Annual Foundation School  I (2021)
Venue:  SAVITRIBAI PHULE PUNE UNIVERSITY. PUNE, Pune, Maharashtra 
Date:  29th, Nov 2021 to 8th, Jan 2022 
Name  Prof. S. A. Katre  Prof. Vinayak Joshi  Dr. Ganesh Kadu 
Mailing Address  Chair Professor, Lokmanya Tilak Chair, C/o Dept. of Mathematics, Savitribai Phule Pune University, Pune – 411007 
Head, Dept. of Mathematics, Savitribai Phule Pune University, Pune – 411007 
Dept. of Mathematics, Savitribai Phule Pune University, Pune – 411007 
List of the actual speakers:
Name of the speaker 
Position 
Affiliation and address 
Hemant Bhate 
(Prof. Emeritus, S. P. Pune University) 
S. P. Pune University) 
Dilip Patil 
formerly IISc, Bangalore 
IISc, Bangalore 
Nitin Nitsure 
formerly TIFR, Mumbai 
TIFR, Mumbai 
List of the actual tutors:
Name of the tutor 
Position 
Affiliation and address 
Syllabus:
Actual syllabus covered by each speaker:
Name of the Speaker 
: 
Detailed Syllabus 
Dilip Patil


Subject: Algebra In later sessions, topics covered included the class equation for pgroups, 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 Gspaces, homogeneous Gspaces and affine Gspaces were introduced.Introduction of Ghomomorphisms between Gspaces, classification of homogeneous Gspaces and affine Gspaces up to Gisomorphisms. The automorphism group of homogeneous Gspaces 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 SkewHermitian 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 

Nitin Nitsure 
TOPOLOGY Definition of metric spaces and those of categories MetDP, MetNE, MetC 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 MetNE, and how it is not a good notion in MetC, completeness and its universal property in MetNE. 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 nonexamples, equivalent formulations of Hausdorffness for topological groups, discrete subgroups, Theorem: Any discrete subgroupof a Hausdorff group is closed, examples and nonexamples. 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. padic norm on rationals. Examples and counterexamples. Compactness definition.The closed interval [o, r] is compact. HeineBorel 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 Function spaces. Sup norm, relationship with uniform convergence, epsilon by three argument, equicontinuity implies uniform equicontinuity on a compact domain, Arzela ArzelaAscoli theorem: Motivation, idea behind the proof, detailed proof Application to holomorphic functions via Cauchy's integral formula for derivative, ArzelaAscoli 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 onepoint compactifications, examples and nonexamples, C[o,r] with sup norm is not locally compact, proper maps and their properties, examples, locally compact spaces, onepoint 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 ndimensional 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. 
Time Table
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