This is the webpage of Math466 For 2024-2 Semester.
Here is a tentative Syllabus.
Week 1)
17.02.2025 (2 Hours) Made an introduction to isometry groups. Introduced the finite dihedral groups and their basic properties.
19.02.2025 (1 Hour) Introduced the infinite dihedral group and its basic properties.
Week 2)
24.02.2025 (2 Hours) Investigated the group Isom(R) and described its elements. Recalled the direct products of groups and automorphism groups.
26.02.2025 (1 Hour) Introduced semi-direct products.
Week 3)
3.3.2025 (2 Hours) Realized the previous examples of groups as semi-direct products. Started investigating Isom(R^n). Introduced orthogonal matrices and their basic properties.
5.3.2025 (1 Hour) Proved that every isometry of R^n that fixes the origin is an orthogonal linear transformation. Deduced that every isometry of R^n is the composition of a translation and an orthogonal matrix. Also deduced that the isometry group of R^n is a semi-direct product of R^n with O(n).
Week 4)
10.03.2025 (2 Hour) Introduced orientations and orientation preserving/reversing isometries of R^n. Started investigating the Eucliedan group (=Isom(R^2) ). Observed that orthgonal linear transformations of R^2 are rotations and refections.
12.03.2025 (1 Hour) Classified isometries of R^2: Proved that an isomtery of R^2 is either a translation, a rotation, a reflection or a glide reflection. Classified these according to being O.R. or O.P.
Week 5)
17.03.2025 (2 Hours) Investigated the possible products of elements of the Euclidean group. Proved that every isometry of R^2 is the product of at most 3 reflections.
19.03.2025 (1 Hour) Proved that the a finite subgroup of the Euclidean group is either finite cyclic or finite dihedral.
(No exercises for Week 5 !!)
Week 6)
24.03.2025 -28.03.2024 Solved exercises
Week 7)
7.04.2025 (2 Hours) Introduced discerete subgroups of E. Gave examples of discrete and non-discrete subgroups of E.
9.04.2025 (1 Hour) Proved that the translation groups of discrete subgroup of E is
either trivial, isomorphic to Z or isomorphic to ZxZ.
Week 8)
14.04.2025 Exam 1
16.02.2025 (1 Hour) Proved that a discrete subgroup of E is finite if and only if it has a fixed point. Stated the Crystallographic Restriction Theorem.
Week 9)
21.04.2025 (2 Hours) Proved that there are seven different frieze patterns
23.04.2025 No Class Due to April 23.
Week 10)
28.04.2025 (2 Hours) Recalled facts about frieze patterns. Introduced wall paper groups (i.e. discrete subgroups of E whose translation subgroup has dimension 2) and classified possible point groups of these.
Started recalling basic fact about group actions.
30.04.2025 (1 Hour) Finished required topics related to group actions. In particular, proved the Orbit-Stabilizer theorem and Burnside’s theorem.
Week 11)
5.05.2025 (2 Hours) Started investigating Isom(R^3). Described 3d rotations and other elements of O(3). Introduced Platonic Solids and proved that there are 5 such solids. Proved that the symmetry group of the tetrahedron is isomorphic to S_4.
7.05.2025 (1 Hour) Investigated rotational symmetries of the tetrahedron. Proved the fact that the symmetry group of the cube is isomorphic to S_4 x Z_2.
Week 12)
Started proving the classification theorem about subgroups of SO(3).
Week 13)
19.05.2025 No class due 19th of May
21.05.2025 (1 Hour) Finished classification of finite subgroups of SO(3).
Week 14)
25.05.2025 (Extra Lecture, 1.5 Hours)
Introduced free groups and group presentations.
Recording of extra online lecture on May 25:
https://drive.google.com/file/d/1Je77q3gMQcQpBDH3MJWiWdMYjCDPmCRw/view
26.06.2025 (2 Hours) Proved certain properties of free groups. Gave examples of group presentations. Introduced Cayley graphs and word metrics on groups.
28.05.2025 (1 Hour) Talked about properties of Cayley graphs and word metrics on groups.