Annual Foundation School -III (2017) - Shillong

 A. Syllabaus: AFS III

• Algebra III : Field Theory and Galois Theory.[Artin] Chapters 13, 14.
• Analysis III : Functional Analysis. [Simmons] Chapters 8,9,10,11.
• Topology III: Algebraic Topology

1. Fundamental Group and Covering Spaces [Munkres] Chapters 9 (Fundamental group) and 13 (Covering spaces). *-ed sections in Ch. 9 may be omitted.
2. Homology: [Spanier] Chapter 3, 4 / or [Greenberg and Harper] Part II.

B Speakers:
1. Lectures and Tutorials on field theory and Galois theory will be shared by

• AMB – Ardeline Mary Buhphang, NEHU, Shillong.
• PS – Parvathi Shastri, Mumbai,
• SL - Shanta Laishram , ISI, Delhi.

2. Lectures and Tutorials on functional analysis will be shared by

• SM – Saikat Mukherjee, NIT Meghalaya.
• ARS – A.R. Shastri, IIT Bombay.
• AC - Anjan Chakrabarty, IIT Guwahati.

3. Lectures and Tutorials on algebraic topology will be shared by

• ATS - Angom Tiken Singh, NEHU, Shillong.
• BS – B. Subhash, IISER, Tirupati.
• HKM - Himadri Kumar Mukerjee, NEHU, Shillong.

 C.Time-Table
Week - One

• FTn : nth lecture in field theory/Galois theory
• TFTn: nth tutorial in field theory/Galois theory
• FAn : nth lecture in functional analysis
• TFAn: nth tutorial in functional analysis
• ATn : nth lecture in algebraic topology
• TATn: nth tutorial in algebraic topology

Basic References:
[Armstrong] M. A. Armstrong, Basic Topology, Springer International Edition.
[Artin] Michael Artin, Algebra, Pearson, 1991, Prentice-Hall of India, New-Delhi, 2003.
[Greenberg and Harper] M. J. Greenberg and J. R. Harper, Algebraic Topology: a first course,
Benjamin/Cummings Pub. Co. 1981.
[Munkres] J. R. Munkres, Topology, II-ed. Prentice Hall.
[Simmons] G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1983.

It was expected that the participants of the AFS-III have undergone the material covered in the AFS-I and AFS- II. Many participants had the requisite background and they have responded quite satisfactorily to the various lectures. A few others did not have prior background, these students faced some problems, extra effort was made for them. Programme was aimed at covering the syllabus of the AFS-III, as best as possible through lectures and tutorial seesins to prpare the students to embark upon higher studies and research in these topics.

 Report of the lectures by individual resource persons:
Saikat Mukherjee: Functional Analysis (FA-1 to FA-5)
As planned along with other speakers of Analysis, the selected candidates were informed to be familiar with Chapter 8 and some basic material from chapters 1-7 from Simmons’ book, before coming to the event. I have given 5 lectures and conducted 5 tutorial sessions during June 12-June 19. Out of five lectures, four lectures were delivered on Chapter 9 on Banach spaces covering basic definitions and examples; bounded linear transformations; dual spaces & reflexive spaces; weak & weak* convergence; Hahn Banach, Open Mapping Theorems & Uniform Boundedness Principle. Remaining one lecture was on Chapter 10 on basic definitions on Hilbert spaces, Complete orthonormal sets, Orthogonal complements.
A set of exercises on these topics were given to the candidates and discussed during tutorial sessions with active participation from the candidates. The last lecture and tutorial session on June 19 was attended by Prof. Anant Shastri. Overall the program was very satisfactory.
Feedback: One hour tutorial session is very short to discuss a good amount of problems. It may be extended by reducing lunch break, other breaks, etc.

Anant R. Shastri: Functional Analysis (FA-6 to FA-11)
As planned, we had informed the selected candidates to be familiar with Chapter 8 and some basic material from chapters 1- 7, before coming to the event.
The first speaker was able to cover the contents on Chapter 9 on Banach spaces modulo some proofs and a little bit of Hilbert space theory. I arrived at the venue two days earlier than the schedule for my
talk and was able to attend one lecture and tutorial hour by Prof. Saikat Mukherjee and got a clear idea of what is happening.

Right in the beginning, I gave the participants a copy of the set of exercises for discussion in the tutorial hour and told them to read Chapter 11 which is on finite dimensional spectral theory in a couple of days so that we also discuss them during the tutorial hour with initiative coming from the participants.I started with recalling Hilbert space theory and was able to cover entire of chapter 10 in 4 lectures. Meanwhile, we were able to cover most part of Chapter 11 through discussion during tutorial hours, which were extended to two hours instead of one. In the last two lectures I covered contents of chapter 12 on Generalities of Banach algebras, thereby laying proper foundation for Gelfand-Neumark Theorem. My last lecture was attended by Prof. Anjan Chakrabarti
who was supposed to give the last five lectures in analysis. I stayed back to attend two of his lectures and assist in tutorials. I must say that I am very happy to have this team of experts.

All in all, the Analysis part of the meeting was extremely satisfying. Even though some students were poor in basic algebra, we were able to cover the basic spirit of Simmons book, which is in the last two chapters. So, it is recommended that the syllabus should include these two chapters also. I also acted as tutor in several of algebra and Topology sessions and attended all the lectures during my stay at the venue i.e., from 18th June to 1st July.

Anjan Kumar Chakrabarty: Functional Analysis (FA-12 to FA-16)
 
I reached NEHU before the noon of 29th June and I could attend one hour of Prof. A. R. Shastri's last lecture on Functional Analysis. I was also present in his last tutorial session on Functional Analysis. In addition to these, I had conversation with
Prof. Shastri on 29th June regarding my lectures and the preparedness of the students. All these helped me to plan my job properly.

Before going to NEHU, I had prepared a problem set based on Chapters 12 and 13 (Banach alge-bras and Gelfand theory) for tutorial sessions. It was distributed to the participants on 29th June.
I started my first lecture on 30th June in which I gave a complete outline of the topics to be covered (and also certain things that are outside the scope of the syllabus for their relevance) including some review of Banach algebras. From the next lecture onwards, I discussed the Gelfand theory for commutative Banach algebras (Chapter 13 of the prescribed book). Going by the background of many participants (and also due to their request), I had to also do in some details weak and weak* topologies, including Banach-Alaoglu theorem, which is required later. I took total five lectures and I could complete the proof of Gelfand-Naimark theorem for commutative Banach algebras.

I took total nine tutorials and most of the problems given for tutorial sessions as well as some additional problems arising out of the lectures could be thoroughly discussed. Overall I am highly satisfied by the active participation of the participants in my lectures and tutorials.
I also attended all the lectures and tutorials in the other two subjects (algebra and topology) during my stay there.
 

A.M.Buhphang : Algebra (Field Theory, Galois theory) (FT-1 to FT-5)
I was given to teach the basic concepts of field theory for the Algebra III course. I started the lectures on 12.06.2017 with some basic facts about rings and fields, such as every field contains a prime field. Then I went on to discuss about algebraic
elements, minimal polynomials and their respective degrees. We stated and prove the tower law and use it to solve some of  the problems given in Artin’s book. We then discussed about multiplicities of roots and using derivatives of polynomials we studied that all the roots of an irreducible polynomial over a field have the same multiplicity. After that we discussed about finite fields and the one-one correspondence between finite fields of characteristic p (p a prime) and the set natural numbers which are powers of p. I ended the lectures with some constructions using ruler and compass on 17.06.2017. Since the prescribed book was Algebra by M. Artin, during the tutorials, apart from clearing the doubts encountered during the morning classes, we tried to solve many of the problems from sections 1 to 6 of Chapter 13 from this book. The participants were attending the classes and the tutorials sincerely and enthusiastically. They made many contributions during the tutorials and I found them very lively and useful.

 Parvati Shastri : Algebra (Galois Theory) (FT-6 to FT-11)

Fundamental Theorem of Algebra (Argand's proof), Splitting fields and algebraic closures, example of algebraic closure of the rational numbers and construction of algebraic closure of finite fields, construction of algebraic closure for arbitrary
fields, uniqueness of splitting field of a polynomial, idea of proof of uniqueness of algebraic closure, number of isomorphisms of a finite extension into the algebraic closure, separable extensions, (simplicity of separable extensions was not done), example of inseparable extension, normal extensions and its equivalence with splitting fields, Galois group and Galois extensions, with a lot of basic examples of quadratic, cubic and quartic extensions, Cyclotomic fields, Galois group of cyclotomic fields and in particular the pth cyclotomic fields, Galois group of finite field is cyclic(by way of exercise), Fundamental Theorem of Galois theory (FTGT), Artin's lemma, proof of FTGT, illustrations, Galois group as subgroup of Sn, discriminants and conditions in terms of discriminant for the Galois group to be subgroup
of An, Galois closure, applications of Galois theory to constructible numbers and constructible regular n-gons. Could not discuss discriminants in detail and also could not do symmetric functions and Galois group of quartic polynomials. Same was reported to the next speaker.
Tutorial sheets were distributed and some exercises were solved completely some were partially done with enough hints. Students participation was not bad. There were at least six students from North East, three of them college teachers,
intending to do PH. D. Over all, most of them were serious and made efforts to learn, but usually in the class room only. Very few were studying after classes. We made a change in the schedule of tutorials. Gave longer break after lunch and conducted tutorial for longer time. My opinion is that this schedule works out well. The students have some time to look at what they were taught that day and also tutorials could be conducted for longer time, which is ideal. In fact tutorials were conducted from 4 to 8 pm with a break of half an hour around 6 pm. Both the tutors could get enough time for discussion.

Suggestion: If possible I suggest that the tutorials could be scheduled between 4:00 to 7:30 pm with a break of half an hour, from 5:30 to 6:00 pm. In all the future AFS. It gives ample time for students interaction and enough time for both the teachers of the day.
A remark about the local organiser: I am very much pleased and admire Dr. Tiken, who worked very enthusiastically, participated in all the lectures of all the subjects, and helped in all the tutorials. I admire his devotion and efficiency, in academics as well as other administration, particularly the way he took care of the participants. This is the first time I am seeing the local organiser being so active and be present in all the lectures. Note on quality of food: Quality of food was poor, substandard and not hygienic. I suggest that this should be brought to the notice of the administrators of the University, in future, while following the rule of selecting the least quotation.
 

Shanta Laishram: Algebra (Galois Theory) (FT-12 to FT-16)
I continued the course on Fields and Galois Theory after the lectures of Buhphang and Shastri. In the course of my _ve
lectures, I covered a number of topics starting with

• Discriminants of polynomials and ways to compute them and proved that Galois group of the splitting field of a polynomial of degree n is a subgroup of either An or Sn according as Discriminant of the polynomials is a square or not.
• Computed the Galois groups of Cubics and Quartics by using Discriminants; the notion of Cubic Resolvents.
• Proof of Primitive Element Theorem.
• Cyclic and Kummer Extensions and Applications.

In the five tutorial sessions, I took up a number of related problems and in my last tutorial session, I introduced Newton Polygons of polynomials and showed how they are used to proving Irreducibility of polynomials and also for computing Galois groups.

 Angom Tiken Singh: Algebraic Topology (AT-1 to AT-5)
During the first and second week in AFS-III held in NEHU from 12th June to 8th July 2018, I delivered five lectures and conducted tutorials on the following topics:
Homotopy, Contractible spaces, Fundamental groups, Covering spaces, Path lifting and Homotopy lifting property of covering map. Retraction and Brouwer fixed-point theorem for the disk, Deformation retracts and Homotopy types,n Fundamental theorem of algebra, Fundamental groups of S .All participants were serious and they could understand the above topics. They were punctual, very interactive and asked many relevant questions during lectures and tutorials. I also attended almost every lecture and every tutorial of other
speakers which were very enjoyable.

B. Subhash: Algebraic Topology (AT-6 to AT-10)

• Lecture 1 : - Recalled covering spaces, discussed some examples, the homotopy lifting property. Started exploring the relations of the fundamental groups with the lifting problem : gave a criteria when loops can be lifted to loops. De_ned normal coverings.
• Lecture 2 :- Solved the lifting problem for covering projections.De_ned equivalence of coverings and gave a criteria for two coverings to be equivalent. Introduced the deck transformation groups, started the discussion to explicitly describe the deck transformation group for normal coverings.
• Lecture 3 :- Completed the description of deck transformation groups for normal covering. Defined universal covering, showed simply connected covers are universal, stated the criteria for existence of simply connected covers, showed that universal cover is unique upto equivalence of coverings. Proved that subgroups of π1 (X) upto conjugation are in bijective correspondence with equivalence classes of connected coverings for a connected, locally path connected, semi locally simply connected spaces.
• Lecture 4 :- Introduced exact sequences, Chain complexes, chain maps, homology of a chain complex, chain homotopy, explained why homotopic chain maps induce the same maps at homology, Computed the homology of direct sum of chain complex, snake lemma.
• Lecture 5 :- Using snake lemma showed that a short exact sequence of chain complex gives rise to a long exact sequence in the homology. Introduced singular chain complex, singular homology, computed the singular homology of point space, showed that for any topological space X; H 0(X) gives the number of path components of X.

Himadri Kumar Mukerjee: Algebraic Topology (AT-11 to AT-16)
I gave the last six lectures on Topology in which I started by stating the Eilenberg-Steenrod axioms of a homology theory and using these several results like, topological invariance, homotopy invariance, homology of retracts and deformations of a space in relation to the homology of the space were derived; kernel sequence corresponding to a map of pairs was discussed and as a consequence reduced homology was defined and reduced homology exact sequence of a pair was established; using the axioms suspension isomorphism of homology was proved and homology of spheres were calculated; using the axioms and the direct sum lemma the homology homomorphism induced by a reflection map on spheres was calculated; it was remarked that the same could be done using the definition of singular homology; exact triads were introduced and Mayer-Vietoris sequence was established; using Mayer-Vietoris sequence a large number of computations of homology of wedge of circles, torus, orintentable surfaces of genus two and more, homology of joints of spaces etc. were done; relation between fundamental group and first integral singular homology group of a space was discussed and a sketch of the proof was given; as an application it was shown (indication) that a singular 1-cycle represented by a continuous map
of a circle, $ f:S^1 \to X $ , is homologous to zero if and only it $ f $ extends to a closed orientable surface of certain genus with boundary $ S^1$; it was indicated using singular homology that the boundary of the Möbius band is not a retract, homology of the Klein bottle as union of two Möbius bands joined along there bounding circles was discussed.Three problem sets were distributed and most of the problems were discussed in the tutorial sessions; the response of the students was reasonable.

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