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Scientific report Mariusz ynel April 22, 2015 Scientific report 2 - - PDF document
Scientific report Mariusz ynel April 22, 2015 Scientific report 2 - - PDF document
Scientific report Mariusz ynel 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
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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.
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[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
- f 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
- f 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 ⊆ 2S. 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
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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 Pk the family of all k-dimensional elements
- f P. Take H ∈ Pk−1 and B ∈ Pk+1, such that H ⊂ B. The set
p(H, B) :=
- U ∈ Pk : H ⊂ U ⊂ B
- (1)
is called a k-pencil. The incidence point-line structure Pk(P) :=
- Pk, Pk(P)
- ,
(2) where Pk(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 Subk(V ) the family of all k-dimensional subspaces of V . In (1), (2) take P = Sub(V ). Then Pk(V ) becomes our Grassmann
- space. It is also called the shadow space of the building associated with V [16, 17].
Grassmann space Pk(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 Pk(V ) is a projective space. In the other cases Pk(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 Pk(V ): stars and tops. Every star and top is, up to an isomorphism, a projective space. In this way we get a covering of Pk(V ) by projective spaces. Stars are sets of the form S(H) = {U ∈ Subk(V ): H ⊂ U}, (3) where H ∈ Subk−1(V ), and tops are sets of the form T (B) = {U ∈ Subk(V ): U ⊂ B}, (4) where B ∈ Subk+1(V ). These are specific cases of interval subspaces, that is sets [Z, Y ]k = {U ∈ Subk(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 Pk(V ) should be called projective Grassmann space due to the fact that Pk(V ) ∼ = Pk−1(P(V )). We say that two points U, W in Pk(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.
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Scientific report 6 Theorem 3.1 (Chow, 1949). Let V be a n-dimensional vector space, where 3 ≤ n < ∞ and let 1 < k < n − 1. Any bijective transformation of the point set Subk(V ) of the Grassmann space Pk(V ) which preserves the adjacency of points in both directions is induced by a collineation of the projective space P(V ), or by a duality of P(V ) in the case where n = 2k. This is one of the first theorems which characterize classes of geometric trans- formations with the absence of regularity conditions: continuity, differentiability or
- affinity. The field of research devoted to this is called characterizations of geomet-
rical transformations under mild hypotheses. In this series of results is Alexandrov’ theorem characterizing Lorentz transformations related to the theory of special rel- ativity (cf. [2]): Theorem 3.2 (Alexandrov, 1950). Any bijective transformation of n-dimensional Minkowski space-time, where 3 ≤ n, which preserves Lorentz-Minkowski distance 0 in both directions (preserves rays of light) is a Lorentz transformation up to a dilatation. There is also in this series a very strong Beckman and Quarles theorem charac- terizing isometries in an Euclidean space (cf. [5]): Theorem 3.3 (Beckman and Quarles, 1953). A mapping of n-dimensional Eu- clidean space into itself, where 2 ≤ n < ∞, which preserves Euclidean distance 1 is a Euclidean motion. In [36], [37] Huang proved that the assumptions to Chow’s Theorem can be weakened: a transformation need not to be a bijection, it suffices that it is an injection preserving adjacency (in one direction as the mapping needs not to be invertible). Another generalization of this theorem, toward applications in polar spaces, is presented by Pankov in [66]. In [57] there is a new result from 2014 achieved by De Schepper and Van Maldeghem generalizing Chow’s theorem for the family of pairs of subspaces of a vector spaces that are in a given distance from each
- ther.
A bit earlier than Chow a similar result in a different context has been achieved by Hua. Namely, in [32], [33] Hua characterized bijections which preserve, in both directions, adjacency of square matrices over complex field. Two matrices are ad- jacent if the rank of the difference is 1 (these matrices are in distance 1 from each
- ther). Grassmann spaces proved to be in a close relationship with spaces of ma-
trices (cf. [91, Ch. 3]). In the literature we can find a lot of applications and successful attempts of carrying Chow’s theorem over to a new ground: [27], [39], [34], [35], [40], [36], [41], [42], [70]. In a recently published work from coding theory [24] Ghorpade and Kaipa apply Chow;s theorem to characterize the group of automorphisms of Grass- mann codes. Next, they prove an analogue of this theorem for Schubert divisors in Grassmannians. These results have been completed in the book by Pankov [67,
- Ch. 6, 6.3]. In [14] there is a new interesting class of linear codes related to polar
Grassmann spaces presented. Chow’s theorem as well as the other theorems in a series of characterizations of transformations under mild hypotheses, are formulated according to Klein’s program
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Scientific report 7 [48]. Based on the theory of definability (cf. [13] and [11], [85]), in particular because
- f:
Fact 3.4. If all the transformations preserving certain geometric property η preserve another geometric property δ, then δ is definable by means of η. Chow’s theorem says that the ternary relation of collinearity in a projective space P(V ) is definable in terms of the binary adjacency of its (k − 1)-dimensional sub-
- spaces. So, we can treat adjacency as a single primitive notion and reformulate
Chow’s theorem: The adjacency of (k − 1)-dimensional subspaces in a (n − 1)-dimensional projective space is a single primitive notion of that projective space. By 3.4 an axiom system expressed in the language of δ can be replaced by an axiom system in the language of η. Instead of invariants of transformations, what Klein’s program [48] suggests, we can equivalently, according to B¨ uchi’s program [13], investigate definability of given geometric properties in terms of other geometric
- properties. Therefore, primitive notions play a fundamental role.
Chow’s theorem can be expressed also in the following way: The ambient projective space P(V ) can be recovered from the Grassmann space Pk(V ). In this thesis we ask a question in what fragments of Pk(V ) the ambient pro- jective space P(V ) can be recovered, or equivalently, what systems of primitive notions on this fragment are sufficient for the geometry of P(V ). Most of the time, as it turns out, a single relation on the family of k-dimensional subspaces is suffi- cient and then we prove a suitable variant of Chow’s theorem. On the family of regular subspaces the relation of orthogonal intersection, ortho-adjacency, is a suffi- cient primitive notion, but we also prove that collinearity as incidence is a sufficient primitive notion. On the other hand in spine spaces a single relation is not always sufficient and the concise system of primitive notions is more complex there. We can get such a system with a nonstandard structure of lines using properties of the covering by strong subspaces and their incidence. For various geometries that come from Grassmann spaces we succeed in achiev- ing a simple characterization in the language of a single relation satisfying Chow’s theorem or in the language of incidence. In any case the automorphisms of the ob- tained system of primitive notions are the expected automorphisms of the ambient projective space. In the sequel we present specific fragments of Grassmann spaces in more details and analyze possible concise systems of primitive notions for them. 3.3.1 Polar Grassmann spaces Let ξ be a nondegenerate, reflexive sesquilinear form on V . The form is reflexive whenever ξ(x, y) = 0 implies ξ(y, x) = 0 for all x, y ∈ V . We denote by m the index
- f the form ξ. For U, W ∈ Sub(V ) we write
U ⊥ W iff ξ(u, w) = 0 for all u ∈ U, w ∈ W,
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Scientific report 8 and U⊥ := {w ∈ V : ξ(u, w) = 0 for all u ∈ U}. We say that a subspace U is
- nonisotropic, if U ∩ U⊥ = 0,
- isotropic, if U ∩ U⊥ = 0,
- totally isotropic, if U ⊆ U⊥.
We write Q for the set of all totally isotropic subspaces, and Qk for the subset of all k-dimensional subspaces in Q. The form ξ determines a polarity, that is an involutory correlation in the projec- tive space P(V ). The image of a point U under this polarity is the hyperplane U⊥, and a point U is selfconjugate if U ⊥ U. The set Q1 (or generally Q) is a quadric in P(V ). Further, we assume that the form ξ is symplectic, that is, ξ(x, x) = 0 for all x ∈ V . Then all the points in P(V ) are selfconjugate. A projective space equipped with a symplectic polarity is called a null system (cf. [17]). Let us recall that a symplectic form can be nondegenerate only if the dimension of V is even. In [15] Chow proved that bijective transformations of Qm which preserve adja- cency in both directions are induced by collineations of the null system, i.e. by au- tomorphisms of V which preserve the relation ⊥. Dieudonn´ e generalized this result in [22], [23] for any reflexive forms. While Huang in [36] shows that the considered transformations need not to be bijective and it is enough they are surjective. Then, in [38] she presents yet another system of conditions, which characterize transfor- mations of Qm. The characterization of transformations of Qm which preserve base subsets is presented by Pankov in [69]. We generalize Chow’s result in [A10] taking Qk for k ≤ m and any polarity, not necessarily symplectic. So, we assume further that V is over a field of odd characteristic, the form ξ need not to be symplectic and consider a polar space M := Q1, Q2, ⊂. The notion of a polar space, as a generalization of the geometry on a quadric with respect to orthogonal, symplectic and unitary polarity comes from Veldkamp who was first to introduce a synthetic characterisation of such a geometry in [90]. Tits extended this characterisation for geometries related to pseudo-quadratic forms in [88]. As Kreuzer notices in [50] both approaches, the one of Veldkamp and the
- ther of Tits, are equivalent.
In [12] Buekenhout and Shult prove that most of Veldkamp’s and Tits’ axioms are implied by one beautiful condition called one-
- r-all axiom: a point not on a line is collinear with one or all the points on that
line (that’s why polar spaces are Gamma spaces) while in [43] Johnson extends the results of Veldkamp and Tits for spaces of any dimension. Note that m − 1 is the dimension of a maximal strong subspace in M. Assume that 1 ≤ k ≤ m. Since k-dimensional subspaces of M are subspaces of dimension k+1 in V , Qk is a fragment of the Grassmann space Pk+1(V ). Moreover, if we take Q as a poset P in (1) and (2), then we get a polar Grassmann space Pk+1(Q) ∼ = Pk(M).
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Scientific report 9 The essential difference in the case where k < m lies in the fact that the join U + W, for U, W ∈ Qk such that U ∩ W ∈ Qk−1, can be totally isotropic or not. If k = m, then this join is never totally isotropic. So, in general case we have to deal with two different adjacency relations. As previously, for U, W ∈ Qk we say that they are adjacent and write U ∼ W whenever U ∩ W ∈ Qk−1. On Qk consider yet another relation. We say that U, W ∈ Q are orthogonal and write U ⊥ W if any two points p, q such that p ⊂ U and q ⊂ W are collinear in M. Now, we can introduce another adjacency relation. For U, W ∈ Qk we write U ∼ ∼ W if U ⊥ W and U ∼ W. Note that ∼ ∼ is the collinearity relation in Pk(Q). Theorem 3.5 ([A10, Theorem 5.1]). Let f be a bijective transformation of the family Qk. Then: (i) If k ≤ m and the transformation f preserves the adjacency ∼, then it is induced by a collineation of the polar space M. (ii) If k < m, either m = 4 or k = 2, and the transformation f preserves the collinearity ∼ ∼, then it is induced by a collineation of the polar space M. To prove the above theorem we first characterize the cliques of ∼, the cliques of ∼ ∼, and strong subspaces of Pk(Q) [A10, Theorem 3.5]. Next, applying properties
- f stars we move from dimension k to dimension 1 and use well known results from
projective geometry. Connectedness of the considered geometry is essential here. One case of the Grassmann space of isotropic lines in a 7-dimensional projective space in which the dimension of totally isotropic subspace is 3 remains open. For orthogonality ⊥ a variant of Chow’s theorem is also true. Theorem 3.6 ([A10, Corollary 5.3]). If k < m, then every bijective transformation
- f Qk which preserves the orthogonality ⊥ is induced by a collineation of the polar
space M. Theorems 3.5 and respectively 3.6 say that under certain assumptions, both adjacency relations, as well as orthogonality relation on the set of k-dimensional subspaces of the polar space M can be a single primitive notion for M. Conse- quently, the polar space M can be recovered from the fragment of a Grassmann space Pk+1(V ) that corresponds to k-dimensional subspaces of the polar space M, equipped with adjacency relation or orthogonality relation. In the next step the projective space P(V ) can be recovered. 3.3.2 Affine polar Grassmann spaces Deleting a hyperplane from a polar space, as Cohen and Shult do in [18], we get an affine polar space. In a standard way we can put the structure of a Grassmann space onto this space and get affine polar Grassmann space. However, in this thesis we prefer a bit different approach. Let V be a vector space over a field of characteristic = 2 and let ξ be a nonde- generate reflexive bilinear form on V . The form ξ determines the relation ⊥. The
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Scientific report 10 family of all k-dimensional cosets in the vector space V , where 0 ≤ k ≤ dim(V ), we denote by Hk =
- a + U : a ∈ V i U ∈ Subk(V )
- .
We identify 0-dimensional cosets with vectors in V and additionally assume that H−1 = {∅}. We say that two cosets a + U, b + W, where U, W ∈ Sub(V ) and a, b ∈ V , are parallel and write a + U b + W when U = W, and we say that they are orthogonal and write a + U ⊥ b + W when U ⊥ W. In a standard way we get an affine space A := V, H1, and metric-affine space (A, ⊥). Taking cosets H instead of linear subspaces Sub(V ), for 0 ≤ k < dim(V ) we construct a Grassmann space Pk(A) = Pk(H) =
- Hk, Pk(H)
- (6)
- ver the affine space A.
Moving from projective to affine space by removing a hyperplane aside pencils with proper vertices pencils with improper vertices will
- appear. For W ∈ Hk, B ∈ Hk+1 when there is a translation t: V −
→ V such that t(W) ⊆ B the pencil of parallel k-subspaces in A is the set of the form p∗(W, B) :=
- U ∈ Hk : W U ⊂ B
- .
(7) Affine Grassmann space is a point-line incidence structure P∗
k(A) := P∗ k(H) =
- Hk, Pk(H) ∪ P∗
k(H)
- ,
(8) where P∗
k(H) is the family of all k-pencils of parallel subspaces of A (cf. [19]). This
is a connected Gamma space. It can be embedded into the projective Grassmann space Pk(V ). Now, consider cosets with respect to totally isotropic subspaces, i.e. the set U := V + Q. We write Uk for the set of k-dimensional cosets in U. The point-line incidence structure, where points are the points of A and the lines are the isotropic lines in A, that is U := V, U1, is an affine polar Grassmann space. In case, where the form ξ is symplectic we have U1 = H1 and consequently A = U. For this reason we assume further that the form ξ is symmetric, 3 ≤ dim(V ), and 0 ≤ k ≤ m, where m is the index of ξ. Our approach to affine polar spaces differs from that of Cohen and Shult [18]. In our view an affine polar space is a point-line incidence structure derived from a metric-affine space. This causes that our affine polar spaces include Minkowskian geometry and, in particular, our result generalizes Alexandrov-Zeeman type theo- rems (cf. [59]) but symplectic geometries, affine polar spaces that arise by deleting nontangent hyperplanes, and all affine polar spaces with nondesarguesian planes are
- excluded. Otherwise, the two approaches define the same geometry, so that they
are equivalent.
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Scientific report 11 Over an affine polar space U, in analogy to (6) and (8), we can build affine polar Grassmann spaces Pk(U) and P∗
k(U). For U, W ∈ Uk consider the following three
adjacency relations: U ∼
− W : ⇐
⇒ U ∩ W ∈ Uk−1, U ∼
+ W : ⇐
⇒ U ∪ W ⊂ X for some X ∈ Uk+1, U ∼ ∼ W : ⇐ ⇒ U ∼
− W i U ∼ + W.
The relation ∼
+ is the binary collinearity in P∗ k(U) and ∼
∼ is the binary collinearity in Pk(U). For both alternatives of affine polar Grassmann space we prove Chow’s theorem. Theorem 3.7 ([A7, Theorem 4.1]). Let ∼ be one of the three relations ∼
−, ∼ + or
∼ ∼. When ∼ = ∼
+, ∼
∼, we assume that 0 ≤ k < m − 1. When ∼ = ∼
−, we assume
that 0 < k < m. Then, the affine polar space U ans the affine space A, are both definable in the structure Uk, ∼. In consequence, both A as well as U are definable in Pk(U) and in P∗
k(U) under the assumption that k < m − 1.
Here, the reasonings in proofs are generally similar to those in [A10] but we use the properties of affine and metric-affine geometry. On the side note, in [76] we present a proof of 3.7 for affine polar spaces in the sense of Cohen and Shult [18]. 3.3.3 Orthogonal intersection in Euclidean geometry In the literature there are various known systems of primitive notions for Euclidean geometry: points, lines, and line perpendicularity [54], tangency of circles in a plane [74], tangency of spheres [63], line intersection [31]. Line intersection can be also a single primitive notion for projective geometry [61]. Bijective transformations of lines in an Euclidean space of dimension at least 3 which preserve perpendicularity have been characterized in [7]. In papers [60] and [65] line perpendicularity is inves- tigated as a single primitive notion for hyperbolic geometry. In [82] Schwabh¨ auser and Szczerba prove that line perpendicularity can be a single primitive notion for Euclidean geometry in dimensions ≥ 4. Our goal in [A9], [A2] is to generalize this result and show that an Euclidean geometry can be expressed as the theory of or- thogonal intersection, or ortho-adjacency, of subspaces of a fixed dimension. Chow’s theorem for ortho-adjacency of lines has been proved in an elliptic geometry [28], [29], in a hyperbolic geometry [56], and in a symplectic geometry [30]. An Euclidean space is a specific fragment of a projective Grassmann space Pk(V ) as it arises by deleting a hyperplane from a projective space P(V ) and by equipping the obtained affine space with a symmetric bilinear form ξ with no isotropic vectors (inner product). The form ξ determines segment orthogonality ⊥ (and segment congruence) in a standard way. So, following the notation of 3.3.2, let our Euclidean space be the structure A = V, H1, , ⊥. Let U, W ∈ H. The intersection of all subspaces containing U ∪ W will be
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Scientific report 12 written as U ⊔ W. Consider the following relations: U ⊥ W : ⇐ ⇒ u1u2 ⊥ w1w2 for all u1, u2 ∈ U and w1, w2 ∈ W, U ⊥
∗ W : ⇐
⇒ U ⊥ W and U ∩ W = ∅, U ⊥
- W : ⇐
⇒ there are U0, W0 ∈ H, such that U0 ⊥
∗ W, W0 ⊥ ∗ U,
(U ∩ W) ⊔ U0 = U, (U ∩ W) ⊔ W0 = W, and U ∩ W = U, W, U ⊥k
k1,k2 W : ⇐
⇒ U ⊥
- W, U ∈ Hk1, W ∈ Hk2, and U ∩ W ∈ Hk.
The relation ⊥ can not be a single primitive notion for an Euclidean geometry (cf. [A9, Fact 1.1]). The relation ⊥
- is auxiliary in some definitions. We prove an
analogue of Chow’s theorem for ⊥
∗ and ⊥k−1 k,k .
Theorem 3.8 ([A9, Theorem 2.10]). If either k = 1 and 4 ≤ n, or 2 ≤ k and k + 2 ≤ n, then the relation ⊥k−1
k,k on the family of all k-dimensional subspaces can
be a single primitive notion for a n-dimensional Euclidean geometry. Theorem 3.9 ([A9, Theorem 2.17]). If 3k + 1 ≤ n, then the relation ⊥
∗ on the
family of all k-dimensional subspaces can be a single primitive notion for a n- dimensional Euclidean geometry. In [A9, Theorem 2.18] there are versions of 3.8 and 3.9 expressed in the language
- f automorphisms, closer to the original wording used by Chow. In [A2] we show that
- rtho-adjacency of subspaces of different dimensions can also be a single primitive
notion for an Euclidean geometry. Theorem 3.10 ([A2, Theorem 2.4]). Assume that 1 ≤ k1, k2 < dim(V ). (i) If 0 ≤ k < k1, k2 and k1 + k2 − k ≤ dim(V ), then the Euclidean space A is definable in the structure
- V, Hk1, Hk2, ⊥k
k1,k2
- .
(ii) The Euclidean space A is definable in the structure
- V, Hk1, Hk2, ⊥
- ∩ (Hk1 × Hk2)
- .
3.3.4 Orthogonal intersection of lines in metric-affine geometry In [A9] we characterized an Euclidean geometry in terms of orthogonal intersection
- f subspaces of a fixed dimension. A natural question arises whether this result
can be generalized to a wider class of geometries. An Euclidean geometry is in the class of metric-affine geometries which also contains the Minkowski geometry. Our methods failed in this level of generality and that is why in [A8] we restrict ourselves to ortho-adjacency of lines. Various, equivalent axiom systems of a metric-affine space as an affine space equipped with a symmetric bilinear form can be found in [52], [53], [81], [80]. A typical notion for metric geometries is reflection, that is an involutory isometry which fixpoints form a subspaces (cf. [3], [55]). In the language of reflections an Euclidean geometry [1] and the Minkowski geometry [84], [92], [93] have been
- characterized. If there is a reflection with respect to some subspace this subspace is
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Scientific report 13 regular (nonisotropic). For this reason we also deal with ortho-adjacency of regular lines in metric-affine geometry in [A8]. Following notation of 3.3.2 i 3.3.3 let A = V, H1, , ⊥, where ⊥ is given by a nondegenerate symmetric bilinear form on V , be our metric-affine space. Theorem 3.11 ([A8, Main Theorem]). Assume that the coordinate field of A is
- infinite. Then the relation ⊥
∗ of orthogonal intersection of lines in the set of all
lines of A as well as in the set of all regular lines of A can be used as a single primitive notion of the metric-affine geometry A if dim(A) ≥ 4. Our reasoning is based on the assumption that the size of a line is large enough, which affects the coordinate field. In dimensions 3 the relation ⊥
∗ in the set of lines
(regular lines or all of the lines) cannot be used as a single primitive notion which has been proved in [A8, Theorem 4.9]. 3.3.5 Orthogonal intersection in metric-projective geometry The relation of orthogonal intersection has been investigated in [A9] and [28]. In [28], [29] Havlicek proves Chow’s theorem for this relation on the lines of an elliptic
- space. In [30] he achieves a similar result for symplectic spaces, while List in [56]
does the same in hyperbolic spaces. In [A4] we generalize results of [28], [30], and [56] by taking ortho-adjacency on k-dimensional subspaces and by applying a common reasoning for elliptic, symplectic as well as hyperbolic geometry. Let ξ be a nondegenerate reflexive sesquilinear form on V . We follow the no- tation of 3.3.1. For k-dimensional subspaces U, W in V we say that U intersects
- rthogonally W, or they are ortho-adjacent, and write
U ∼ ⊥ W, iff U ∼ W i U ∩ W ⊥ = 0. By S we denote the set of all nonisotropic (regular) subspaces w.r.t. ξ. The Grass- mann space of regular subspaces Pk(S), a fragment of the Grassmann space Pk(V ), is defined in a standard way. The goal of [A4] is to recover the ambient metric-projective space (P(V ), ⊥) from the structure Sk, ∼ ⊥ of nonisotropic k-dimensional subspaces with ortho-adjacency. It is worth to point out the three cases here. When the form ξ is symplectic, then the relation ∼ ⊥ on Sk is empty. When the coordinate field is of characteristic 2, then the set of selfconjugate points (the quadric) w.r.t. ξ in the projective space P(V ) is a hyperplane. Then, there are nonisotropic lines which contain no regular points (nonisotropic 1-dimensional subspaces). In the case where dim(V ) = 2k the form ξ determines an automorphism of Sk, ∼ ⊥ which is not induced by an automorphism
- f (P(V ), ⊥).
Theorem 3.12 ([A4, Theorem 1.1]). If the coordinate field is not of characteristic 2, the form ξ is not symplectic, and 1 ≤ k ≤ n, then ortho-adjacency ∼ ⊥ on the set Sk of all nonisotropic k-dimensional subspaces can be used as a single primitive notion for metric-projective geometry (P(V ), ⊥). In the vein of Chow’s theorem this result can be put in the following way:
SLIDE 14
Scientific report 14 Theorem 3.13 ([A4, Theorem 1.2]). Under assumptions of 3.12, every bijective transformation of the set Sk of all nonisotropic k-dimensional subspaces which pre- serves ortho-adjacency ∼ ⊥ in both directions is induced by a collineation of the am- bient projective space P(V ) that preserves ⊥. 3.3.6 Grassmann spaces of regular subspaces In papers [A10] and [A7] we have investigated, generally speaking, the geometry
- f a quadric, or more precisely, the structure of subspaces of a fixed dimension in
projective and affine polar spaces. In [83] Shult asked a question if it is possible to characterize a fragment of a projective space outside a given quadric. This gave rise to the study of Delta spaces, called also copolar spaces. In [20] the structure
- f selfconjugate points and secants, i.e. nonisotropic lines that intersect a quadric
in two points, is examined. The paper [21] deals with the points outside a quadric and lines tangent to that quadric are, while [25] and [26] deal with points and lines outside a quadric. Another reason, mentioned in 3.3.4, to deal with a quadric complement is reflexion geometry (cf. [3], [4], [79]). Axes of reflexions are regular subspaces, that is subspaces outside a quadric. In [A8] and [A4] we examine structures of regular subspaces endowed with ortho- adjacency and recover the ambient metric-affine or metric-projective space. Here we want to reconstruct the ambient geometry from the linear structure of a Grassmann space of regular subspaces, both in metric-affine as well as in metric-projective space. Let V be a vector space over an infinite field of characteristic = 2 and let ξ be a nondegenerate symmetric bilinear form on V . We use notions and notations from 3.3.1, 3.3.2, and 3.3.5. Let us recall that Pk(S) is a projective Grassmann space of regular subspaces. For this fragment of the projective Grassmann space Pk(V ) we prove a counterpart of Chow’s theorem. Theorem 3.14 ([A5, Theorem 3.12]). Assume that dim(V ) = 2k. The metric- projective space (P(V ), ⊥) is definable in the projective Grassmann space of regular subspaces Pk(S). Theorem 3.15 ([A5, Theorem 3.18]). Automorphisms of the projective Grass- mann space of regular subspaces Pk(S) are induced by automorphisms of the regular metric-projective space S1, S2, ⊂, that is, by collineations of the projective space P(V ) which preserve ⊥ or, exclusively in case dim(V ) = 2k, also by products of such collineations with the correlation ⊥. We have an analogous result for the metric-affine space (A, ⊥), where A = V, H1, . The set of all regular subspaces in (A, ⊥) is R := V +S and Rk is the set
- f k-dimensional subspaces in R. We prove Chow’s theorem for affine Grassmann
space of regular subspaces. Theorem 3.16 ([A5, Theorem 4.5]). The metric-affine space (A, ⊥) is definable in both the Grassmann space of regular subspaces Pk(R) and in affine Grassmann space of regular subspaces P∗
R(k). Automorphisms of Pk(R) and P∗ R(k) are induced
by collineations of the affine space A which preserve ⊥.
SLIDE 15
Scientific report 15 In addition to analytic proofs of 3.14, 3.15, and 3.16 we provide also an axiomatic approach to recovering the ambient metric space from a Grassmann space of regular subspaces in [A5]. 3.3.7 Primitive notions of spine spaces So far, a set of subspaces of V which satisfy a certain condition with respect to a fixed sesquilinear form on V was chosen as a fragment of the Grassmann space Pk(V ). In contrast, a spine space is a fragment of a Grassmann space chosen so that it consists of subspaces of V which meet a fixed subspace in a specified way. The concept of spine spaces was introduced in [73] and developed in [75], [77], [78], [B4]. Let W be a fixed subspace of V and let m be an integer with k − codim(W) ≤ m ≤ k, dim(W). From the points of the Grassmann space Pk(V ) we take those which as subspaces of V meet W in dimension m, that is: Fk,m(W) := {U ∈ Subk(V ): dim(U ∩ W) = m}. As new lines we take those lines of Pk(V ) which have at least two new points: Gk,m(W) := {L ∩ Fk,m(W): L ∈ Pk(V ) and |L ∩ Fk,m(W)| ≥ 2}. On a line L of Pk(V ) either, there are no new points, all the points on L are new, or all except one are new. Respectively, in the second case a new line M = L∩Fk,m(W) will be called projective and in the last case it will be called affine. Note that, for affine lines we can define parallelism in a natural way: those lines are parallel which share a point outside the new universe. So, we have a horizon here. The class of affine lines is denoted by A and projective lines fall into two disjoint classes Lα and Lω. The distinction comes from how these lines can be extended to maximal strong subspaces. The point-line structure: A = Ak,m(V, W) :=
- Fk,m(W), Gk,m(W)
- will be called a spine space. This is a Gamma space. Its specific case, depending on
k, m and dim(W) can be: a projective space, a slit space (cf. [45], [46]), an affine space or the space of linear complements (cf. [9], [78]). On the other hand, A is a quite regular fragment of the Grassmann space Pk(V ). In the thesis we examine its possible systems of primitive notions. Adjacency of points Two points of A are adjacent, if they are adjacent in Pk(V ). In the language of the adjacency ∼, a ternary relation of collinearity L can be defined for points U1, U2, U3
- f A in a following way:
L(U1, U2, U3) : ⇐ ⇒ ∼(U1, U2, U3) ∧ (∃ Y1, Y2)
- Y1, Y2 ∼ U1, U2, U3,
Y1 ∼ Y2
- .
This is the key definition in a proof of Chow’s theorem for spine spaces:
SLIDE 16
Scientific report 16 Theorem 3.17 ([B5, Corollary 4.11]). Assume that there are no projective lines in A of a given type or such lines can always be extended to maximal strong projective
- subspaces. Then, the binary adjacency relation of points can be a single primitive
notion for the geometry of the spine space A. Line intersection Not every two intersecting lines determine a plane in A but through every point of A there go at least two noncoplanar lines. These two observations are critical in the proof of the following fact: Theorem 3.18 ([A6, Corollary 4.5]). Binary line intersection, taken on the family
- f all the lines, or on the family of projective lines only, can be a single primitive
notion for the geometry of the spine space A. Affine lines with parallelism Consider a substructure of the spine space A taking affine lines only: Aτ := Fk,m(W), A, . The horizon A∞ of A is closely related in a natural way to parallelism. Theorem 3.19 ([A6, Corollary 4.13]). Under assumptions of 3.17 the structures A and Aτ are definitionally equivalent. The universe of affine lines alone with no parallelism is essentially weaker and it is impossible to prove something similar for it. The closure of a spine space Let us consider the structure Aτ endowed with projective lines of type σ ∈ {α, ω} together with its projective closure: Nσ := Fk,m(W) ∪ Fk,m+1(W), Lτ ∪ Lσ ∪ L−σ
1 ,
where Lτ is the set of affine lines completed with their points at infinity and L−σ
1
is the set of lines on the horizon A, i.e. the set of directions of σ-semiaffine planes. Theorem 3.20 ([A6, Corollary 4.15]). If there are projective lines of both types on the horizon of A, then the structures A and Nσ are definitionally equivalent. Stars and tops as primitive notions As one would expect, maximal strong subspaces of the spine space A are restrictions
- f maximal strong subspaces of the ambient Grassmann space Pk(V ). These are, up
to isomorphisms, projective spaces or slit spaces (in other words semiaffine spaces). So, consider the family SW of all nontrivial intersections of stars (3) of Pk(V ) with the point set Fk,m(W) of our spine spaces A, and the incidence structure of points and stars Astar
k,m(V, W) :=
- Fk,m(W), SW
- .
SLIDE 17
Scientific report 17 Dually, TW is the family of all nontrivial intersection of tops (4) of Pk(V ) with the point set Fk,m(W), and A
top
k,m(V, W) :=
- Fk,m(W), TW
- .
As it turns out the points and stars of A together with the natural incidence relation ∈ can be a sufficient system of primitive notions to recover the entire spine space A. The same holds true if we take tops instead of stars. Theorem 3.21 ([A6, Fact 3.4]). The spine space A is definable in the spaces of stars Astar
k,m(V, W) (respectively in the space of tops A
top
k,m(V, W)) iff either, there are
no ω-projective lines, or every ω-projective line can be extended to an ω-projective subspace (respectively, there are no α-projective lines, or every α-projective line can be extended to an α-projective subspace). Two distinct stars can share at most a single point. We say that two stars are adjacent if they share a point. Analogously we define top adjacency. We have strengthened 3.21 and recover A using stars and star adjacency as primitive notions, and dually, using tops and top adjacency. Theorem 3.22 ([A6, Corollary 4.20]). If there are α-projective lines in A, ω- projective subspaces on A∞ and α-projective subspaces on (A∞)∞ (respectively, there are ω-projective lines in A, α-projective subspaces on A∞ and ω-projective subspaces
- n (A∞)∞), then the spine space A is definable in terms of star adjacency (respec-
tively, top adjacency), that is in terms of line adjacency in Astar
k,m(V, W) (respectively
in A
top
k,m(V, W)).
Summary If the spine space A is neither a linear space nor a Grassmann space, then the systems of primitive notions gathered in Table 1 are sufficient for A.
SLIDE 18
Scientific report 18 system of primitive notions assumptions (i) points and point adjacency for σ = α, ω either, there are no σ- projective lines in A, or there are σ-projective subspaces (ii) lines and line adjacency none (iii) projective lines and their adjacency there are projective lines in A (iv) points, affine lines and parallelism as in (i) (v) proper and improper points, the union of σ-projective lines, affine lines, and directions
- f
σ-affine planes there are σ-affine and (−σ)-affine planes in A (vi) points as in (v) and their adjacency w.r.t. lines from (v) there are σ-projective subspaces and σ-affine planes in A (vii) points and stars as in (i) with σ = α only (viii) points and tops as in (i) with σ = ω only (ix) stars and star adjacency there are α-projective lines in A, ω- projective subspaces on A∞ and α- projective subspaces on (A∞)∞ (x) tops and top adjacency as in (ix) with α instead of ω and vice versa Table 1: Possible systems of primitive notions in spine spaces. 3.3.8 Orthogonality and correlations of Grassmann spaces In all of geometries that have been considered so far, the orthogonality was realized by a bilinear form. In [A3] we introduce a synthetic relation for possibly wide class
- f geometries so that it resembles a correlation of a projective space. This relation,
called conjugacy, is a symmetric binary relation defined by 7 axioms (C0) – (C6) in the set of points of a partial linear space. The main criterion for formulation of these axioms was to obtain a correlation compatible with a Grassmann space, that is, the analytical representation of such a correlation should be the same as in case of the correlation of the underlying projective space. It was necessary to add yet more axiom (Ck
7) which requires that
a dimension function is defined for the family of subspaces of the underlying space. Actually, it is a typical condition for Grassmann spaces of k-dimensional subspaces. The main result of [A3] is a characterization of this axiomatically defined relation: Theorem 3.23 ([A3, Theorem 4.24]). Let dim(V ) < ∞. Consider a relation ⊥ ⊂ Subk(V ) × Subk(V ). The following conditions are equivalent: (i) Pk(V ) equipped with ⊥ satisfies the conditions (C0) – (Ck
7).
(ii) If dim(V ) = 2k, then ⊥=⊥ξ for some nondegenerate reflexive sesquilin- ear form ξ on V . If dim(V ) = 2k, then ⊥=⊥ξ, where ξ as previously, or ⊥= ϕ∗
k, where
SLIDE 19
Scientific report 19 ϕ: V − → V is an injective semilinear map such that ϕ∗
k is an involutory
collineation of Pk(V ). 3.3.9 Complements of intervals in projective Grassmannians The procedure of deleting a hyperplane, called affinization (cf. [71]), through an analogy of deleting a hyperplane from a projective space to get an affine space, is quite common in geometry. In result of affinization we can get an affine Grassmann space [19] and an affine polar space [18]. Instead of a hyperplane we can delete any subspace. That way from a projective space we get a slit space (cf. [45], [46]) mentioned earlier in 3.3.7. In a projective space the points of a complement of a fixed subspace of codimension 2 together with lines which do not intersect that subspace form so called partial geometry, which has been examined in [87]. In [89] it is proved that the complement of a line on a finite affine plane can be embedded into a projective plane of the same rank. This result has been generalized in [58], where the authors investigate configurations that emerge by deleting a pencil of lines from a finite affine plane. In a Grassmann space subspaces determined by intervals of the lattice into which this space is embedded play a very important role. We call these subspaces interval subspaces. In [A3, Theorem 2.6] an essential property of interval subspaces has been found. Namely, interval subspaces are those subspaces, and only those, which have the structure of a Grassmann space. In [A1] we delete from a projective Grassmannian a substructure which resembles an interval subspace. Assume that 0 < k < dim(V )−1 and consider a structure similar to a Grassmann space Gk(V ) := Subk(V ), Subk+1(V ), ⊂, which we call a Grassmannian. The points U1, U2 ∈ Subk(V ) are collinear in Gk(V ) if there is B ∈ Subk+1(V ) with U1, U2 ⊂ B. It is easy to verify that the Grassman- nian Gk(V ) is a Veblenian partial linear space, but it is not a Gamma space. If k = 1, then Gk(V ) is a projective space. Although, except the trivial case where k = 1, Grassmannian Gk(V ) is not embeddable into a Grassmann space Pk(V ), but these are mutually definable structures. As a line of Gk(V ) is not a set of points it is better to deal with closed sub- structures instead of subspaces. Let M = S, L, |, where S and L are any sets and | ⊆ S × L is an incidence relation, be a partial linear space. We say that M′ = S′, L′ is a closed substructure in M whenever: (A) if a line l has two of its incident points in M′, then l is a line of M′, (B) if a point a is incident with two lines of M′, then a is a point of M′. Let Z and W be fixed subspaces in V so that the interval [Z, W] = {U ∈ Sub(V ): Z ⊆ U ⊆ W} is nonempty. Consider a set D :=
- U ∈ Sub(V ): U /
∈ [Z, W]
- = {U ∈ Sub(V ): Z U or U W}
SLIDE 20
Scientific report 20
- f outer subspaces w.r.t. that fixed interval. To have D = ∅ we assume that Z =
0 or W = V . The Grassmannian of outer subspaces Gk(D) := Dk, Dk+1, ⊂. is a substructure of the Grassmannian Gk(V ). We single out the known cases and assume that 2 ≤ k ≤ n − 2. The elements of [Z, W] uniquely determine both Z and W, contrariwise to the elements of [Z, W]k. To avoid technical problems we introduce: Zmax =
- [Z, W]k ∪ [Z, W]k+1
- and
Wmin =
- [Z, W]k ∪ [Z, W]k+1
- .
We prove that the underlying projective space can be defined in terms of the com- plement of an interval in a projective Grassmannian. Theorem 3.24 ([A1, Theorem 3.2]). The underlying projective space P(V ) and the interval [Zmax, Wmin] can be recovered from the Grassmannian Gk(D). An automorphism of a Grassmannian Gk(D) is a pair of mappings (f, g) where f : Dk − → Dk, g: Dk+1 − → Dk+1, and f(U) ⊂ g(Y ) iff U ⊂ Y. Theorem 3.25 ([A1, Theorem 3.3]). Each automorphism F = (f, g) of the Grass- mannian Gk(D) can be extended to an automorphism F ′ = (f′, g′) of the projective Grassmannian Gk(V ) such that f′ preserves the interval [Z, W]k and g′ preserves the interval [Z, W]k+1. Hence f and g are both induced by a semilinear map on V which preserves [Zmax, Wmin].
4 The other scientific results
4.1 Papers not contained in the scientific achievement, published after PhD
[B1] K. Petelczyc, M. Żynel, The complement of a point subset in a projective space and a Grassmann space, J. Appl. Logic 13 (2015), no. 3, 169–187,
DOI: 10.1016/j.jal.2015.02.002.
[B2] K. Petelczyc, M. Prażmowska, K. Prażmowski, M. Żynel, A note on char- acterizations of affine and Hall triple systems, Discrete Math. 312 (2012), no. 15, 2394-2396,
DOI: 10.1016/j.disc.2012.03.037.
[B3] K. Prażmowski, M. Żynel, Segre subproduct, its geometry, automorphisms and examples, J. Geom. 92 (2009), no. 1-2, 117-142,
DOI: 10.1007/s00022-009-1951-9.
[B4] K. Prażmowski, M. Żynel, Extended parallelity in spine spaces and its geometry,
- J. Geom. 85 (2006), no. 1-2, 110-137,
DOI: 10.1007/s00022-005-0032-y.
SLIDE 21
Scientific report 21
[B5] M. Pankov, K. Prażmowski, M. Żynel, Transformations preserving adjacency and base subsets of spine spaces, Abh. Math. Sem. Univ. Hamburg 75 (2005), 21-50,
DOI: 10.1007/BF02942034.
[B6] K. Prażmowski, M. Żynel, General projections in spaces of pencils, Beitr. Algebra
- Geom. 46 (2005), no. 2, 587-608.
4.2 Summary of the other scientific results
Besides 10 papers comprising the thesis there are 22 papers including 5 preprints
- n arXiv.org and 11 papers published before PhD. Furthermore, two new papers
are currently under review. My research before and after PhD concentrates around problems related to the geometry of Grassmann spaces in its possibly wide meaning. This subject prevails since graduation. The very first results published in 2 papers are related to finite Grassmannians [94] and to the proof of classical Pascal-Brianchon theorem [95]. In the paper [73] Prażmowski introduced an interesting generalization of an affine Grassmann space called a spine space, which embraces a wide class known
- geometries. This gave rise to a new research trend devoted to the study of properties
- f spine spaces. The first result in this area is a jointly obtained characterization of
automorphisms of these spaces [77]. Verification of classical theorems of projective and affine geometry has been done in [75], while in [78] a specific case of spine spaces: a space of linear complements has been examined. Those spaces have been studied by Blunck and Havlicek [9]. A part of my PhD dissertation devoted to projections in spine spaces could be recognized as a contribution to the development of the theory of such geometries. After PhD, strictly in this research thread yet two more papers have been published. In [B6] the original results concerning projections in Grassmann spaces have been gathered. In [70] Pankov introduced a notion of a base subset in a Grassmann space. Such a set resembles a projective frame. Pankov proved that transformations of a Grassmann space which preserve base subsets are induced by collineations, or by correlations in a selfdual case, of the ambient projective spaces. He established a similar result for null systems [69]. In this context a question about the meaning of bases subsets in spine spaces arises: is it possible to repeat Pankov’s result in these more general and more complex settings of spine spaces. The effect of cooperation with Pankov is [B5], where we prove Chow’s theorem for adjacency and for base subsets in spine spaces Where complements of deleted substructures are considered, parallelism plays the key role in the ambient spaces recovery (cf. [A1]). As far as classically parallel lines are those which meet on the horizon, in [B4] we deal with extended parallelism which means that two lines are parallel if their points at infinity can be joined with a path on the horizon. In the context of spine spaces such a parallelism has many interesting properties. The outcome of the analysis of those properties is a characterization of dilatation group in [B4]. The results obtained there have been applied in [A6] to recover the horizon. Dealing with Segre products of Grassmann spaces we have noticed that a slight modification to the definition of the product results in a very wide class of geometric structures as subproducts of the Segre product of partial linear spaces. Specific cases
SLIDE 22
Scientific report 22
- f these subproducts are spine spaces, Schubert varieties, and spaces of linear com-
- plements. Besides finding general properties of Segre subproducts we characterized
automorphism groups of these subproducts in [B3]. The subject of the thesis is by no means exhausted. In the time of writing these lines a new paper has been published [B1], which is a continuation of the research upon the possibility of recovering the ambient projective spaces and the ambient Grassmann spaces from their reducts. In order to recover the ambient geometry a relation imitating affine parallelism is applied. This is just one of the possible approaches and not always can be utilised. In the case of a ruled quadric it is known that the ambient projective space can be recovered, but other methods are
- applied. The methods we propose are connected with bundle spaces studied in [44]
and [51]. In addition to the main research stream of primitive notions in reducts of Grass- mann spaces and in derived geometries also other aspects of these geometries have been widely considered. Interesting results have been obtained for the family of reg- ular and tangent subspaces in a symplectic copolar space. Affinization of symplectic polar space leads to a nice generalization of alternating forms and to affine semipo- lar spaces which are induced by those forms. These results need to be completed so, they have been published temporarily as preprints on arXiv.org. Working in the Department of Foundations of Geometry I am also taking an active part in joint endeavors of the whole team. The undertaken research sometimes falls outside the scope of my own interests. An example would be analysis of finite geometry and configuration, resulting in a paper about Steiner triple systems [B2].
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