TEW Complex Analysis (2013)

Proposed Scientific Plan

In the proposed TEW, our theme would be Complex Analysis (from a geometric viewpoint). We propose to have three mini-courses in this program consisting of around 4 lectures, each of 1.5 hrs duration. The rest of the time will be devoted to group discussions and tutorial. The aim of this course would be to depict a flow of ideas along with absolute mathematical precision wherever possible. We hope the courses will provide a geometric realization of several topics of complex analysis to the audience.
Later in the summer or winter of 2014, we plan to have a follow up of this workshop with advanced courses followed by a mini-conference consisting of survey talks that will connect the subject to cutting edge research.

Syllabus of the Workshop.

The syllabus of the workshop has been designed in a way so that the major part of the workshop will be devoted to foundational aspects of complex analysis. A salient feature of the proposed workshop is that besides foundational aspects, the participants will also be exposed to some advanced topics like Mobius geometry or Introductory Complex Dynamics. These topics are mostly missing in the syllabus of colleges/universities though they are foundational to learn many modern aspects of Mathematics.  In addition to the mini-courses, we shall also have two survey talks on the last day of the program. The survey talks will be given by known experts in the subject from the region. In the following we summarize the syllabus of the courses, along with the names of the speakers for the respective courses. We have also mentioned the number of contact hours for each course.

1. Course Title: Arithmetic of Complex Numbers.       (3 hrs)
   Speaker:  Brundaban Sahu (NISER Bhubaneswar)
   Course outline:  (i) Arithmetic of complex numbers, Exponential functions, Holomorphic functions, Harmonic Functions, Relation between holomorphic and harmonic functions. (ii) The Exponential and complex trigonometric functions, Polynomials, Logarithmic function, Power series.
2. Course Title: Geometric Complex Analysis  (7 hrs)
   Speaker:  Kingshook Biswas (RKMVU Belur)
   Course Outline: Complex Integration: Cauchy’s theorem, Cauchy’s formula, General contour integration; Application of Cauchy’s formula: zeros of a holomorphic function. Liouville’s theorem; Maximum modulus principle. Isolated Singularities, Riemann’s removable singularity theorem, Residue theorem, Poles and essential singularity, Meromorphic function, Analytic continuation. 
3. Course Title: Mobius Geometry   (4.5 hrs)
   Speaker: Krishnendu Gongopadhyay (IISER Mohali)
   Course outline: Ideas of conformal transformation, Euclidean similarity geometry using complex numbers, introduction to hyperbolic plane, Mobius groups (groups of Mobius transformations of the sphere), action of Mobius transformations on hyperbolic plane, Algebraic classification of Mobius transformations; Finite groups of Mobius transformations. 
4. Course Title:  Dynamics on the Riemann Sphere   (3 hrs)
   Speaker:  Tarakanta Nayak (IIT Bhubaneswar)
   Course outline: Rational functions on the sphere, The Fatou and the Julia set of rational functions, Classification of Fatou components, Properties of the Julia sets, Critical values and their dynamical importance, Iteration of Transcendental map, Baker domains, Wandering domains, Singular values and their importance, Some open questions.
   On the last day, there will be two survey talks:
   (i) Title: Survey on Nevanllina Theory.
   Speaker: Indrajit Lahiri (Kalyani University, WB)
   (ii) Title: Characterization of maximal ideals in the ring of entire functions.
   Speaker: K. C. Chattopadhyay (Burdwan University, WB)
   

Participants

Participants who would be provided local hospitality would be strictly Under-Graduate College teachers. Preferences will be given to UG-teachers from the region. However, if there are slots, we may include a few more from neighboring places of West Bengal. We are expecting around 35 College Teachers to participate. If there are more than 35 participants, we may ask each participants for some registration fees, if any, that would be required.  Some local doctoral and post doctoral students may also be included in the workshop. However, it will be made clear to them that the RMS-TEW is primarily for UG-college teachers.

Objectives

The objective is same as that of the proposed RMS-TEW. That is (i) to promote student-initiative activities at the UG-PG levels. (ii) To increase the knowledge of the fundamentals of the college teachers, unity of mathematics, and problem-solving abilities. Emphasis will be given on basic examples.

The program will be largely interactive. The process of learning is described as "1/4th from teacher's teaching, 1/4th from one's own efforts, 1/4th from discussion with the like-minded people, and 1/4th from sleeping over the material for a long time". The program will follow the second and third steps in this process.