All the books are available in electronic form in multiple places.
= available in
the university library
A brief presentation:
A BRIEF SYNOPSIS
(each class lasted approximately 1.5h unless specified otherwise)
Class 1 February 9, 2026
Organizational issues. Refresher: linear spaces as central objects in linear
algebra, their importance, idea of linearity. Affine space: definition,
examples, elementary properties. Affine maps. Affine maps with equal linear
parts.
(Slides, pp.3-15; Kostrikin-Manin, pp.195-200; Mac Lane-Birkhoff, pp.564-565;
Postnikov, pp.46-48).
Class 2 February 16, 2026
Composition of affine maps is an affine map. Isomorphism of affine spaces.
Affine coordinates, barycentric combination, barycentric coordinates.
Characterization of affine maps as maps preseving barycentric combinations.
(Slides, pp.15-21; Kostrikin-Manin, pp.197-203; Mac Lane-Birkhoff, pp.566-569; Postnikov, pp.49-50; Vinberg, pp.240-241).
Class 3 February 23, 2026
Characterization of affine maps as maps preseving barycentric combinations
(finish). Any affine space is isomorphic to the space of the form (V,V).
Affine group. Affine extension of a group.
(Slides, pp.21,23,31; Kostrikin-Manin, pp.198-200,203-204;
Mac Lane-Birkhoff, pp.568-572; Postnikov, p.49).
Class 4 March 2, 2026
Affine extension of a group (recap). Affine subspace, parallel subspaces.
Affine span.
(Slides, pp.25-27,31; Kostrikin-Manin, pp.204,207-209).
Class 5 March 9, 2026
Affine span (finish). Characterization of affine subspaces as sets of zeroes of
affine maps. Euclidean affine spaces.
(Slides, pp.27-28,30; Kostrikin-Manin, pp.209-210).
Class 6 March 16, 2026
Motions. The group of motions of an Euclidean affine space is isomorphic to the
affine extension of the orthognal group. Geometric interpretation of motions in
low-dimensional spaces. Computation of distance between two affine subspaces.
(Slides, pp.31-33; Kostrikin-Manin, pp.204-207,211-213;
Reventos Tarrida, pp.159-162; Mac Lane-Birkhoff, pp.586-589;
Postnikov, pp.144-146).
Class 7 March 23, 2026
Linear programming.
(Slides, pp.35-37; Kostrikin-Manin, pp.215-218).
Class 8 March 30, 2026
Projective spaces: definition, realizations, examples. The Fano plane.
Projective subspaces.
(Slides, pp.45-49; Kostrikin-Manin, pp.222-226,241;
Mac Lane-Birkhoff, pp.592-595; Onishchik-Sulanke, pp.2-3,12).
Class 9 April 13, 2026
Projective span. Formula for dimensions of intersection and projective span of
two projective subspace, its consequences.
(Slides, p.49; Kostrikin-Manin, pp.226-227; Mac Lane-Birkhoff, p.592).
Class 10 April 20, 2026
Projective duality. Projective group.
(Slides, pp.51-54; Kostrikin-Manin, pp.228-229,233-234;
Mac Lane-Birkhoff, pp.595-596; Onishchik-Sulanke, pp.65-71; Vinberg, p.285).
