MTH204
MTH204 Basic Structures of Mathematics
(January-April 2019)
Instructor : Dr Anupam Singh
Audience : 2nd year BS-MS students at IISER Pune
Tutors : TBA
Schedule : Lecture Thursday 12-1.
Tutorials Friday at 10:50 B1 (TA Pratima in 105) and B4 (TA Uday in 106)
at 12 B2 (TA Pratima in 105) and B3 (TA Saikat in 106)
Evaluation : Test I - 20%, mid-sem - 30%, Test II - 20%, end-sem - 30%
Prerequisite : Enthusiasm towards science
Goal of the course : To learn what mathematics at higher level looks like and how it interacts with other branches of science.
Proposed Content : Groups: Groups and symmetries of geometric objects, linear groups, permutation groups, rigid transformations, subgroup and Lagrange’s theorem, SU(2), SL(2, R), regular objects in 2d and 3d, platonic solids and groups of symmetry.
Metric spaces: Definition and basic properties, examples, closed sets, open sets, continuity, sequences in a metric space, compactness and connectedness.
Hilbert spaces: Banach space, examples, shapes of unit balls under various norms, operators on Banach spaces, Hilbert spaces, examples, orthonormal basis, inequalities (Bessel, Cauchy-Schwartz), operators and spectral theorem.
Computational Mathematics: Discrete structures, algorithms, computing, examples and applications.
Text books and reference material :
Algebra : Artin
Symmetry: Tapp
Introduction to topology and modern analysis: Simmons
Discrete Mathematics: Lovasz, Pelikan, Vesztergombi
Weekly topics covered :
3 January 2019 - Definition and examples of Group, Euclidean metric on R^n, Isometry.
4 January 2019 - Tutorial
10 January 2019 - Group of Symmetries, Symmetric bilinear form.
11 January 2010 - Tutorial
17 January 2019 - Orthogonal group, Classification theorem of isometries of R^n.
18 January 2019 - Tutorial
24 January 2019 - SO(3), Platonic solids and their symmetry
25 January 2019 - Tutorial
Some interesting videos on QUATERNIONS
https://www.youtube.com/watch?v=d4EgbgTm0Bg
https://www.youtube.com/watch?v=zjMuIxRvygQ
31 January 2019 - Quaternions and 3D geometry
1 February 2019 - Test I at 10:50 am - 1 pm
For understanding quaternions see the book NUMBERS Chapter 7.
7 February 2019 - Metric space, open sets, limit points, closed sets.
8 February 2019 - Tutorial
14 February 2019 - properties of open and closed sets, more examples
15 February 2019 - Tutorial
19 February 2019 Mid Sem exam at 10AM
28 February 2019 - Continuous maps, homeomorphism, examples: Cantor set, sphere, projective plane
(Reference: Massey, Algebraic Topology Chapter 1)
1 March 2019 - Tutorial
Some interesting videos
https://www.youtube.com/watch?v=AmgkSdhK4K8
https://www.youtube.com/watch?v=yuVqxCSsE7c
https://www.youtube.com/watch?v=0z1fIsUNhO4&t=17s
7 March 2019 - 2-dimensional surfaces (torus, sphere, Klein bottle, projective plane), n-sphere, n-projective space
8 March 2019 - Tutorial
https://www.youtube.com/watch?v=XlQOipIVFPk
14 March 2019 - Normed linear space, Banach Space, Inner product space, Hilbert space.
15 March 2019 - Tutorial
21 March 2019 - Holiday
22 March 2019 - Tutorial
28 March 2019 - Discrete maths, Graphs, examples, Cayley graph
29 March 2019 - tutorial
https://www.youtube.com/watch?v=VvCytJvd4H0&t=329s
4 April 2019 - Konigsberg bridge problem, Euler's formula (Chapter 7 and 12 from Discrete Mathematics: Lovasz, Pelikan, Vesztergombi).
5 April 2019 - TEST - II at 10:50 AM up to 1 PM
11 April 2019 - Computational Group Theory, SAGEmath
12 April 2019 - Tutorial / re-TEST at 11AM
22 April 2019 END-SEM EXAM at 10AM