Scientific report Mariusz ynel April 22, 2015 Scientific report 2 - PDF document

Scientific report Mariusz Żynel April 22, 2015

Scientific report 2 Contents 1 Scientific degrees 3 2 Employment 3 3 Scientific achievement 3 3.1 The title of scientific achievement . . . . . . . . . . . . . . . . . . . . 3 3.2 Papers that constitute scientific achievement . . . . . . . . . . . . . 3 3.3 Summary of scientific achievement . . . . . . . . . . . . . . . . . . . 4 3.3.1 Polar Grassmann spaces . . . . . . . . . . . . . . . . . . . . . 7 3.3.2 Affine polar Grassmann spaces . . . . . . . . . . . . . . . . . 9 3.3.3 Orthogonal intersection in Euclidean geometry . . . . . . . . 11 3.3.4 Orthogonal intersection of lines in metric-affine geometry . . 12 3.3.5 Orthogonal intersection in metric-projective geometry . . . . 13 3.3.6 Grassmann spaces of regular subspaces . . . . . . . . . . . . . 14 3.3.7 Primitive notions of spine spaces . . . . . . . . . . . . . . . . 15 3.3.8 Orthogonality and correlations of Grassmann spaces . . . . . 18 3.3.9 Complements of intervals in projective Grassmannians . . . . 19 4 The other scientific results 20 4.1 Papers not contained in the scientific achievement, published after PhD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Summary of the other scientific results . . . . . . . . . . . . . . . . . 21

Scientific report 3 1 Scientific degrees • 2004: Doctor of Philosophy (PhD), Warsaw University of Technology, Faculty of Mathematics and Information Science, dissertation: Projections in the lattice of subspaces of a vector space , supervisor: prof. nzw. dr hab. Krzysztof P. Belina-Prażmowski-Kryński. • 1996: Master of Science (MSc), Warsaw University, Białystok Branch, Institute of Mathematics, subject: Finite Grassmannian geometries , supervisor: prof. nzw. dr hab. Krzysztof P. Belina-Prażmowski-Kryński. 2 Employment • 2005 - : adiunkt, University of Białystok. • 1997 - 2005: asystent, University of Białystok. • 1996 - 1997: asystent, Warsaw University, Białystok Division. 3 Scientific achievement 3.1 The title of scientific achievement Concise systems of primitive notions for geometry of fragments of projective Grass- mann spaces 3.2 Papers that constitute scientific achievement [A1] M. Żynel , Complements of Grassmann substructures in projective Grassmannians , Aequationes Math. 88 (2014), no. 1-2, 81-96, DOI: 10.1007/s00010-013-0210-1 . [A2] J. Konarzewski, M. Żynel , A note on orthogonality of subspaces in Euclidean geometry , J. Appl. Logic 11 (2013), no. 2, 169-173, DOI: 10.1016/j.jal.2013.01.001 . [A3] M. Żynel , Correlations of spaces of pencils , J. Appl. Logic 10 (2012), no. 2, 187- 198, DOI: 10.1016/j.jal.2012.02.002 . [A4] K. Prażmowski, M. Żynel , Orthogonality of subspaces in metric-projective geom- etry , Adv. Geom. 11 (2011), no. 1, 103-116, DOI: 10.1515/advgeom.2010.041 . [A5] M. Prażmowska, K. Prażmowski, M. Żynel , Grassmann spaces of regular sub- spaces , J. Geom. 97 (2010), no. 1-2, 99-123, DOI: 10.1007/s00022-010-0040-4 .

Scientific report 4 [A6] K. Prażmowski, M. Żynel , Possible primitive notions for geometry of spine spaces , J. Appl. Logic 8 (2010), no. 3, 262-276, DOI: 10.1016/j.jal.2010.05.001 . [A7] M. Prażmowska, K. Prażmowski, M. Żynel , Affine polar spaces, their Grass- mannians, and adjacencies , Math. Pannon. 20 (2009), no. 1, 37-59. [A8] M. Prażmowska, K. Prażmowski, M. Żynel , Metric affine geometry on the universe of lines , Linear Algebra Appl. 430 (2009), no. 11-12, 3066-3079, DOI: 10.1016/j.laa.2009.01.028 . [A9] M. Prażmowska, K. Prażmowski, M. Żynel , Euclidean geometry of orthogonal- ity of subspaces , Aequationes Math. 76 (2008), no. 1-2, 151-167, DOI: 10.1007/s00010-007-2911-9 . [A10] M. Pankov, K. Prażmowski, M. Żynel , Geometry of polar Grassmann spaces , Demonstratio Math. 39 (2006), no. 3, 625-637. 3.3 Summary of scientific achievement Let us start with some remarks as to the subject of the thesis itself. So, we in- tensionally use an imprecise term of ”concise” for we do not classify systems of primitive notions, neither for the number of relations, the number of arguments to specific relations, nor the complexity of axioms. We avoid the term ”the minimal system of primitive notions” which suggests such a classification and requires to specify criteria. The word ”concise” is intended to mean here that such a system of primitive notions is possibly simple and independent as well as intuitive and ele- mentary, known from classical geometries or close to them. The fragment of a Grassmannian, or in fact the fragment of a Grassmann space, is a well and naturally defined subset of the point set in the Grassmann space together with lines the intersections of which with that subset are sufficiently large. The term ”projectively embeddable Grassmann space” does not mean here that the given space can be embedded by a collineation into a suitable exterior power of some vector space. It rather means that the Grassmann space is defined over some projective space, that is, its points are subspaces of that projective space. As the dimension of considered projective spaces is more then 3, we can equivalently, without loss of generality, take vector spaces. Such and approach lets to utilize a convenient algebraic apparatus tied to vector spaces. Let us specify the notion of a Grassmann space substantial to our considerations and other essential notions. Let S be any set and L ⊆ 2 S . The elements of the set S we call points and the elements of the set L we call lines . Then the incidence structure A = � S, L� , where the incidence relation between points and lines is im- plicitly the membership relation ∈ , is a partial linear space whenever two distinct lines meet in at most one point. The point-line space A is connected , when every two of its points can be joined with a path, i.e. with a sequence of lines where every two consecutive lines intersect each other. We call a subset X ⊆ S a subspace of A if every line that joins two distinct points in X is entirely contained in X . A subspace is said to be strong if every two of its points are collinear. We say that A is a linear space when every two points in A are collinear. Among partial linear spaces there are Gamma spaces characterized by a condition called none-one-or-all

Scientific report 5 axiom : a point not on a line is collinear with none, one or all the points on that line . Now, let � P, ⊆� be a poset and let dim: P − → { 0 , 1 , . . . , n } be a dimension function. For 0 ≤ k ≤ n , we denote by P k the family of all k -dimensional elements of P . Take H ∈ P k − 1 and B ∈ P k +1 , such that H ⊂ B . The set p ( H, B ) := � � U ∈ P k : H ⊂ U ⊂ B (1) is called a k -pencil . The incidence point-line structure � � P k ( P ) := P k , P k ( P ) , (2) where P k ( P ) is the family of all k -pencils, is a Grassmann space (cf. [8], [68], [86]). At this level of generality not much can be said about the properties of this structure. In the classical approach P is the family of subspaces of some projective space, or equivalently, of a vector space when 3 ≤ n . Let V be a (left) vector space over some division ring. We denote by Sub( V ) the family of all the subspaces, and by Sub k ( V ) the family of all k -dimensional subspaces of V . In (1), (2) take P = Sub( V ) . Then P k ( V ) becomes our Grassmann space. It is also called the shadow space of the building associated with V [16, 17]. Grassmann space P k ( V ) is a partial linear space, and even more, it is a connected Veblenian Gamma space (cf. [17]). If k = 1 , or k = dim( V ) − 1 provided that dim( V ) < ∞ , then P k ( V ) is a projective space. In the other cases P k ( V ) is a proper partial linear space, that is there are pairs of noncollinear points in it. In the study of such geometries maximal strong subspaces play an important role. There are two families of such subspaces in P k ( V ) : stars and tops . Every star and top is, up to an isomorphism, a projective space. In this way we get a covering of P k ( V ) by projective spaces. Stars are sets of the form S ( H ) = { U ∈ Sub k ( V ): H ⊂ U } , (3) where H ∈ Sub k − 1 ( V ) , and tops are sets of the form T ( B ) = { U ∈ Sub k ( V ): U ⊂ B } , (4) where B ∈ Sub k +1 ( V ) . These are specific cases of interval subspaces , that is sets [ Z, Y ] k = { U ∈ Sub k ( V ): Z ⊆ U ⊆ Y } , (5) where Z, Y ∈ Sub( V ) i Z ⊆ Y . A projective space, where points are one-dimensional subspaces of V and lines are two-dimensional subspaces of V will be written as P ( V ) . Actually P k ( V ) should be called projective Grassmann space due to the fact that P k ( V ) ∼ = P k − 1 ( P ( V )) . We say that two points U, W in P k ( V ) are adjacent and write U ∼ W , whenever dim( U ∩ W ) = k − 1 . The classical result characterizing collineation of a Grassmann space is Chow’s theorem (cf. [15]) which can be viewed as the generalization of the Fundamental Theorem of Projective Geometry.