Annual Foundation School – I (2016)

Department of Mathematics, IIT Guwahati

1^st - 28^th Dec, 2016

School Convener(s)

Speakers and Syllabus 

1. Cartan-Dieudonne Theorem: Every nxn orthogonal matrix is a product of atmost n reflections.
2. Discussed the relation of SO(2) and SO(3) with rotations of plane and space. In particular, Euler's Theorem about 3x3 orthogonal matrices was proved.
3. Characterization of the finite subgroups of SO(3) was discussed. Calculation of the groups of rotations symmetries of the Platonic Solids. A sketch of proof of the fact that the finite subgroups of SO(3) are either cyclic or dihedral or the groups of symmetries of the Platonic solids was given.

1. Group action, Sylow's theorem (along with proof).
2. Semi-direct product, Groups of order 12, Simplicity of alternating groups.
3. Universal property and construction of Free groups and Free abelian groups
4. Presentation of a group by generators and relators (only introduction).

1. Sesquilinear functions ; non-degeneracy, complete duality, Gram matrix ( finite dimensional case),Gram's criterion for complete duality.
2. Symmetric bilinear forms and quadratic forms.
3. Complex hermitian forms, Orthogonality.
4. Isotropic and anisotropic forms. Basic computations with standard dot product and the Lorentz form on finite dimensional real vector spaces and with complex hermitian forms.
5. Diagonalisation of symmetric bilinear and complex hermitian forms. SYLVESTER's Law of inertia.

Lecture-I
Lecture-II
Lecture-III
Lecture-IV

1. The story of Classification of Finite Simple Groups (CFSG), action of GL(n,k) and SL(n,k) on projective space.
2. Simplicity of PSL(n,k) when either n>2 or |k|>3.
3. Hamilton's quaternion algebra, the double cover map of SO(3) and SU(2).
4. Orthogonal and Symplectic groups, Structure of Linear group GL(n,k), Flags, Borel and parabolic subgroups, Bruhat decomposition.

1. Complex Differentiable Functions, Holomorphic Functions, Properties of Holomorphic Functions
2. Complex Integration, Goursat’s Theorem
3. Local existence of Primitives, Cauchy’s Integral Formula, Cauchy’s Inequality, Liouville’s Theorem
4. Fundamental Theorem of Algebra, Uniqueness Theorem (Theorem 4.8, Corollary 4.9), Morera’s Theorem, Maximum Modulus Principle, Schwarz Lemma

1. Zeroes and Singularities, Classification of Isolated Singularities, Holomorphic Functions defined in terms of integrals, Revisited Cauchy Integral Formula and Power Series:
2. Taylor Series, Laurent Series, Isolated Singularities & Laurent Series, Residues, Computing Residues
3. Casorati-Weierstrass Theorem (Statement and Proof), Argument Principle (Statement), Rouche’s Theorem (Statement and an application)
4. Homotopies, Simply Connected domains, Logarithm, Multi-valued functions and how they come about due to choice of paths, Definition of Size, Tan functions as Inverses of Multi-valued functions

Lecture-I
Lecture-II
Lecture-III
Lecture-IV

1. The argument principle and applications (Rouche's theorem, Open Mapping theorem and Maximum Modulus principles), The complex logarithm
2. Applications of the residue formula in evaluating real integrals
3. Sequences of holomorphic functions (discussed Riemann zeta function for Re(s)>1 as the prime example) and Holomorphic functions defined in terms of integrals (discussed the Gamma function for Re(s)>0 as the prime example).
4. Schwarz reflection principle and Runge's theorem.

1. History of Fourier analysis, Fourier transform of functions of moderate decrease
2. Decay properties of Fourier transform in terms of analytic properties of functions, Inversion formula, application to differential equations, Poisson summation formula
3. Holomorphic functions of several complex variables, Cauchy's integral formula and Power series expansion
4. Analytic continuation, Hartogs's extension phenomenon

1. Quotient topology and Identification Spaces - Mobius strip, Torus, Projective spaces, etc.
2. Topological Groups.
3. Path homotopy and the fundamental group.
4. Covering spaces and the fundamental group of the circle.
5. Some applications: Brouwer's Fixed Point Theorem, Borsuk-Ulam Theorem and Ham-sandwich Theorem.

1. Triangulations, Simplices, Simplicial Complexes and their polyhedrons, examples.
2. Topology of polyhedra of simplicial complexes, barycentric subdivision, examples, topological and simplicial consequences of barycentric subdivision.
3. Barycentric subdivision (continued), Simplicial approximation theorem statement and examples.
4. Introduction to surfaces (without boundary), closed surfaces, statement of the classification theorem of closed surfaces, orientation in triangulated surfaces.
5. Introduction to homology groups, preliminary examples.