# AFS-III - Annual Foundation School - III (NISER 2019)

## Speakers and Syllabus

Syllabus to be covered in terms of modules of 4 lectures each of 1.5 hrs

 Name of the Speakers with their affiliation. No. of Lectures Detailed Syllabus Dr. Tanusree Khandai (TK) IISER, Mohali 4 Field extension and examples. Algebraic and transcendental elements, minimal polynomial. Degree of a field extension, finite and infinite extensions. Simple extensions.Transitivity of finite/algebraic extensions. Compositum of two fields. Ruler and compass constructions. Characterization of constructible numbers via square root towers of fields. Impossibility of squaring the circle, trisection of angles and duplication of cubes by ruler and compass, Gauss’ criterion of constructible regular polygons. (Wantzel’s characterization of constructible regular p-polygons. Richmond’s construction of a regular pentagon). Examples of symmetric polynomials. The fundamental theorem of symmetric polynomials. Newton’s identities for power sum symmetric polynomials. Discriminant in terms of power-sum symmetric polynomials. Discriminant of a cubic. Existence of a splitting field of a polynomial. Fundamental theorem of algebra via symmetric polynomials Dr. Jaban Meher (JM), NISER, Bhubaneswar 4 Splitting fields and algebraic closures; existence and isomorphisms, Criterion for multiple roots of polynomials in terms of derivatives. Characterization of perfect fields of positive characteristic.Separable and inseparable extensions. Transitivity of separable extensions,Roots of an irreducible polynomial have equal multiplicity. Separable degree. Finite fields: existence and uniqueness, algebraic closure. Finite subgroup of the multiplicative group of a field is cyclic. Gauss’ formula for the number of monic irreducible polynomials of a given degree over a finite field. Factorization of polynomials over finite fields. Primitive element theorem.Finite separable extensions have a primitive element. Normal extensions and their examples. Characterization of normal extensions in terms of embeddings and splitting field. Dr. Senthil Kumar (SK), NISER, Bhubaneswar 4 Galois extensions. Galois groups of finite extensions od finite fields and quadratic extensions. Galois groups of biquadratic extensions. Galois groups of a separable cubic polynomials. Fundamental Theorem of Galois theory (FTGT).Artin’s Theorem about fixed field of a finite group of automorphisms.Behavior of Galois group under isomorphisms. Normal subgroups of the Galois groups and their fixed fields. Fundamental theorem of algebra via FTGT. Gauss’ criterion for constructible regular polygons. Symmetric rational functions. Galois group of some binomials. Roots of unity in a field. Galois group of x roots of unity. Irreducibility of the cyclotomic polynomial Φn(x) over Q. A recursive formula for the cyclotomic polynomial. Discriminant of Φp(x). Subfields of Q(ζp). Kronecker-Weber theorem for quadratic extensions of Q. Algorithms for construction of primitive elements of subfields of Q(ζp). Subfields of Q(ζ7), Q(ζ13) and Q(ζ17). Infinitude of primes p ≡ 1(mod n). Inverse Galois problem for finite abelian groups. Structure of some cyclic extensions. n − a over a field having n-th Prof. Parvati Shastri (PS), Univ. Mumbai 4 Cyclic extensions of degree p over fields with characteristics p. Solvable groups. Simplicity of An. Galois group of composite extensions Galois closure of a separable field extension. Radical extensions. Solvability by radicals and solvable Galois groups. A quintic polynomial which is not solvable by radicals. Cardano’s method for roots of cubic equations. Lagrange’s method for roots of quartic equations. Ferrari’s method for roots of quartic equations.Galois group as a group of permutations. Irreducibility and transitivity.Galois groups of quartics. The norm and the trace function. Multiplicative form of Hilbert’s Theorem 90. Cyclic extensions of degree n: Additive version of Hilbert’s 90. Cyclic extensions of prime degree: Artin-Schreier Theorem Dr. Manas R. Sahoo (MS), NISER, Bhubaneswar 4 Algebraic Systems, Linear spaces, The dimension of a linear spaces, Linear transformations, Alegbras, Banach Spaces: The definition and some examples, Continious linear transformations Dr. Dinesh Keshari (DK), NISER, Bhubaneswar 4 Banach Spaces: The Hann-Banach Theorem, The natural embeddings of N in N**, The open mapping Theorem, The conjugate of an operator Dr. Lingaraj Sahu (LS), IISER, Mohali 4 Hilbert Spaces: The definition and some properties, Orthogonal complements, Orthogonal sets, The conjugate space H*, The adjoint of an operator, Self-adjoint operators, Normal and unitary operators, Projections Dr. Sutanu Roy (SR), NISER, Bhubaneswar 4 Finite-dimensional spectral Theory: Matrices, Determinnats and the spectrum of an operator, The spectral Theorem Dr. Kasi Viswnadh (KV), IISER, Berhampur 4 Statements of basic problems in algebraic topology extension problems and lifting problems; homotopy, relative homotopy, deformation, contraction, retracts etc. Typical constructions: Adjunction spaces, Mapping cones, Mapping cylinder, Smash- product, reduced cones reduced suspension etc. Categories and Functors. Definition and examples. Equivalence of functors, adjoint functors, examples. Computation of fundamental group of the circle and applications. Dr. Ajay Singh Thakur (AT), IIT, Kanpur 4 Covering spaces and Fundamental groups lifting properties,relation with fundamental group. Classification of covering spaces(proof of existence may be skipped), Computation of Fundamental groups: simpler cases of Van-Kampen theorem. Effect of attaching n-cells Dr. Archana S. Morye (AM), Univ. Hyderabad 4 CW-Complexes and Simplicial complexes basic topological properties of CW complexes. Products of CW complexes (especially the CW-structure on X ×[0, 1]). Homotopy theoretic properties of CW complexes. Abstract simplicial complexes and geometric realization, barycentric sub division and simplicial approximation theorem. Applications: cellular Approximation theorem, Brouwer’s invariance of domains etc. Prof. Anant Shastri (AS), IIT, Mumbai 4 Singular and Simplicial Homology Chain complexes, exact sequences of complexes, snake lemma, four lemma and five lemma,homology long exact sequence. Axioms for homology, construction of singular chain complex, verification of axioms (except homotopy axiom and excision axiom). Simplicial and singular simplicial homologies. Statement of equivalence of all these homologies. Computations and applications: Separation theorems, Invariance of Domain. Euler characteristic

References:

1. M. Artin: Algebra, 2nd Edition, Prentice Hall of India, 2011. Chapter 13, 14
2. Introduction to Toplogy and Modern Analysis: G. F. Simmons (Chapters: 8, 9, 10, 11)
3. J. R. Munkres: Topology 2nd Edition, Prentice Hall of India. (Chapter 9 and 13)
4. Basic Algebraic Topology--by A. R. Shastri, CRC Press.
5. E. H. Spanier: Algebraic Topology, Tata McGraw-Hill (Chapter 3, 4) or
6. M. J. Greenberg and J. R. Harper, Algebraic Topology: a first course, Benjamin/Cummings Pub.1981 (Part II).
7. S. Lang: Algebra, Revised 3rd Edition, Springer.
8. N. Jacobson: Basic Algebra, Vol.1 and Vol.2, Dover,2011.
9. E. H. Spanier: Algebraic Topology, Tata McGraw-Hill.
Names of the confirmed tutors:
1. Balesh Kumar (IMSc, Chennai)
2. Abhash Kumar Jha (IISc, Bengaluru)
3. Anindya Ghatak (ISI, Bengaluru)
4. Bikramaditya Sahu (IISc, Bengaluru)
5. Moni Kumari (TIFR, Mumbai)
6. Prem P. Pandey (IISER, Berhampur)
7. Sujeet Kumar Singh (NISER, Bhubaneswar)
8. Amit Kumar (NISER, Bhubaneswar)
9. Nabin Meher (NISER, Bhubaneswar)
10. Atibur Rahaman (NISER, Bhubaneswar)
11. Abhrojyoti Sen (NISER, Bhubaneswar)
12. 1Rahul Kumar Singh (NISER, Bhubaneswar)

## Time Table

Tentative time-table mentioning the names of speakers with their affiliation:

 Day Date Lecture 1 (9.30–11.00) Tea (11.05–11.25) Lecture 2 (11.30–1.00) Lunch (1.05–2.25) Tutorial (1 2.30 – 3.30) Tea (3.35-355) Tutorial 2 4.00 – 5.00 Snacks 5.05-5.30 (name of the speaker) (name of the speaker) (name of the tutor) (Name of the tutor) Mon 01-07-2019 TK MS Algebra Algebra Tues 02-07-2019 MS KV Analysis Analysis Wed 03-07-2019 KV TK Topology Topology Thu 04-07-2019 TK MS Algebra Algebra Fri 05-07-2019 MS KV Analysis Analysis Sat 06-07-2019 KV TK Topology Topology SUNDAY: HOLIDAY Mon 08-07-2019 JM DK Algebra Alegbra Tue 09-07-2019 DK AT Analysis Analysis Wed 10-07-2019 AT JM Topology Topolgy Thu 11-07-2019 JM DK Algebra Algebra Fri 12-07-2019 DK AT Analysis Analysis Sat 13-07-2019 AT JM Topology Topology SUNDAY: HOLIDAY Mon 15-07-2019 SK LS Alegbra Algebra Tue 16-07-2019 LS AM Analysis Analysis Wed 17-07-2019 AM SK Topology Topology Thu 18-07-2019 SK LS Algebra Algebra Fri 19-07-2019 LS AM Analysis Analysis Sat 20-07-2019 AM SK Topology Topology SUNDAY: HOLIDAY Mon 22-07-2019 PS SR Algebra Algebra Tue 23-07-2019 SR AS Analysis Analysis Wed 24-07-2019 AS PS Topology Topology Thu 25-07-2019 PS SR Algebra Algebra Fri 26-07-2019 SR AS Analysis Analysis Sat 27-07-2019 AS PS Topology Topology
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