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 3h unless specified otherwise)
Class 1 February 13, 2024 (2h 20min)
Organizational issues. Refresher: linear spaces as central objects in linear
algebra, their importance, idea of linearity. Affine space: definition, examples
and non-examples, elementary properties. Affine maps.
(Slides, pp.3-14; Kostrikin-Manin, pp.195-199; Mac Lane-Birkhoff, pp.564-565;
Postnikov, pp.46-48).
Class 2 February 20, 2024 (2h 20min)
Composition of affine maps is an affine map. Affine maps with equal linear
parts. Affine coordinates, barycentric combination, barycentric coordinates,
characterization of affine maps as maps preseving barycentric combinations.
(Slides, pp.15-20; Kostrikin-Manin, pp.197-203; Mac Lane-Birkhoff, pp.566-569;
Postnikov, pp.49-50; Vinberg, pp.240-241).
Class 3 February 27, 2024
Isomorphisms and automorphisms of affine spaces. Any affine space is isomorphic
to the space of the form (V,V). Refresher: general linear group, semidirect product. Affine
group.
(Slides, pp.21,23; Kostrikin-Manin, pp.198-199,203-204;
Mac Lane-Birkhoff, pp.568-572; Postnikov, pp.48-49).
Class 4 March 5, 2024
Discussion of homeworks 1-7. Affine group (continuation). Affine extension of a
group. Euclidean affine spaces.
(Slides, pp.30-31; Kostrikin-Manin, p.204; Mac Lane-Birkhoff, p.570 onwards).
Class 5 March 19, 2024
Discussion of homework 8. 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.
(Slides, pp.31-32; Kostrikin-Manin, p.204 onwards;
Mac Lane-Birkhoff, p.586 onwards; Postnikov, pp.144-146).
Class 6 March 26, 2024
Discussion of homeworks 9-11. Affine subspace, parallel subspaces.
Characterization of affine subspaces as sets of zeroes of affine maps.
(Slides, pp.25-26,28; Kostrikin-Manin, pp.207-208,210).
Class 7 April 2, 2024
Discussion of homeworks 12-13.
Counting the number of non-parallel lines in an affine space.
Affine span. Computation of distance between two affine subspaces.
(Slides, pp.27-28; Kostrikin-Manin, pp.209,211-214;
Reventos Tarrida, pp.159-162).
Class 8 April 9, 2024
Computation of distance between two affine subspaces (continuation), examples.
Linear programming.
(Slides, pp.35-37; Kostrikin-Manin, pp.212-213,215-218;
Reventos Tarrida, pp.159-162).
Class 9 April 16, 2024
Projective space: definitions, realizations, examples. The Fano plane. Counting
subspaces in the finite vector spaces. Projective subspaces, projective span.
(Slides, pp.45-49; Kostrikin-Manin, pp.222-226,241;
Mac Lane-Birkhoff, pp.592-595; Onishchik-Sulanke, pp.2-3,12).
Class 10 April 23, 2024
Counting in finite vector spaces (continuation). Projective span (continuation).
Formula for dimensions of intersection and projective span of two projective subspace, its consequences. Projective duality.
(Slides, pp.49-52; Kostrikin-Manin, pp.226,229; Onishchik-Sulanke, pp.65-71).
Class 11 May 7, 2024
Projective group.
(Slides, pp.54-55; Kostrikin-Manin, pp.233-236; Mac Lane-Birkhoff, pp.595-596;
Vinberg, p.285).
