Annual Foundation School - III (2019)

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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

name of the speaker
(abbr)

name of the speaker
(abbr)

name of the speaker
(abbr)

name of the speaker
(abbr)

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)