Annual Foundation School - I (2021)

School Convener(s)

Name of the speaker

Position

Affiliation and address

Hemant Bhate

(Prof. Emeritus, S. P. Pune University)

Dilip Patil

formerly IISc, Bangalore

Nitin Nitsure

formerly TIFR, Mumbai

Name of the tutor

Position

Affiliation and address

Name of the Speaker

:

Detailed Syllabus

 Dilip Patil

Subject: Algebra
The total syllabus covered in the AFS -l for this course contains the topics as follows:
The group action and the correspondence between a group action and the action homomorphism with some examples of group actions, namely, the left regular action and
the conjugation operation. The orbit- stabilizer theorem and the class equation of a group action on a set. Some special kind of group actions, namely, transitive group action, simply transitive group action, free action and faithful action. The conjugation group action of a group on itself and the class equation in this case.

In later sessions, topics covered included the class equation for p-groups, the Cauchy's theorem. Every group action gives rise to a faithful action of another group on the same set. Further, an example of a group action of symmetric group of a set on the power set of the same set was discussed and the orbits and isotropy subgroups were discussed. The conjugation operation of a group on the power set of itself was explained. Further, the category of G-spaces, homogeneous G-spaces and affine G-spaces were introduced.Introduction of G-homomorphisms between G-spaces, classification of homogeneous G-spaces and affine G-spaces up to G-isomorphisms. The automorphism group of homogeneous G-spaces were discussed, affine spaces and affine maps between affine spaces. Introduction of affine group and relation between the affine group of an affine space and the automorphism group of the corresponding vector space. Also, the example of affine group of a vector space was discussed.

Introduction of semi direct product of groups. Permutation Groups and Alternating Groups,Signature and Alternating Groups, Simplicity of the Alternating Groups of more than or equal to 5 elements and Proof of Simplicity of the alternating groups for n greater than or equal to 5.

The statement of Sylow Theorems, Proof of the Sylow Theorems, proof of third Sylow theorem and Application of Sylow Theorem, elementary matrices and example of Projective linear groups, matrix Groups and projective spaces, Action of Projective Special linear Groups on the projective spaces, Pseudo reflections, Transvections and Dialations,Simplicity of Projective special linear groups over finite fields (PSr._n (F_q) where (n, F_q) is not (2, F_z) and (2, F_l)).

Sesquilinear functions and forms, examples and duality, non degeneracy and complete duality, trace form, Hermitian and Skew Hermitian forms, Polarization identity, example of the Lorentz form. Classification of Hermitian and Skew-Hermitian forms. The Decomposition theorem, Type and Signature of Hermitian forms, Sylvester's law of inertia,Hurwitz's criterion, positive definite and negative definite forms.

 Hemant Bhate

SUB]ECT NAME: COMPLEX ANALYSIS
This course in AFS -I started with holomorphic functions, Cauchy Riemann equations Power series, complex integration. Later on the topics covered in this course contained conformality, Mobius transformations, symnetry, elliptic, hlperbolic, loxodromic transformations, homogeneous coordinates, Goursats theorem, Cauchy integral formula,Luioville's theorem, fundamental theorem of algebra, Schwarz reflection principle,Residues, argument principle, evaluation of some integrals, argument principle, Rouches theorem, mean value property, homotopy of curves, biholomorphic maps between some regions, automorphisms of H, equicontinuity, normal families, Riemann mapping theorem,Fourier transform, gamma function, zeta function, elliptic functions and theta functions

 Nitin Nitsure

TOPOLOGY
The syllabus covered in the AFS -l for this course included the topics such as review of some set theory, empty set, ordered pairs, maps, construction of natural numbers as sets, binary products and coproducts, products and coproducts of families of sets, initial and terminal sets, universal properties, the notion of being unique up to unique isomorphism,Introduction to some basic notions of categories via examples of Sets, Groups, commutative rings and various categories of metric spaces, initial and terminal objects, products,coproducts, universal properties, uniqueness up to unique isomorphism.

Definition of metric spaces and those of categories Met-DP, Met-NE, Met-C whose objects and metric spaces and arrows are respectively distance preserving maps, non expanding maps and continuous maps, discussion of isomorphisms and the existence and construction of initial and terminal objects, products, coproducts in these, comments on how these dier among the three categories.

The notion of Cauchy sequence and completeness in Met-NE, and how it is not a good notion in Met-C, completeness and its universal property in Met-NE. Construction (hence existence) of completion via Cauchy sequences. Baire's theorem and examples of applications.

Fixed point theorem for contraction mappings and reference to its use for existence and uniqueness theorem for ODEs, definition of topological spaces and continuous maps. The categoryTop of topological spaces. Initial and final objects. Construction of binary products and coproducts in Top, verifications of their universal properties. The diagonal morphism for an object. Hausdorff topological spaces. Hausdorff and the diagonal (separatedness).Equivalent characterizations of hausdorffness. Basic properties of   Hausdorffness.Continuity of addition, multiplication, inverse, etc. Proving continuity of various maps by combination of the above together with diagonal maps. Examples of topological spaces.

Configuration of a clock, and its motion as a curve on a torus. Subspaces, embeddings.Examples. Diagonal, fibres. Quotient morphisms. Examples. Projections. The exponential map Universal property of embeddings. Universal property of quotients. Comparision of these properties across the categories Sets, Groups, Topological Spaces. Construction of quotient topology. Construction of a quotient by an equivalence relation in topological spaces, and the verification of its universal property. Embeddings and quotients in various categories (Continued). Inheritance of various properties under embeddings or quotients.Criterion for Hausdorffness of quotients in terms of closedness of the graph of the equivalence relation. Examples. The category of topological groups and continuous group homomorphisms. Examples. General linear group. Unit circle in complex numbers.Continuous homomorphisms from reals to reals, from reals to circle, from circle to circle, Automorphisms, Homogeneous metric spaces under distance preserving automorphisms,Homogeneous topological spaces, examples and non-examples, equivalent formulations of Hausdorffness for topological groups, discrete subgroups, Theorem: Any discrete subgroupof a Hausdorff group is closed, examples and non-examples.

Automorphisms of various objects in different categories. Automorphism groups. Examples.Equivalent formulations of Hausdorffness for topological spaces and topological groups.Connectedness. The closed interval and the real line are connected. Path connectedness.Path connectedness implies connectedness. Connectedness, path connectedness,connected components. Behavior under maps, products, coproducts. Totally disconnectedspaces. p-adic norm on rationals. Examples and counterexamples. Compactness definition.The closed interval [o, r] is compact. Heine-Borel theorem. Compactness for metric spaces.Total boundedness. Sequential compactness. Completeness. Equivalent formulations of compactness for metric spaces.

Equivalent formulations of compactness for metric spaces: proofs of some of the equivalences. Uniform continuity, Examples and nonexamples. Compactness of domain
implies uniform continuity for any continuous map. Families of functions. Equicontinuity, Uniform equicontinuity, Examples and nonexamples, Compactness for metric spaces -revision.

Function spaces. Sup norm, relationship with uniform convergence, epsilon by three argument, equicontinuity implies uniform equicontinuity on a compact domain, Arzela-
Ascoli theorem, Proof via total boundedness, Function spaces, Sup norm: motivation.Generalities on norm, inner product.

Arzela-Ascoli theorem: Motivation, idea behind the proof, detailed proof Application to holomorphic functions via Cauchy's integral formula for derivative, Arzela-Ascoli theorem,Equicontinuity via mean value theorem, application to holomorphic functions via Cauchy's integral formula for derivative, contraction mapping theorem (existence and uniqueness of a fixed point for a contraction mapping on a complete metric space), application to existence and uniqueness of a solution to an Ordinary Differential Equation.

Definition of a proper map, discussion of when is the map to a singleton point proper,definition of a proper maps, proper maps are closed, compactifications, proof that the projection frorn X to a point is proper if and only if X is compact in the category of Hausdorff spaces that admit compactifications, definition of locally compact Hausdorff spaces, construction of one-point compactifications, examples and non-examples, C[o,r] with sup norm is not locally compact, proper maps and their properties, examples, locally compact spaces, one-point compactifications and their examples, proper maps, locally compact spaces, and their propert1', M.S. Narasimhan's proof that the field of complex numbers is algebraically closed, action of a topological group on a topological space, proper action, Hausdorffness of quotients, any action of a cornpact group on a Hausdorff space is proper, action of the permutation group S_n on the n-dimensional complex space C^n, elementary symmetric function map is the quotient and that it is both proper and open. The elementary symmetric function map F from C^n to C^n. Precise formulation of the idea that 'a small variation in the coefficients of a complex monic polynomial will produce only a small variation in its roots'. Proof via properness of F and proof via complex analysis.