Annual Foundation School - III (2019)

Venue:	National Institute of Science Education and Research, Bhubaneswar, Odisha(Orissa)
Date:	1st, Jul 2019 to 27th, Jul 2019

School Convener(s)
Name	Dr. Brundaban Sahu	Dr. Jaban Meher
Mailing Address	Associate Professor National Institute of Science Education and Resaerch, Bhubaneswar, Via: Jatni, Khurdha 752050, Odisha	Reader F National Institute of Science Education and Resaerch, Bhubaneswar, Via: Jatni, Khurdha 752050, Odisha

Speakers with their affiliations:

Name of the speaker	Position	Affiliation
Tanusree Khandai	Assistant Professor	IISER, Mohali
Jaban Meher	Reader F	NISER, Bhubaneswar
Senthil Kumar	Assistant Professor	NISER, Bhubaneswar
Parvati Shastri	Retired Professor	University of Mumbai
Lingaraj Sahu	Assistant Professor	IISER, Mohali
Manas Ranjan Sahoo	Reader F	NISER, Bhubaneswar
Sutanu Roy	Assistant Professor	NISER, Bhubaneswar
Dinesh Keshari	Assistant Professor	NISER, Bhubaneswar
Prem Prakash Pandey/ Kasi Viswanadh	Assistant Professor	IISER, Berhampur
Ajay Singh Thakur	Assistant Professor	IIT, Kanpur
B. Subash	Assistant Professor	IISER, Tirupati
Anant R. Shastri	Retired Emeritus Professor	IIT, Bombay

Tutors :

Name of the tutor	Position	Affiliation
Abhash Kumar Jha	Postdoctoral Fellow	HRI, Prayagraj
Bikramaditya Sahu	Postdoctoral Fellow	IISC, Bengaluru
Anindya Ghatak	Postdoctoral Fellow	ISI, Bengaluru
Moni Kumari	Postdoctoral Fellow	TIFR, Mumbai
Balesh Kumar	Postdoctoral Fellow	NISER, Bhubaneswar
Nabin Meher	Postdoctoral Fellow	NISER, Bhubaneswar
Rahul Singh	Postdoctoral Fellow	NISER, Bhubaneswar
Sujit Kumar Singh	Phd student	NISER, Bhubaneswar
Abhrojyoti Sen	Phd student	NISER, Bhubaneswar
Atibur Rehman	Phd student	NISER, Bhubaneswar
Gorekh Sena	Phd student	NISER, Bhubaneswar
Nilakantha Das	Phd student	NISER, Bhubaneswar

Syllabus:

The topics covered:

Name of the Speaker	Detailed Syllabus
Tanusree Khandai	Lecture 1. Chapter 13 (of the old edition) Section 1 and parts of section 2. Recalled topics from Ring Theory (Chapter 10 and Chapter 11 from Artin) on field of fractions, irreducibility criterion, algebraic and transcendental extensions. Lecture 2. Chapter 13, Section 2 of Artin was completed. Besides, simple extensions and multiple extensions was done and the main theorem proved was that every finitely generated algebraic extension of Q is a simple extension. Lecture 3. Chapter 13, Section 3 was discussed. Defined algebraic numbers and proved that the set of algebraic integers in a finite extension of Q forms a ring. The result and proof was then compared with the result that the set of algebraic elements in an extension K over Q is a subfield of K. Degree of extensions of a field and the related problems were discussed. Lecture 4. Chapter 13, Section 4 was discussed. Starting from the basic ruler and compass construction, all the results in Artin was discussed. Then the Gauss-Wantzel's Theorem was proved and it was mentioned that the converse should be regarded as an exercise once the main theorem of Galois is proved.
Jaban Meher	Followed Artin's Algebra (old version). Lecture 1: Symbolic adjunction of roots, Multiple roots (some portion) [Section 13.5] Lecture 2: Multiple roots (remaining portion), Finite Field (some portion) [Section 13.6] Lecture 3: Finite Field (remaining portion), Calculating the number of irreducible polynomials over $\mathbb{F}_p$ [Section13.6] Lecture 4: Algebraically closed fields, Algebraic closure, Fundamental Theorem of Algebra, One construction of Algebraic closure for $\mathbb{F}_p$ [Section 13.9]
K. Senthil Kumar	Lecture 1: started with the classical problem of solving polynomial equations by radicals and informed the students that we shall see at the end of my lectures that this problem has a negative answer. Then I went on to discuss: existence and uniqueness of splitting fields (up to isomorphism), existence and uniqueness of algebraic closures (up to an isomorphism). And examples. Lecture 2: Here I discussed the problem of embedding algebraic extensions into an algebraically closed field. This leads to the study of separable degree and its transitive property, separable extensions and few properties of the class of separable extensions (like the separability of a tower of separable extensions, separability of compositum). And examples. Lecture 3: This lecture was mostly dedicated to normal extensions. We gave three equivalent definitions for normal extensions. We explained that unlike separable extensions, the tower of normal extensions need not always lead to normal extensions. We also introduced Galois extensions and Galois groups. And examples. Lecture 4: In this lecture, I stated and proved the fundamental theorem of Galois theory. Constructed Galois groups of cubic extensions, Artin's proof of C is algebraically closed. We also have shown that there are polynomials for which their Galois group is S_n. And examples.
Parvati Shastri	Lecture 1. Brief introduction to the problem of solving polynomials by radicals. Review of Galois extensions, Galois group, with a lot of examples. Quadratic, cubic polynomials, biquadratic extensions. Finite extensions of finite fields are Galois. Cyclotomic fields over the rationals. Cyclotomic polynomials. Abelian extensions, cyclic extensions, structure of Galois group of a cyclotomic extension of the rationals. \\ Lecture 2. Fundamental theorem of Galois theory, proof using Artin?s lemma. Examples. Galois group as subgroup of the symmetric group. Criterion for Galois group to be contained in the alternating group in terms of the discriminant. Possible Galois groups of cubics and quartics. Examples of each possibility were given in tutorials.\\ Lecture 3. Construction of regular polygons, revisited. Full proof of Gauss- Wantzel Theorem. Example of extensions of degree 4 which has no quadratic subfield. Statement of Galois criterion for solvability by radicals. Definition and examples of solvable groups. Example of a polynomial over the rationals whose Galois group is isomorphic to the symmetric group $S_5$. (Partially done in exercises). Extended idea of constructing polynomials over the rationals with Galois group $S_p$, where $p$ is prime bigger or equal to 5. \\ Lecture 4. Fields having enough roots of unity, equivalence of radical and cyclic extensions over such fields, Kummer Theory. Proof of Galois criterion. Abel Ruffini Theorem. Mentioned Inverse Galois problem.
Lingaraj Sahu	Started with vector space, Linearly independence, Algebraic ( Hamel) basis and some examples of finite and infinite dimensional; spaces. \\ Recalled the structure of the order field R , gave definition of normed linear / Banach spaces with examples, Schauder basis, example of non-separable norm spaces. Discussed uncountability of Hamel basis in a Banach space. Recalled the definition of Hilbert space, Cauchy- Schwartz inequality, Parallelogram Law, Polarisation Identity , orthogonality, existence of perpendicular vector. Discussed Riesz's lemma and characterisation of finite dimensionality in term of compactness . Discuss about Quotient space , Quotient norm and its completeness, Equivalence of all norms in a finite dimensional norm space, Recalled the definition of Linear transformations, continuity, Example of bounded / un-bounded linear transformation. Defined norm on the algebra of bounded linear transformations, equivalence definition of norm Recalled the dual of norm spaces, some examples
Manas Ranjan Sahoo	Lecture-1: Hahn- Banach Theorem, examples and application(Banach Limit, An abstract approach to Poissons integral) Lecture -2: Uniqueness and non-uniqueness of Banach Extension, Reflexive Banach space Lecture 3: Banach- Steinhaus theorem and application to Fourier series Lecture-4: Open mapping theorem and application.
Sutanu Roy	Lecture-1: Hilbert Spaces: The definition, Orthogonal complements; Lecture-2: The conjugate space H*, The adjoint of an operator, Riesz Representation Theorem; Lecture-3: Self-adjoint operators, Normal and unitary operators, Projections; Lecture-4: Compact operator, multiplication operator, comparison of the statements of the Spectral Theorem for compact self adjoint and self-adjoint operators using multiplication operator.
Dinesh Keshari	Lecture 1: Finite-dimensional spectral Theory: Matrices, Determinants and the spectrum of an operator, The spectral Theorem. Lecture 2 and Lecture 3: Spectral theorem for compact normal operators; The Montel-Hely Selection Principle. Lecture 4: Spectral theorem for self-adjoint operators.
Prem Prakash Pandey/ Kasi Viswanadh	Lecture 1. Fundamental (classification ) problem of topology, path homotopy, loops in a topological space, fundamental group of a topological space, Lecture 2. functorial property of fundamental group, lifting of path homotopies, covering space, computation of fundamental group of circle. Lecture 3- Definition and examples of categories and Functors, Maps between functors, Equivalence of functors. Lecture 4- Natural transformation between functors. Relative homotopy, Deformation, Contraction and Retraction, and some of their properties.
Ajay Singh Thakur	Lectures - 1 and 2: Equivalence of Covering spaces. Proof of the general lifting lemma, Group actions, Group of covering Transformations, A brief sketch on the construction of universal covering space, The classification of covering spaces. Lectures - 3 and 4: Free Groups, Proof of the Seifert-van Kampen Theorem, Fundamental Groups of the wedge of circles, Adjoining two cells and itseffect on Fundamental groups, Fundamental groups of Torus, Klein Bottle, Projective planes were discussed.
B. Subash	Lecture 1 : Recalled Quotient spaces, Introduced attaching cells, Introduced CW complex Lecture 2 : CW Structure on the spheres and projective spaces, CW structure on the product of CW complex, Introduced Simplicial complex. Lecture 3: introduced Triangulation of spaces, Introduced Barycentric subdivision of a simplicial complex, Prism construction was done. Lecture 4 : Geometric Realisation of a Simplicial Complex, proved that Geometric realisation of a simplicial complex and its barycentric subdivision are homeomorphic, Stated the simplicial Approximation theorem and gave an outline of the proof.
Anant R. Shastri	Lecture 1: Basic homological algebra; snake lemma, four and five lemma. Lecture 2. Algebra of Euler characteristic, Lefchetz number. Construction of Singular Homology. Lecture 3. Basic properties of Singular homology. computations. Additional hour: Simplicial homology and the relation with singular homology. Lecture 4. Applications of homology. Brouwer Fixed point theorem Lefchetz Fixed point theorem, Jordan-Brouwer Separation theorem, Brouwer's Invariance of domain.

Time Table

Day	Date	Lecture 1 09.30-11.00AM	Tea 11.00 AM	Lecture 2 11.30-1.00PM	Lunch 1.00 PM	Tutorial 1 2.30-3.30PM	Tea 3.30 PM	Tutorial 2 4.00-5.00PM	Snacks 5.00 PM
Day	Date	name of the speaker (abbr)	Tea 11.00 AM	name of the speaker (abbr)	Lunch 1.00 PM	name of the speaker (abbr)	Tea 3.30 PM	name of the speaker (abbr)	Snacks 5.00 PM
Mon	01-07-2019	TK		LS		SKS, BKS, TK		SKS, BKS, TK
Tues	02-07-2019	LS		PPP		AS, AJ, LS		AS, AJ, LS
Wed	03-07-2019	PPP		TK		PPP, BK, ND		PPP, BK, ND
Thu	04-07-2019	TK		LS		SKS, BKS, TK		SKS, BKS, TK
Fri	05-07-2019	LS		KV		AS, AJ, LS		AS, AJ, LS
Sat	06-07-2019	KV		TK		PPP, BK, KV		PPP, BK, KV

Mon	08-07-2019	JM		MRS		SKS, BKS, JM		SKS, BKS, JM
Tue	09-07-2019	MRS		AST		AS, MK, MRS		AS, MK, MRS
Wed	10-07-2019	AST		JM		ND, BK, AST		ND, BK, AST
Thu	11-07-2019	JM		MRS		SKS, BKS, JM		SKS, BKS, JM
Fri	12-07-2019	MRS		AST		AS, MK, MRS		AS, MK, MRS
Sat	13-07-2019	AST		JM		ND, BK, AST		ND, BK, AST

Mon	15-07-2019	KS		SR		AJ, GS, KS		AJ, GS, KS
Tue	16-07-2019	SR		BS		AR, AG, SR		AR, AG, SR
Wed	17-07-2019	BS		KS		RS, NM, BS		RS, NM, BS
Thu	18-07-2019	KS		SR		AJ, GS, KS		AJ, GS, KS
Fri	19-07-2019	SR		BS		AR, AG, SR		AR, AG, SR
Sat	20-07-2019	BS		KS		RS, NM, BS		RS, NM, BS

Mon	22-07-2019	PS		DK		AJ, GS, PS		AJ, GS, PS
Tue	23-07-2019	DK		ARS		AR, AG, DK		AR, AG, DK
Wed	24-07-2019	ARS		PS		RS, NM, ARS		RS, NM, ARS
Thu	25-07-2019	PS		DK		AJ, GS, PS		AJ, GS, PS
Fri	26-07-2019	DK		ARS		AR, AG, DK		AR, AG, DK
Sat	27-07-2019	ARS		PS		RS, NM, ARS		RS, NM, ARS

Full forms for the abbreviations of speakers and tutors :

Tanusree Khandai (TK), Lingaraj Sahu (LS), Prem P. Pandey (PPP)/Kasi Viswanadh (KV),
Jaban Meher (JM), Manas Ranjan Sahoo (MRS), Ajay Singh Thakur (AST),
K. Senthil Kumar (KS), Sutanu Roy (SR), B. Subash (BS),
Parvati Shastri (PS), Dinesh Keshari (DK), Anant R. Shanstri (ARS).

Abhash K. Jha (AJ), Bikramaditya Sahu (BKS), Anidya Ghatak (AG), Moni Kumari (MK),
Balesh Kumar (BK), Nabin Meher (NM), Rahul Singh (RS), Sujeet Kumar Singh (SKS),
Abhrojyoti Sen (AS), Atibur Rehman (AR), Gorekh Sena (GS), Nilakantha Das (ND).

List of actual Participants

Sr	SID	Full Name	Gender	Affiliation	Position in College/ University	University/Institute M.Sc./ M.A.	Year of Passing M.Sc./ M.A	Ph.D. Deg. Date
1	26504	Mr. Mohd Shanawaz Mansoori	Male	Aligarh Muslim University, Aligarh	PhD	MJP Rohilkhand University, Bareilly	2014
2	26828	Mr Mohammad Iliyas	Male	Aligarh Muslim University	Ph.D. Student	Aligarh Muslim University	2017
3	26868	Mr. Anirban Kundu	Male	Siksha-Bhavana, Visva-Bharati	Ph.D. Student	IIT Madras	2014
4	26895	Mr. Mohd. Imran Idrisi	Male	Aligarh Muslim University	PHD STUDENT	Jamia Millia Islamia	2015
5	26958	Mr. Ayush Bartwal	Male	HNBGU	Ph. D.	HNBGU	2014
6	26998	Mr Tamilarasan B	Male	Madurai Kamaraj University	PhD	Madurai Kamaraj University	2014
7	27174	Mr. Dharminder Mann	Male	The LNM Institute of Information Technology	Research scholar	Dcrust	2012
8	27206	Mr. Shivajee .	Male	IIT Gandhinagar	Phd scholar	Banaras Hindu University	2017
9	27236	Mr. Himanshu Setia	Male	IIT, Ropar	PhD	University of Delhi	2017
10	27405	Ms Swati Antal	Female	H.N.B. Garhwal University	Ph.D scholar	M.Sc	2016
11	27409	Ms Ayantika Laha	Female	IIT Ropar	PhD	university of calcutta	2016
12	27421	Mr Rahul Maurya	Male	IIIT- Allahabad	Ph.D.	Banaras Hindu University	2014
13	27443	Mr Sujeet Kumar	Male	IIT Bhubaneswar	PhD	Central University of South Bihar, Gaya	2016
14	27518	Mr. Abhinay Kumar Gupta	Male	IIT Delhi	PhD Student	IIT Delhi	2018
15	27531	Ms. Devichandrika V	Female	University Of Hyderabad	Ph.D	Central University Of Kerala	2018
16	27596	Mr. Mohit Pal	Male	Indian Institute of Technology Jammu	PhD	IIT Kharagpur	2016
17	27602	Mr. Mohit Kumar Baghel	Male	I.I.T. DELHI	Ph.D. Student	Dayalbagh Educational Institute	2018
18	27605	Mr. Sushil Singla	Male	Shiv Nadar University	Phd	Shiv Nadar University	2016
19	27786	Mr. Priyaranjan Mallick	Male	IISER, Berhampur	PhD	Utkal University, Bhubaneswar	2013
20	27802	Ms. Manideepa Saha	Female	Presidency University	PhD	University Of Calcutta	2014
21	27867	Mr. Sworup Kumar Das	Male	P.G. Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar	Ph. D. Research Scholar	Utkal University, Vani Vihar	2014
22	27976	Mr Abhishek Shrivastava	Male	Aligarh Muslim University	PhD	University of Delhi	2016
23	-	Mr. Shesh Pandey	Male	University of Delhi	PhD	-	-

How to reach

How to Reach NISER Campus:
Via Train: From the Khurda Road Railway Station (KUR) one can hire an auto-rickshaw (or taxi) to reach the Institute. Distance: 3 km (approx)
Via Air route: From Biju Pattanaik airport (BBI) one can hire a taxi or auto-rickshaw to reach NISER campus - Jatni. There is a prepaid taxi counter at the Bhubaneswar Airport. Distance: 25 km (approx)