ATMW Topology of Manifolds and Group Actions (2017)

Venue: University of Delhi,  Delhi, Delhi
Date:  9th, Nov 2017 to 22nd, Dec 2017

 


 

School Convener(s)

Name Prof. Satya Deo Dr. Hemant Kumar Singh Dr. Satya Prakash Tripathi
Mailing Address Harish- Chandra Research Institute,
Chhatnaag Road, Jhusi,
Allahabad- 211019
Assistant Professor,
Dept. of Mathematics,
University of Delhi, Delhi-110007
Associate Professor
Dept. of Mathematics,
Kirori Mal College,
University of Delhi, Delhi-110007

 


 

Speakers and Syllabus 

This was an advanced workshop in four different themes in harmonic analysis. First set of lectures by Professor J Faraut focused on orbital measures and spline functions. The action of unitary group on the space of Hermitian matrices can be described in terms of the results of Olshanksii, Okunkov etc and thus involve harmonic analysis on compact Lie groups. Horns theorem describing the projection of an orbit in the space of Hermitian matrices under the action of the unitary group and the orbital measure was explained. Second set of lectures by Professor Linda Saal dealt with generalized Gelfand pairs associated to the Heisenberg group. Spherical analysis associated to these pairs was discussed in detail. Third set of lectures by Professor Shobha Madan and Professor Saurabh Srivastava focused on sparse operators and a proof of the A2 conjecture. Finally, the lectures by Professor Swagato Ray dealt with harmonic analysis on symmetric spaces. In particular, Kunze-Stein phenomenon and boundedness of convolution operators were established.

 Speakers and their affiliations with detailes of lecture

Speakers Details of lectures delivered
Dr. Angom Tiken Singh
North Eastern Hill University, Shillong

Lecture1:-Topological manifolds, smooth manifolds, manifolds with boundary, smooth maps,
diffeomorphism.
Lecture2:-Vector bundles, canonical line bundle, local and global sections of vector bundles, local and
global frame s of vector bundles, bundle homomorphisms and isomorphisms.
Lecture3:-Restriction of a vector bundles, induced bundles, Cartesian products and Whitney sums of
vector bundles.
Lecture4:-Subbundles, Euclidean vector bundles, orthogonal complement of a subbundle and quotient
of a subbundle.

Dr. Satya P. Tripathi
Kirori Mal College, University of Delhi

Lecture1:-Concepts in general topology, examples of topological groups
Lecture2:-Group actions, examples of topological groups and their topological properties

Prof Tej B. Singh
University of Delhi, Delhi

Lecture1:-Simplicial Complexes, Simplicial Homology and Cohomology.
Lecture2:-Direct Limit, Cech Cohomology
Lecture3:-Covering Spaces and Fibrations
Lecture4:-Fiber Bundles and Spectral Sequences

Prof S. S. Khare
North Eastern Hill University, Shillong

Lecture1:-Derivatives, tangent vectors & tangent space, derivative of a smooth map between two
smooth manifolds, immersion, submersion, embedding.
Lecture2:-Inverse function theorem, constant rank theorem, local diffeomorphism verses immersion and
submersion.
Lecture3:-Local immersion theorem, local submersion theorem, global rank theorem, local embedding
theorem, r-slice, embedded (immersed) sub manifold.
Lecture4:-Constant rank-level set theorem, submersion level set theorem, regular points, critical points,
regular values, critical values, regular level set theorem, Whitney embedding theorem,
Whitney immersion theorem
Lecture5:-Tangent bundle, parallelizable manifolds, Riemannian metric on a smooth manifold,
Riemannian manifold, normal bundle of a smooth manifold embedded in R^n, normal bundle
of a sub manifold embedded in another manifold.

Prof Himadri Mukerji
North Eastern Hill University, Shillong

Lecture1:-Smooth Lie group action on smooth manifolds, representations, various associated definitions
and examples, equivariant maps, equivariant rank theorem, applications.
Lecture2:-Proper actions and quotient of manifolds by group actions, examples, statement and sketch of
proofs of the quotient manifold theorem.
Lecture3:-Covering manifolds, group of covering transformations, discrete group actions on connected
smooth manifolds, examples, homogeneous spaces, examples, homogeneous spaces
construction and characterization theorems.
Lecture4:-Further examples of homogeneous spaces, applications to sets with Transitive Lie group
actions, connected components of familiar lie groups.

Prof Satya Deo
Harish-Chandra Research Institute, Allahabad

Lecture1:-P.A.Smith fixed point theorems- their statements and the elementary proof as given by
G.E.Bredon for the p-groups and toral p-groups using simplicial cohomology of simplicial
complexes.
Lecture2:-Will continue the matter of first lecture, which will occupy this lecture also.
Lecture3:-Smith theorems for finitistic spaces for p-groups and for toral p-groups. Will try to cover the
well- known counterexamples to Smith theorems for other groups.
Lecture4:-Equivariant cohomology and the Borel’s method for studying the structure of fixed point sets
under compact Lie group actions.
Lecture5:-Localization theorem and its applications in the study of the structure of fixed point sets under
p-groups and toral p-groups.

Abstract of Special Lectures

Speakers Details of lectures delivered
Prof Ajejandro Adem
University of British Columbia, Canada
http://www.math.ubc.ca/~adem
Finite Groups from a Topology Perspective
Understanding the symmetries of a topological space is a classical problem in mathematics. In
this talk we will discuss how methods from algebraic topology can be used to approach this.
We will review classical results such as those describing the finite groups that can act without
fixed points on a sphere. This will be used as a motivation for the modern notion of "group
actions up to homotopy", which leads to interesting interactions between topology, group
theory and representation theory. A number of examples will be provided to illustrate this.
Prof Satya Deo
Harish-Chandra Research Institute, Allahabad
http://www.hri.res.in/~sdeo/

On Kneser Conjecture
Let us take a set X having 2n+k points. Suppose σ denotes the set of all subsets of X having n
elements. Partition the set σ into k+1 disjoint classes. Then Kneser conjecture says that there is a class
which has a pair of two disjoint n-sets. In this talk we will present a clever beautiful proof given by I
Barany using the Borsuk-Ulam theorem and a theorem due to Gale.

Prof Parameswaran Sankaran
Institute of Mathematical Sciences, Chennai
https://www.imsc.res.in/~sankaran/

Homeomorphisms of the circle and their rotation numbers
Poincare dened the notion of the rotation number of a homeomorphism of a circle. It is
invariant under topological semi-conjugacy. This notion leads to an important classication
theorem for homeomorphisms of the circle. I will talk about the classication theorem and some
related results.

 


 
Time Table

 

  Date 8:45-9:15 9:15-9:30 09:30-10:00 10:00- 11:30 11:30 -1:00 1- 2:30 2:30- 3:30 3:30- 4:30 4:30- 5:00
Day-1 09/11/17 Thursday Registration Inauguration Tea Lecture-1 SPT-1 Lecture-2 ATS-1 Lunch Tutorial SPT, JKM Tutorial ATS, HKS Tea

 

    Lecture-1 Tea Lecture-2 Lunch TutorSpecial ial or Lecture Tea & Snacks
  Date 9:30-11:00 11:00- 11:30 11:30 -1:00 1-2:30 2:30-3:30 3:30-4:30 4:30- 5:00
Day-2 10/11/17 Friday SPT-2 T ATS-2 L SPT, JKM ATS, HKS T
Day-3 11/11/17 Saturday ATS-3 E TBS-1 U SPT, JKM ATS, HKS E
        SUNDAY        
Day-4 13/11/17 Monday ATS-4 A TBS-2 N SPT, JKM ATS, HKS A
Day-5 14/11/17 Tuesday SD-1 B SSK-1 C SPT, JKM ATS, HKS B
Day-6 15/11/17 Wednesday SD-2 R SSK-2 H SPT, JKM ATS, JK R
Day-7 16/11/17 Thursday SSK-3 E TBS-3   ATS, JK Special Lecture-AA E
Day-8 17/11/17 Friday SD-3 A SSK-4 B Special Lecture-SD ATS, JK A
Day-9 18/11/17 Saturday SD-4 K TBS-4 R SPT, JKM ATS, HKS K
        SUNDAY        
Day-10 20/11/17 Monday SSK-5   HM-1 E JK, JKM ATS, HKS  
Day-11 21/11/17 Tuesday HM-2   HM-3 A JK, JKM Test/ Assessment  
Day-12 22/11/17 Wednesday SD-5   HM-4 K Special Lecture-PS Feedback/ Velidictory Function  

 

 

AA : Alejandro Adem, University of British Columbia, Canada
SD : Satya Deo, Harish-Chandra Research Institute, Allahabad
SSK : S S Khare, North Eastern Hill University,, Shillong
HM : Himadri Mukerjee, North Eastern Hill University, Shillong
PS : Parmeswaran Sankaran, Institute of Mathematical Sciences, Chennai
TBS : Tej Bahadur Singh, University of Delhi
ATS : Angom Tiken Singh, North Eastern Hill University, Shillong
JKM : Jitendra Kumar Maitra, R D University, Jabalpur
SPT : Satya Prakash Tripathi, Kirori Mal College, University of Delhi
HKS : Hemant Kumar Singh, University of Delhi
JK : Jaspreet Kaur, St. Stephen’s College, University of Delhi

 


 

Actual Participants 

Sr

SID

Full Name

Gender

Affiliation

Position in College/ University

Year of Passing M.Sc./ M.A.

1

14872

Mr. Arijit Mukherjee

Male

Hyderabad Central University

Ph.D. Student

2014

2

14875

Mr. Debashis Bhowmik

Male

IIT Patna

Ph.D. Student

2014

3

14861

Mr. Umar Mahammad Khan

Male

Aligarh Muslim University

Ph.D. Student

2015

4

14627

Ms. Baljinder Kaur

Female

Central University of Punjab

Ph.D. Student

2013

5

14599

Mr. Sachin Pathak

Male

IIT Patna

Ph.D. Student

2015

6

13598

Mr. Santu Dey

Male

Jadavpur University

Ph.D. Student

2012

7

13381

Mr. Pankaj Kumar Singh

Male

MNIT, Allahabad

Ph.D. Student

2014

8

13980

Mr. Chandan Kumar Mondal

Male

The University of Burdwan

Ph.D. Student

2015

9

14217

Mr. Sumanjit Sarkar

Male

Jadavpur University

Ph.D. Student

2014

10

13945

Mr. Pradip Mandal

Male

Burdwan University

Ph.D. Student

2016

11

13965

Ms. Priyanka Singh

Female

Dr. Hari Singh Gour Central University

Ph.D. Student

2015

12

14817

Mr. Sk Rabiul Islam

Male

IIT Patna

Ph.D. Student

2016

13

14034

Ms. Karishma Shravan

Female

Tripura University

Ph.D. Student

2014

14

13339

Dr. Jitender Singh

Male

Gurunanak Dev University

Assistant Professor

2002

15

14097

Ms. M Rowthri

Female

Sri Sarda College for Women, Fairlanda, Salem

Ph.D. Student

2014

16

15044

Dr Saranya Subbaiyan

Female

Vignan’s University, Guntur

Assistant Professor

2009

17

14274

Mr. Rahul Dutta

Male

Tezpur University

M.Sc. Student

Awaited

18

13494

Mr. Ajay Raj

Male

MS University of Baroda

M.Sc. Student

Awaited

19

13533

Mr. Dipen Ganguly

Male

University of Calcutta

M.Sc. Student

Awaited

20

13359

Mr. Nirmal Kotal

Male

IIT Kharagpur

M.Sc. Student

Awaited

21

13515

Mr. Manpreet Singh

Male

IIT Delhi

Ph.D Student

2014

22

13372

Ms. Jyoti Sharma

Female

University of Delhi

Ph.D Student

2015

23

13357

Ms. Anju Kumari

Female

University of Delhi

M.Phil Student

2014

24

14548

Ms. Gunjan Singh

Female

University of Delhi

Ph.D Student

 

25

14023

Mr. Abdul Gaffar Khan

Male

University of Delhi

Ph.D Student

2016

26

13893

Ms. Aliya Naaz Siddiquis

Female

Jamia Millia Islamia

Ph.D Student

201

27

13347

Mr. Harshit Mathur

Male

University of Delhi

Ph.D Student

2013

28

13894

Ms. Pooja Bansal

Female

Jamia Millia Islamia

Ph.D Student

2014

29

14099

Dr. Majid Ali Choudhary

Male

Zakir Husain Delhi College

Assistant Professor

2008

30

14970

Mr. Beenu Singh

Male

University of Delhi

PhD

2015

31

14972

Mr. Sheetal Luthra

Male

University of Delhi

PhD Student

2015

32

14909

Ms. Barkha Kashyap

Female

University of Delhi

PhD Scholar

2013

33

13578

Mr. Ankush Goel

Male

Kirorimal College

M.Sc

2017

34

13524

Mr Tushar Bag

Male

IIT Patna

Ph.D. Student

2014

35

15014

Mr Abhyendra Prasad

Male

IIT Patna

Ph.D. Student

2014

36

14090

Ms. Tamilmani Jambu

Female

Sri Sarda College for Women, Fairlanda, Salem

Ph.D. Student

2014

37

14980

Mr. Puneet Kumar Gupt

Male

University of Lucknow

Ph.D. Student

2012

38

14076

Ms Nandhini Rajamannan

Female

Sri Sarda College for Women, Fairlanda, Salem

Ph.D. Student

2014

 

 


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