1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Geometry

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Geometry - opis przedmiotu

Cel przedmiotu

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

Wymagania wstępne

Zapisz zmiany

Linear algebra 2.

Zakres tematyczny

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. Orthogonal transformations and their matrices relative to the orthonormal basis. Decomposition of space into minimal invariant subspaces: rotations, reflections. Canonical form of the orthogonal mapping matrix. Space orientation.Euclidean point spaces: distance, ball, classification of isometries. (8h)
3. 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)
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. (4h)

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

Metody kształcenia

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

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

The condition for taking the exam is a positive grade from the exercises. The condition for passing the course is a positive grade in the exam. The grade for the course consists of the grade for the exercises (50%) and the grade for the exam (50%). The instructor may increase this grade in recognition of the student's special merits.

Literatura podstawowa

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

Literatura uzupełniająca

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.

Uwagi

The description has been prepared originally by dr hab. Krzysztof Przesławski, prof. UZ.

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 21-02-2024 16:06)