Annual Foundation School  Part II (2015)  Shiv Nadar Univ
Venue:  Shiv Nadar Univ, Gautam Buddha Nagar, UP 
Dates:  4^{th}  30^{th }May, 2015 
Convener(s)  Speakers, Syllabus and Time table  Applicants/Participants 
Name  Amber Habib  Rajendra Bhatia 
Mailing Address  Head, Department of Mathematics. Shiv Nadar University, NH91, Tehsil Dadri, Gautam Buddha Nagar , Uttar Pradesh, 201314 
Distinguished Scientist Theoretical Statistics and Mathematics Unit, Indian Statistical Institute 7, S.J.S. Sansanwal Marg New Delhi 110 016 
Resource Persons
Differential Topology  Functional Analysis  Ring Theory 
Lecturers:  Lecturers:  Lecturers: 
Archana S Morye 
Rajendra Bhatia Affiliation: Indian Statistical Institute, Delhi 
Gurmeet K Bakshi Affiliation: Panjab University, Chandigarh 
N Raghavendra 
Tanvi Jain Affiliation: Indian Statistical Institute, Delhi 
Maneesh Thakur Affiliation: Indian Statistical Institute, Delhi 
Anant R Shastri Affiliation: IIT Bombay 
Sachi Srivastava Affiliation: University of Delhi 
Anupam Singh Affiliation: IISER, Pune 
Jaydeb Sarkar Affiliation: Indian Statistical Institute, Bangalore 
Dinesh Khurana Affiliation: Panjab University, Chandigarh 

Tutors:  Tutors:  Tutors: 
Pradip Mishra Affiliation:Shiv Nadar University 
Sneh Lata Affiliation: Shiv Nadar University 
Sugandha Maheshwary Affiliation:Panjab University, Chandigarh 
Rahul Singh Affiliation: HarishChandra Research Institute, Allahabad 
Priyanka Grover Affiliation:Shiv Nadar University 
Anirban Bose Affiliation:Institute of Mathematical Sciences, Chennai 
Kushal Lalwani Affiliation:Delhi University 
Niteesh Sahni Affiliation:Shiv Nadar University 
A Satyanarayana Reddy Affiliation:Shiv Nadar University 
Satyajit Guin Affiliation: Indian Statistical Institute 
Anjana Khurana Affiliation: Panjab University, Chandigarh 
Syllabus Covered by the Resource Persons
 Functional Analysis
 Rajendra Bhatia (May 49): Banach spaces (Sequence spaces, Lebesgue spaces), algebraic and topological bases, local compactness, quotient spaces, bounded linear operators, dual space, HahnBanach theorem, Uniform Boundedness Principle, Open Mapping Theorem, Inverse Mapping Theorem, Closed Graph Theorem.
 Tanvi Jain (May 1116): Duals of classical spaces lpn, lp, c and c0, Lp[0,1], Riesz Representation Theorem for C[0,1], Montel Helly Selection Principle, weak topology, nets, second dual, reflexive and nonreflexive spaces, weak* topology, BanachAlaoglu Theorem, Annihilators, Hilbert spaces, Subspaces, Projections, Direct sums, orthonormal bases, separable Hilbert spaces.
 Sachi Srivastava (May 1823): Topologies on operators (norm, strong, weak), inverse, adjoint, self adjoint operators, positive operators, normal operators, unitary operators, orthogonal projections, resolvent, spectrum, spectral radius, subdivisions of the spectrum, spectra of normal operators.
 Jaydeb Sarkar (May 2530): Square roots and polar decomposition, compact operators, Spectral theorem for selfadjoint compact operators, brief Fredholm theory, (very) brief view of spectral measure and spectral theorem.
 Ring Theory
 Gurmeet Bakshi (May 49): Modules, submodules, quotient modules, free modules, modules over integral domains, modules over PIDs, rational canonical form, Jordan canonical form.
 Maneesh Thakur (May 1116): Commutative rings, prime and maximal ideals, nil radical, Jacobson radical, localization, projective modules, Noetherian rings, primary decomposition.
 Anurag Singh (May 1823): Integral extensions of rings, Going up and going down theorems,finiteness of integral closure, discrete valuation rings, Krull's normality criterion, Noether normalization lemma, Hilbert's Nullstellensatz.
 Dinesh Khurana (May 2530): Semisimple rings, Wedderburn's Theorem, Rings with chain conditionsand Artin's theorem, Wedderburn's main theorem.
 Differential Topology
 Archana Morye (May 49):Lectures 1 and 2: Review of differential calculus of several variables, culminating with the inverse and implicit function theorems (without proof).
Lectures 3 and 4: Differentiability over subsets of Rn and submanifolds with and without boundary. Special attention was paid to manifolds arising from regular values of smooth functions.  N Raghavendra (May 1116):Lecture 1: Topological manifolds, examples, spheres, projective spaces, basic topological properties of topological manifolds.
Lecture 2: Constructing manifolds from charts, Grassmanians.
Lecture 3: Smooth manifolds, examples, manifolds with boundary.
Lecture 4: Smooth functions on manifolds, partitions of unity, gluing manifolds.  Anant Shastri (May 1830):Lecture 1: Review, introduction of immersion, submersion, regular and critical values.
Lecture 2: Measure zero sets, Sard’s theorem, sources for obtaining manifolds: Regular level sets, Jacobian Criterion, Gluing.
Lecture 3: Study of Gluing continued with emphasis on Hausdorffness of the quotient space.Statement of classification of 1dimensional manifolds with a sketch of the proof.
Lecture 4: Orientation on vector spaces and manifolds. Different criterion for orientation. Induced orientations on the boundary.
Lecture 5: Regular values and Preimage theorem. Manifolds with boundary and proper (neat) submanifolds. Application to Brouwer fixed point theorem.
Lecture 6: Transversality as an extension of notion of submersion. Statement of extended transverse homotopy theorem.
Lecture 7: Oriented (as well as unoriented) intersection theory basic properties. Degree of a map. Degree of the antipodal map. Application to nonexistence of vector fields on even dimensional spheres.
Lecture 8: Statement and Proof of JordanBrouwer Separation Theorem. Remarks on some related results such as embeddings on other codimension. Statement and Proof of BorsukUlam theorem.
Program Schedule
Lecture 1 9:3011  Tea  Lecture 2 11:301  Lunch  Tutorial 1 2:303:30  Tea  Tutorial 2 4:005:00  Snacks  
Mon  4May  F. Analysis  D. Topology  F. Analysis  F. Analysis  
Tues  5May  F. Analysis  Ring Theory  Ring Theory  Ring Theory  
Wed  6May  F. Analysis  D. Topology  D. Topology  D. Topology  
Thu  7May  F. Analysis  Ring Theory  F. Analysis  F. Analysis  
Fri  8May  Ring Theory  D. Topology  Ring Theory  Ring Theory  
Sat  9May  Ring Theory  D. Topology  D. Topology  D. Topology  
Mon  11May  D. Topology  F. Analysis  F. Analysis  F. Analysis  
Tue  12May  F. Analysis  Ring Theory  Ring Theory  Ring Theory  
Wed  13May  D. Topology  Ring Theory  D. Topology  D. Topology  
Thu  14May  D. Topology  F. Analysis  F. Analysis  F. Analysis  
Fri  15May  Ring Theory  F. Analysis  Ring Theory  Ring Theory  
Sat  16May  D. Topology  Ring Theory  D. Topology  D. Topology  
Mon  18May  D. Topology  Ring Theory  Ring Theory  Ring Theory  
Tue  19May  D. Topology  Ring Theory  D. Topology  D. Topology  
Wed  20May  Ring Theory  F. Analysis  F. Analysis  F. Analysis  
Thu  21May  Ring Theory  F. Analysis  Ring Theory  Ring Theory  
Fri  22May  D. Topology  F. Analysis  D. Topology  D. Topology  
Sat  23May  D. Topology  F. Analysis  F. Analysis  F. Analysis  
Mon  25May  D. Topology  Ring Theory  D. Topology  D. Topology  
Tue  26May  D. Topology  F. Analysis  F. Analysis  F. Analysis  
Wed  27May  D. Topology  Ring Theory  Ring Theory  Ring Theory  
Thu  28May  D. Topology  F. Analysis  D. Topology  D. Topology  
Fri  29May  Ring Theory  F. Analysis  F. Analysis  F. Analysis  
Sat  30May  Ring Theory  F. Analysis  Ring Theory  Ring Theory 
Actual Applicants 
The following participants completed the School and received certificates:
Sr.n  Name  Gender  Institution  State  Degree Program 
1  Kirandeep Kaur  F  IIT Ropar  Punjab  PhD 
2  Priyanka Kumari  F  BITS Pilani  Rajasthan  PhD 
3  Reena  F  Delhi University  Delhi  MPhil 
4  Surinder Kaur  F  IIT Ropar  Punjab  PhD 
5  Aadil Hussain Dar  M  Aligarh Muslim University  Uttar Pradesh  PhD 
6  Anil Kumar Gupta  M  Guru Ghasidas Vishwavidyalaya  Chhattisgarh  PhD 
7  Chiranjeev Kumar Yadav  M  Guru Ghasidas Vishwavidyalaya  Chhattisgarh  PhD 
8  Kedarnath Buda  M  IISER Mohali  Punjab  PhD 
9  Lalit Vaishya  M  Banaras Hindu University  Uttar Pradesh  MSc 
10  Prince Kanhya  M  IIT Indore  PhD  
11  Prashanta Garain  M  IIT Kanpur  Uttar Pradesh  PhD 
12  Shiva Sharma  M  IIT (BHU)  Uttar Pradesh  PhD 
13  Sajad Ahmad Pary  M  Aligarh Muslim University  Uttar Pradesh  PhD 
14  Santosh Kumar  M  Aligarh Muslim University  Uttar Pradesh  PhD 
15  Soumitra Das  M  NorthEastern Hill University  Meghalaya  PhD 
16  Vijay Kumar Patel  M  IIT (BHU)  Uttar Pradesh  PhD 
17  Vishvesh Kumar  M  IIT Delhi  Delhi  PhD 
How to Reach SNU
Please visit the SNU website (contact us) for the address and for maps that give general directions from NOIDA as well as a detailed local schematic map.
By Air: The nearest airport is Indira Gandhi International Airport, New Delhi. From IGIA drive to Dhaula Kuan and take the Ring Road towards NOIDA. From NOIDA join the expressway to Greater NOIDA. From Pari Chowk in Greater NOIDA follow the instructions in the schematic map whose link is given above.
By Rail: From the Delhi Stations (Delhi/New Delhi/Nizamuddin) proceed to NOIDA and then follow the directions given above. Another option is to disembark at Ghaziabad (if you are on that line) and drive down on NH91 to Dadri. SNU is about 3 km after Dadri on the right hand side.
Faculty Feedback
 Archana Morye: Students had not attended AFS I and were not comfortable with calculus of several variables. Hence review of calculus took extra time and concepts of partition of unity and richness of smooth maps could not be discussed.
 Rajendra Bhatia: Almost all of them said they have had a course on FA. However, as is usual, they do not seem to have been taught much. Certainly, they had difficulties with the elementary exercises, such as how one might show one function f(x) is bigger than g(x)... On the positive side, most of them do seem interested in learning. We insisted that they must attempt the problems themselves, and in the second tutorial session, some of them could be prodded to come up with some solutions.
 Anant Shastri: Students were not working outside class, and did not utilise opportunity to interact with faculty in the evening and night.