary and sufficient condition is the acyclicity of F. I will present some classical methods which lead to the following theorem. A compact manifold F can be realized as the fixed point set of a symplectic action of the circle group on a disk of sufficiently large dimension if and only if it is symplectic with convex boundary.
|12.20-13.00||M. Mroczkowski (UG), Minimal crossing number of classical links under the projection with the Hopf map
Let h from S^3 to S^2 be the Hopf map and L a link in S^3. We study the minimal crossing number of h(L), denoted by ch(L). This link invariant was considered by T. Fiedler [F], where some lower bounds for ch were found for a family of algebraic knots. We will introduce arrow diagrams of links and some corresponding Reidemeister moves. As ch(L) is the minimal number of crossings for arrow diagrams of L, it is possible to evaluate ch(L) for specific links and knots. We present examples of such links and some estimates of ch depending on classical (i.e. non arrow) diagrams of links.
[F]: Thomas Fiedler, Algebraic links and the Hopf fibration, Topology 30(2), 259-265
|14.35-15.15||M. Zynel (UwB), Geometry on the lines of spine spaces I PDF|
|15.20-16.00||K. Petelczyc, Geometry on the lines of spine spaces II|
|16.15-16.55||M. Czarnecki (UŁ), Bisectors in complex hyperbolic space
Bisectors or equidistants from pair of distintct points in real space forms are totally geodesic hypersurfaces. In the complex hyperbolic space, i.e. complex projectivization of the set of negative (with respect to the Hermitian form) vectors, such an equdistant is not totally geodesic but possesses interesting geometric properties which could be algebraically studied. We shall give some examples containing families which foliate complex hyeprbolic space.
Czwartek, 22.06 (sala 209)
|09.00-9.40||V. Shevchishin (UWM Olsztyn), Higher dimensional projective geometry. Lines and quadrics
The subject of study of the „classical” geometry are simple planar objects: lines, triangles, polygons, circles, conics. Trying to generalize those notions and theorems to higher dimensions, one comes naturally to foundations of projective algebraic geometry.
In my talk I present two simple, but still very beautiful theorems about lines and quadrics in CP3.
|09.45-10.25||A. Matraś (UWM Olsztyn), Distant graphs and Cayley graphs PDF|
|10.45-11.25||E. Bartnicka (UWM Olsztyn), The classification of distant graphs of projective lines over finite rings. PDF|
|11.30-12.10||M. Pankov (UWM Olsztyn), Zigzags in triangulations PDF (joint work with Adam Tyc)|
|12.15-12.55||M. Niebrzydowski (UG), Homology of ternary algebras yielding invariants of knots and knotted surfaces
We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move in knot theory. Adding some simple axioms leads to homological invariants of knots and knotted surfaces.
|14.35-15.15||W. Kozlowski (UŁ), Geometry of Bott type connection joint work with J. Kalina i A. Pierzchalski.|
|15.20-16.00||A. Zastrow (UG) The comparison of topologies related to various concepts of generalized covering spaces (joint work with: Ziga Virk, IST Vienna)
The idea to generalize covering-space theory beyond the class of semilocally simply connected spaces is fairly old, and apparently the first paper making already a suggestion in this direction is from the sixties. Various non-equivalent definitions for generalized covering spaces have been suggested, depending on which properties of classical covering spaces should been maintained and which can be given up. One of the concepts is the idea to use in principle the same construction as in the classical case via the „universal path space” [BS], but being happy with covering spaces satisfying weaker conditions. Even for this concept of generalizing covering spaces different propositions for how to define the topology on the covering spaces have been made, which in the classical locally path-connected and semilocally simply connected case all give the topology of the classical covering spaces. The subspace of the universal path space consisting of those paths that return to the base point is the fundamental group, which, starting with a paper from 2002 by Biss ([Biss]) is meanwhile also considered as an object which apart from its algebraic structure has a topological structure. Biss’ paper, although it contains several mistakes, may therefore be considered as influential; and the object proposed by him, the topological fundamental group, has since been discussed in several papers and has been generalized to higher dimensions, although it is by now known to be only a quasitopological group. We are aware of at least five different topologies that allow to topologize the universal path path space with the aim to obtain generalized covering spaces.
The first three of them coming from the papers [Biss], [FZ] & [BDLM] have already been compared in our paper [VZ], and, wherever this was not proposed in the original source, have been extended from the fundamental group to the universal path space. The main aim of this talk will be the definition of the forth topology, its extension to the universal path space, and its comparison with the topologies that have already been discussed in [VZ]. This forth topology was constructed by Brazas in [Br] as a weaker topology than those of Biss, but amongst them as the richest topology, that makes sure that the fundamental group will still be a topological group. The results of [VZ] will be, wherever necessary, repeated, and, if time should allow, I might also add to this talk my original motivation for constructing generalized covering spaces, which was a problem from geometric group theory.
[Biss] D. K. Biss, „The topological fundamental group and generalized covering spaces”, Topology Appl., Vol. 124 (2002), 355–371.
[BS] W. A. Bogley, A. J. Sieradski, „Universal path spaces”, preprint; http://oregonstate.edu/~bogleyw/.
[FZ] H. Fischer, A. Zastrow, „Generalized universal covering spaces and the shape group”, Fund. Math., Vol. 197 (2007), 167–196.
[BDLM] N. Brodsky, J. Dydak, B. Labuz, A. Mitra, „Covering maps for locally path connected spaces”, Fund. Math., Vol. 218 (2012), 13–46.
[VZ] Z. Virk, A. Zastrow, „The comparison of topologies related to various concepts of generalized covering spaces”, Topology Appl., Vol. 170 (2014), 52–62.
[Br] J. Brazas, „The fundamental group as a topological group”, Topology Appl., Vol. 160 (2013), 170–188.
Instytut Matematyki Uniwersytetu Gdańskiego
ul. Wita Stwosza 57, Gdańsk
Wszystkie spotkania w dniu 21 czerwca odbędą się w auli nr 2, natomiast 22 czerwca – w sali 209.
Andrzej Szczepański: firstname.lastname@example.org