Advanced Instructional School on Complex Analysis (AIS-Complex) is being organised in Pune in June 2008 on behalf of NBHM.

Members of the Local Organising Committee
Bhaskaracharya Pratishthana: C. S. Inamdar (Custodian), R. R. Simha, Anandateertha Mangasuli

University of Pune: B .N. Waphare (HOD, Maths.), S. A. Katre, H. Bhate

Speakers

• A. R. Shastri: See the detailed syllabus
• Dinesh Thakur: See the detailed syllabus
• A. Mangasuli:
  1. Covering space theory (3 lectures)
  2. De Rham Cohomology and Hodge Theory (An overview) (1 lecture)
• Ravi Raghunathan: The convexity principle and applications to functional analysis and number theory.
• R. R. Simha: Riemann Surfaces
• S. A. Katre: Doubly periodic functions
• Kaushal Verma: (6 lectures)
  1. Statement of Picard type theorems on the plane.
  2. Definition of Fatou-Bieberbach domains, their properties and the theory of normal forms for local holomorphic automorphisms with fixed points.
  3. Fatou and Bieberbach's construction.
  4. Theorems of Rosay-Rudin for constructing Fatou-Bieberbach type domains.
  5. Examples and connections with complex dynamics in higher dimensions.
• Ravi Kulkarni:
  1. Geometry of Complex Numbers
  2. Branched Covering Space Theory
  3. The modular Group
• S. R. Ghorpade: Hardy-Ramanujan-Rademacher formula for partitions
• Ajit Iqbal Singh: Weierstrass' Product Theorem and Mittag-Leffler's Theorem from Complex-valued to the Banach Algebra set-up.
  Please refer to the link: The Mittag-Leffler Theorem: The Origin, Evolution, and Reception ...
• Sanjay Pant: Complex dynamics.

Detailed Syllabus (to be updated)

 1  Contour Integration  1
1.1  Path Connectivity . . . . . . . . . . . . . . . . . . . . . . 1
1.2   Definition and Basic Properties of Contour Integration . . . . . . . 6
1.3  Existence of Primitives . . . . . . . . . . . . . . . . . . . . . . . 17
1.4  Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . 21
1.5  * Cauchy’s Theorem via Green’s Theorem . . . . . . . . . . . . . . . 27
1.6   Cauchy’s Integral Formulae . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7  Analyticity of Complex Differentiable Functions . . . . . . . . . . . . . 33
1.8   A Global Implication: Liouville . . . . . . . . . . . . . . . . . . . . 37
1.9   Mean Value and Maximum Modulus . . . . . . . . . . . . . . . . . . 39
1.10 Miscellaneous Exercises. . . . . . . . . . . . . . . . . . . . . 41

 2   General Form of Cauchy’s Theorem   45
2.1   Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2   Homotopy and Simple Connectivity . . . . . . . . . . . . . . . . . 51
2.3  Homology Form of Cauchy’s Theorem . . . . . . . . . . . . . . . . 55
2.4  Miscellaneous Exercises . . . . . . . . . . . . . . . 60

 3 Convergence in Function Theory  63
3.1   Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2   The Exponential and Trigonometric Functions . . . . . . . . . . . . . 76
3.3   Sequences of Holomorphic Functions . . . . . . . . . . . . . . . . . 85
3.4 Convergence Theory for Meromorphic Functions . . . . . . . . . . . . . 89
3.5 Partial Fraction Development of π cot πz. . . . . . . . . . . . . . . . . . 94
3.6 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.7 Runge’s Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . 104

 4   Normal Families and Conformal Mappings 115
4.1  Metric on Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2   Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3   Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4   Families of Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . 121
4.5   Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6   Miscellaneous Exercises . . . . . . . . . . . . . . . . . . . . . . 128

 5 Harmonic Functions 129
5.1   Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2   Application to Potential Theory . . . . . . . . . . . . . . . . . . . . . . . 138
5.3   Mean Value Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4   Harnack’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.5   Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.6   Perron’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7   Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.8   Multi-connected Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.9   Miscellaneous Exercises . . . . . . . . . . . . . . . . . . . . . . 169

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