1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Geometry

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Geometry - course description

Aim of the course

The course has to main goals: developing skills of ‘geometrizing’ mathematical problems, solving geometric problems by algebraic methods.

Prerequisites

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Linear algebra 2.

Scope

Lecture

Affine and Euclidean point spaces.
1. Affine combination of points; affine independency; examples of affine spaces; isomorphisms of affine spaces; a standard model of an affine space. Affine mappings. (4h)
2. Affine subspaces: hyperplane, line. Particular subsets of an affine space: line segments, convex sets, simplices. Sets of points in a general postion. Convex hull, polytope as a convex hull of a finite set. Caratheodory’s theorem. Radon’s theorem, Helly’s theorem. (6h)
3. Euclidean point spaces: distance, ball, classification of isometries. (2h)
4. Halfspaces: geometric interpretation of linear inequations. Paralleotopes, cubes. (2h)
5. Closed convex sets; the distance of a point from a convex set and a hyperplane. (2 godz.)
6. Volume of a set – volume of a parallelotope and a simplex; Brunn–Minkowski inequality; John ellipsoid. (8h)
Projective spaces
1. Definition, basic properties, projective mappings(2h)
Quadric surfaces
1. Classification of conics and general quadrics.(4h)

Class
1. Exercises in elementary geometry (4h)
2. Elements of spherical geometry, spherical polytopes (formulas to be derived as exercises). Euler’s formula for convex and spherical polytopes. Applications. (4h)
3. Compositions of isometries of the plane and the space. (3h)
4. Applications of Helly’s theorem. (2h)
5. Finding the distance from a point to a set. (2h)
6. Finding the Minkowski’s sum of convex figures and estimation of the volume of the sum – isoperimetric inequality. (2)
7. Minkowski’s theorem on lattice points (the proof as a series of exercises); applications (2h)
8. Discussion over essays. (2h)
9. Informal introduction to the Euler characteristics– counting the Euler characteristic of selected set (e.g. closed surfaces). (2h)
10. Properties of conics and quadric surfaces (4h)
11. Class test (2h)

Teaching methods

Traditional lecturing, solving problems under the supervision of the instructor, preparing presentations or essays (collaborative effort).

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

1. Preparation of the students and their active participation is assessed during each class by their instructor.
2. Class test with problems of diverse difficulty helping to assess whether a student achieved minimal outcomes.
3. Essay or presentation prepared by a team.
4. Written examination: It consists of three theoretical problems (sth need to be proved or explained) and four practical problems (sth need to be calculated, checked or found).
Final grade = 0.4 x class grade + 0,6

Recommended reading

1. M. Berger, Geometry I and II, Universitext, Springer.

Further reading

1. H. Hopf, Differential Geometry in the Large, LNM 1000, Springer, 1989.
2. J. Matoušek, Lectures on discrete geometry, Springer, 2002.
3. M. Aigner, G. M. Ziegler, Proofs from the BOOK, Springer 2004.

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 19-09-2019 13:45)