IST - Transformations & Isometries (2020)
Speakers and Syllabus
|Name and affiliation of the speaker||Number of lectures||Detailed Syllabus|
Associate Professor, IISER Kolkata
|8||Isometries of Linear Spaces
This part of the school deals with distance and angle preserving maps in R 2 . We shall discuss isometries in R 2 and complex formula of plane isometries, orientations, finite group of plane isometries as well as isometries in R n We will talk about matrix groups, permutation groups and classification of isometries. We will discuss (orthogonal) projection maps and its properties. Finally we discuss finite group of isometries in R 3 and classification of conics under isometries.
References: Geometry - Ancient & Modern by J. R. Silvester
(Assistant Professor, IISER Kolkata)
This part of the school deals with hyperbolicity of planar domains. We start with classical Schwarz lemma, Schwarz-Pick lemma and computation of the automorphism group of the unit disk. We discuss Poincaré metric on D in details. Schwarz lemma will be revisited from the viewpoint of the Poincaré metric and its curvature. Some complex analysis results like Liouville’s Theorem and Montel’s Theorem will be presented from a geometric viewpoint. A proof of the big Picard Theorem will be discussed.
We shall also discuss some new invariant metrics, i.e., Caratheodory and Kobayashi metrics and their isometries on planar domains. We shall also analyze whether auto-morphisms fo the domain give rise to isometries. Hyperbolicity of the domain will be discussed in detail.
References: Complex Analysis: The Geometric Viewpoint by S. G. Krantz
(Associate Professor, IISER Kolkata)
|8||Isometries & Gaussian Curvature
This part of the school deals with isometries of surfaces embedded in R3.We start with the basics of curves and surfaces in R3,revisiting osculating circles (and planes),Serret-Frenet formula. We discuss curvature (Gaussian and mean) of surfaces in various forms. The first and second fundamental form will be discussed, followed by various implications of the sign of the Gaussian curvature. We shall talk about Gauss-Bonnet theorem and the local-global phenomenon it illustrates. We discuss isometries of a surface, geodesics and some relevant examples (for the three basic prototypes).
References: Differential Geometry of Curves & Surfaces by M. Do Carmo, Differential Geometry: A First Course inCurves and Surfaces by T. Shifrin.
- Sachchidanand Prasad (PhD student, IISER Kolkata) (tutor for topics B & C)
- Sanjoy Chatterjee (PhD student, IISER Kolkata) (tutor for topics A & C)
- Golam Mondal (PhD student, IISER Kolkata) (tutor for topics A & B)
Prerequisites: We will assume familiarity with the basic syllabus covered in BSc and MSc on linear algebra, complex analysis, calculus in several variables and basic topology.
|8/06/2020||Lecture A.1||Lecture B.1||L||Tutorial A.1||Tutorial B.1|
|9/06/2020||Lecture B.2||Lecture A.2||U||Tutorial B.2||Tutorial A.2|
|10/06/2020||Lecture C.1||Lecture A.3||N||Tutorial C.1||Tutorial A.3|
|11/06/2020||Lecture A.4||Lecture B.3||C||Tutorial A.4||Tutorial B.3|
|12/06/2020||Lecture C.2||Lecture A.5||H||Tutorial C.2||Tutorial A.5|
|13/06/2020||Lecture B.4||Lecture A.6||Tutorial B.4||Tutorial A.6|
|14/06/2020||Lecture A.7||Lecture C.3||L||Tutorial A.7||Tutorial C.3|
|16/06/2020||Lecture B.5||Lecture A.8||U||Tutorial B.5||Tutorial A.8|
|17/06/2020||Lecture C.4||Lecture B.6||N||Tutorial C.4||Tutorial B.6|
|18/06/2020||Lecture B.7||Lecture C.5||C||Tutorial B.7||Tutorial C.5|
|19/06/2020||Lecture C.6||Lecture C.7||H||Tutorial C.6||Tutorial C.7|
|20/06/2020||Lecture B.8||Lecture C.8||Tutorial B.8||Tutorial C.8|
The lecture titled Lecture X.n means nth lecture on topic X, where X is one of the topics listed in the syllabus. Tutorial X.n will be taken by the speaker for topic X long with the two tutors assigned for topic X.