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AFS-I being organised in Chandigarh in December 2010 and is funded by NBHM.
Speakers and Syllabus
Speakers
| Algebra |
SKK-Sudesh Kaur Khanduja
Panjab University, Chandigarh |
PS-Parvati Shastri
University of Bombay |
GKB-Gurmeet Kaur Bakshi
Panjab University, Chandigarh |
| Analysis |
AK - Ajay Kumar
University of Delhi |
SB-Savita Bhatnagar
Panjab University, Chandigarh |
RBJ-Renu Bajaj
Panjab University, Chandigarh |
| Differential
Geometry &Topology |
CSA - C.S. Arvinda
TIFR, Banglore |
SD-Satya Deo
HRI, Allahabad |
ARS-A.R. Shastri
IIT Bombay |
| UM-Lecture
Series |
I.B.S.P-I.B.S. Passi
Panjab University, Chandigarh |
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Syllabus
| Speaker
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Lectures
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Detailed Syllabus
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| Algebra
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| Prof Parvati
Shastri |
8 |
Modules over commutative
rings, Modules over a PID, Matrices over a PID and Structure Theorem,
canonical forms |
| Prof Sudesh
Khanduja |
8 |
Review of basic field
theory, Separable and normal extensions, algebraically closed fields,
splitting fields, Norm and Trace and their properties. Hilbert Theorem
90, The idea behind Galois theory , Fundamental theorem of Galois theory.
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| Dr Gurmeet
K. Bakshi |
8 |
Cyclotomic Extensions &
Cyclic Extensions, Application of the Cyclotomic extensions and Galois
theory to the constructability of regular polygons. Historical
aspects of the solvability of polynomial equations, Solvability of polynomial
equations by radicals, The inverse problem of Galois theory. Computation
of Galois groups over Q for small degree polynomials.
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| Analysis
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| Prof Savita
Bhatnagar |
12 |
Measures, Integration,
Normed spaces, Baire category theorem.Open mapping theorem, Closed graph
theorem, Uniform boundedness theorem.
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| Prof Ajay
Kumar |
6 |
Review
of Fourier series, Basic Fourier Analysis on groups Z of integers, unit
circle T and the real line R generalized Reimann lebesgue lemma,
The Gibb’s Phenomenon.
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| Dr Renu
Bajaj |
6 |
Existence of local solutions
for first order systems, maximal time of existence, finite time blow-up,
global solutions. Gronwall inequality,Continuous dependence on initial
data and on the vector field on bounded intervals. Examples of
linearsystems, Fundamental solutions.
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| Topology
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| Prof
A R Shastri |
12 |
Revision
of Calculus of n-variables, smoothness of maps defined on arbitrary
subsets of Rn Inverse and implicit function theorem.
Manifolds in Rn(as `submanifolds'), tangent space,
induced map on the tangent spaces. Regular and critical points,
Sard's theorem(statement only), regular inverse image. Classification
of 1-dim. Manifolds.
Richness
of smooth functions, smooth partition of unity, immersions,submersions,
embeddings. Abstract manifolds, different definitions of tangent space,
examples Orientability.
Embedding
theorems for manifolds in to Euclidean spaces. Normal bundleand tubular
nbd theorem. Homotopy and stability. Transversality, Vector fields and
isotopies.
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| Prof C
S Aravinda |
6 |
Frenet-Serret
Theory of Curves, congruence of curves, classification of compact surfaces
with boundary (examples, and statement), tangent and cotangent bundles,
Riemannian metrics, differential forms, first and second fundamental
form of surfaces in E3, normal curvature, Codazzi equations,
congruence of surfaces, Gauss Bonnet theorem(statement). |
| Prof Satya
Deo |
6 |
Transversality, Oriented
intersection theory, Brauwer degree Winding
number and Brauwer separation
theorem, Borsuk-Ulam Theorem
Hopf Degree
theorem Lefschetz Theory of vector fields, Poincare-Hopf Theorem.
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