Annual Foundation School - I (2015) - Tezpur University - Speakers and Syllabus

Names and affiliation of speakers and course associates.

[1] Prof. Alok Goswami , ISI Kolkata, Speaker as well as Course Associate

[2] Prof. A. R. Shastri, IIT Bombay, Speaker as well as Course Associate.

[3] Prof. P. Shastri, University of Mumbai, Speaker as well as Course Associate.

[4] Prof. Goutam Mukherjee, ISI Kolkata, Speaker as well as Course Associate

[5] Prof. Ramji Lal, Allahabad University and HRI, Allahabad, Speaker as well as Course Associate.

[6] Dr. Shamindra Kumar Ghosh, ISI, Kolkata, Speaker as well as Course Associate.

[7] Dr. Paramita Das, ISI, Kolkata, Speaker as well as Course Associate.

[8] Dr. Partha Pratim Ghosh, ISI, NE Centre, Speaker as well as Course Associate.

[9] Dr. Mrinal Kanti Das , ISI, Kolkata, Speaker as well as Course Associate.

[10] Dr. Swagato Ray, ISI, Kolkata, Speaker as well as Course Associate.

 Syllabus to be covered in terms of modules of 6 lectures each.

(1) Group Theory.

[1] Basic examples of groups such as cyclic groups, dihedral and quaternion groups, matrix groups and permutation groups. reviewof normal subgroups and isomorphism theorems, internal and direct products.

[2] Isometries of R^n and plane. group actions, _nite subgroups ofSO(2) and SO(3):

[3] Sylow Theor , classi_cation of _nite groups of order 12, simplicity of the alternating groups and

[4] Linear groups, Classical groups, SU(2), latitudes and longitudes on the 3-sphere, simplicity of

(2) Real Analysis.

[1] Basics Abstract measure spaces and the concept of measurability, simple functions, basic properties of measures, Lebesgue integration of positive functions and complex values functions, measure zero sets, completion of a measure and outer measure.

[2] Positive Borel Measures: Topological preliminaries on locally compact Hausdorff spaces, Riesz representation theorem (outline of the proof), Borel measures, Lebesgue measure on R^k, comparison with Riemann integration. Approximation by continuous functions,Generalized Riesz representation theorem.

[3] Differentiation: Maximal functions, Lebesgue points, I Fundamental Theorem of integral calculus, Absolutely continuous functions, II fundamental theorem of integral calculus. Change of variable formula.

[4] Integration on products Monotone classes, algebra on products, product measure, Fibini, completion, convolution.

 (3) Topology.

 [1] Review of Multivariable Differential Calculus (8 hours) Differentiability of functions on open subsets of R^n, relation with partial/ directional derivative, Taylor's theorem etc. Inverse and implicit function theorems, rank theorem, Diffrentiability of functions on arbitrary subsets of subsets of R^n, diffeomorphisms, s mooth version of invariance of domain.Richness of smooth functions, smooth partition of unity and consequences on subspaces of R^n such as approximation of continuous functions by smooth functions.Sard's theorem for smooth functions Rn ! Rm and some applications.

 [2] Basic point-set topology part (a) (8 hours)
Open sets and closed sets, limit points, closure and boundary points, subspace. Bases and subbases.Continuous functions, open functions, closed functions, homeomorphisms. Separation axioms: Hausdor_ness regularity and normality. Urysohn's lemma and Tietze
 extension theorem. Compactness and Lindel• o_ property, local compactness. I and II countability separability.Path connectedness, connectedness, local connectedness.

[3] Basic point-set topology part (b) (8 hours)
Induced and coinduced topologies.Quotient topology, separation axioms under quotient topology, criterion for a restriction of a quotient map to be a quotient map, examples such as cones, cylinders, Mobius strips etc.Paracompactness and partition of unity, Stone's theorem (paracompactness of metric topology). Topological groups and orbit spaces. Examples from matrix groups. Function spaces, compact-open-topology and exponential correspondence.

(c) Names of speakers who will cover each module of 6 lectures (in case of Topology, 8 lectures).

Group Theory

[1] Dr. Paramita Das

[2] Prof. Parvati Shastri

[3] Dr. Mrinal Kanti Das

[4] Prof. Ramji Lal

Analysis

[1] Prof. Alok Goswami

 [2] Dr. Partha Pratim Ghosh

[3] Dr. Swagato Ray

[4] Prof. Alok Goswami

Topology

[1] Dr. Shamindra K. Ghosh

[2] Prof. Anant R. Shastri

[3] Prof. Gautam Mukherjee

 A tentative time-table mentioning the names of speakers and course associates.

Week-I
  09:30 11:00 11:30 1:00 2:30 3:30 4:00 5:00
01/12/15 Tuesday Group Thy- L1 (P. Das) T Analysis- L1 (A. Goswami) L Group Thy -T1 (P. Das) T Group Thy -T2 (P. Das) S
02/12/15 Wednesday Topology-L1 (S. K.Ghosh)   Group Thy - L2 (P. Das) U Analysis-T1 (A. Goswami)   Analysis-T2 (A. Goswami) N
03/12/15 Thursday Analysis- L2 (A. Goswami) E Topology-L2 (S. K.Ghosh) N Topology-T1 (S. K.Ghosh) E Topology-T2 (S. K.Ghosh) A
04/12/15 Friday Group Thy- L3 (P. Das)   Analysis- L3 (A. Goswami) C Group Thy -T3 (P. Das)   Group Thy -T4 (P. Das) C
05/12/15 Saturday Topology-L3 (S. K.Ghosh) A Algebra- L4 (P. Das) H Analysis-T3 (A. Goswami) A Analysis-T4 (A. Goswami) K
07/12/15 Monday Analysis- L4 (A. Goswami)   Topology-L4 (S. K.Ghosh)   Topology-T3 (S. K. Ghosh)   Topology-T4 (S. K.Ghosh) S
Week-II
  09:30 11:00 11:30 1:00 2:30 3:30 4:00 5:00
08/12/15 Tuesday Group Thy- L5 (P. Shastri) T Analysis- L5 (P. P. Ghosh) L Group Thy -T5 (P. Shastri) T Group Thy -T6 (P. Shastri) S
09/12/15 Wednesday Topology-L5 (A.R. Shastri)   Group Thy - L6 (P. Shastri) U Analysis-T5 (P. P. Ghosh)   Analysis-T6 (P. P. Ghosh) N
10/12/15 Thursday Analysis- L6 (P. P. Ghosh) E Topology-L6 (A.R. Shastri)   Topology-T5 (A.R. Shastri) E Topology-T6 (A.R. Shastri) A
11/12/15 Friday Group Thy- L7 (P. Shastri)   Analysis- L7 (P. P. Ghosh) N Group Thy -T7 (P. Shastri)   Group Thy -T8 (P. Shastri) C
12/12/15 Saturday Topology-L7 (A.R. Shastri)   Algebra- L8 (P. Shastri) C Analysis-T7 (P. P. Ghosh) A Analysis-T8 (P. P. Ghosh) K
14/12/15 Monday Analysis- L8 (P. P. Ghosh) A Topology-L8 (A.R. Shastri) H Topology-T7 (A.R. Shastri)   Topology-T8 (A.R. Shastri) S
Week-III
  09:30 11:00 11:30 1:00 2:30 3:30 4:00 5:00
15/12/15 Tuesday Analysis – L9 (S. Ray) T Topology – L9 (A.R.Shastri) L Analysis – T9 (S. Ray) T Analysis – T10 (S. Ray) S
16/12/15 Wednesday Topology- L10 (A.R. Shastri)   Group Thy- L9 (M. K. Das) U Topology – T9 (A.R.Shahstri)   Topology – T10 (A.R. Shastri) N
17/12/15 Thursday Group Thy-L10 (M. K. Das) E Analysis – L10 (S. Ray)   Group Thy –T9 (M. K. Das) E Group Thy–T10 (M. K. Das) A
18/12/15 Friday Analysis - L11 (S. Ray)   Topology- L11 (G.Mukherjee) N Analysis-T11 (S. Ray)   Analysis-T12 (S. Ray) C
19/12/15 Saturday Group Thy–L11 (M. K. Das)   Analysis – L12 (S. Ray) C Group Thy–T11 (M. K. Das)   Group Thy–T12 (M. K. Das) K
21/12/15 Monday Group Thy–L12 (M. K. Das) A Topology-L12 (G.Mukherjee) H Topology-T11 (G. Mukherjee) A Topology-T12 (G. Mukherjee) S
Week IV
  09:30 11:00 11:30 1:00 2:30 3:30 4:00 5:00
22/12/15 Tuesday Group Thy-L13 (Ramji Lal) T Analysis- L13 (A. Goswami) L Group Thy-T13 (Ramji Lal) T Group Thy-T14 (Ramji Lal) S
23/12/15 Wednesday Topology-L13 (G.Mukherjee)   Group Thy-L14 (Ramji Lal) U Analysis-T13 (A. Goswami)   Analysis-T14 (A. Goswami) N
24/12/15 Thursday Analysis- L14 (A. Goswami) E Topology-L14 (G. Mukherjee)   Topology-T13 (G. Mukherjee) E Topology-T14 (G. Mukherjee) A
25/12/15 Friday Group Thy-L15 (Ramji Lal)   Analysis- L15 (A. Goswami) N Group Thy-T15 (Ramji Lal)   Group Thy-T16 (Ramji Lal) C
26/12/15 Saturday Topology-L15 (G.Mukherjee)   Group Thy-L16 (Ramji Lal) C Analysis-T15 (A. Goswami)   Analysis-T16 (A. Goswami) K
28/12/15 Monday Analysis- L16 (A. Goswami) A Topology-L16 (G. Mukherjee) H Topology-T15 (G. Mukherjee) A Topology-T16 (G. Mukherjee) S
29/12/15 Tuesday Valedictory