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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