AIS Algebraic Topology (2016)
|Venue:||North-Eastern Hill University, (NEHU) Campus, Shillong|
|Dates:||13th June to 2nd July, 2016|
|Convener(s)||Speakers, Syllabus and Time table||Applicants/Participants|
|Name||Himadri Kumar Mukerjee||Goutam Mukherjee||Angom Tiken Singh|
|Mailing Address||Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-
793022, Meghalaya, India.
|Indian Statistical Institute,
203 B.T. Road, Kolkata- 700108,
West Bengal, India.
|Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-
793022, Meghalaya, India.
Speakers with their affiliations
|S. No||Name of Associates||Affiliation|
|1.||Swagata Sarkar (SS)||CEBS, Mumbai|
|2.||Keerti Vardhan (KV)||Punjab University, Chandigarh|
|S. No||Name of Resource persons||Affiliation|
|1.||Somnath Basu (S/B)||RKMV Univ. Belur, Howrah|
|2.||Samik Basu (SNB)||RKMV Univ. Belur, Howrah|
|3.||Amiya Mukherjee (AM)||ISI Kolkata|
|4.||Mahuya Dutta (MD)||ISI Kolkata|
|5.||Goutam Mukherjee (GM)||ISI Kolkata|
|6.||Himadri Kumar Mukerjee (HKM)||NEHU, Shillong|
|7.||Angom Tiken Singh (ATS)||NEHU, Shillong|
Topics actually covered:
|Topic 1 (HKM/SS)||6 (90 minute)||Cohomology, cup products, orientations, Poincare duality. Steenrod operations. Generalized cohomology theories. K-theory as an example of a generalized cohomology theory.|
|Topic 2 (ATS/KV)||6 (90 minute)||Definition of homotopy groups, Fibrations and induced exact sequences. Some elementary differential topology : smooth manifolds, regular values, critical values, Sard's theorem. Computation of \pi_k(S^n) for k<n. A short introduction to the theory of orientations of manifolds, degree and an idea of the Hopf degree theorem.|
|Topic 3 (AM/GM)||6 (90 minute)||Spectral sequences, the Serre spectral sequence associated to a fibration, some elementary computation. Proof of Serre's theorem on the finiteness of the homotopy groups of spheres. Eilenberg MacLane spaces. The Steenrod algebra. Computation of H^*(K(Z/p,n);Z/p).|
|Topic 4 (MD)||6 (90 minute)||(Morse theory) Non-degenerate smooth functions on a manifold (existence). Determination of the homotopy type in terms of the critical values. Examples. Morse inequalities. The Lefschetz theorem on hyperplane sections. Other applications.|
|Topic 5 (SB)||6 (90 minute)||Adams spectral sequences. Easy computations with K-theory. The Hopf invariant one theorem. Computation of cobordism groups - unoriented, oriented, complex. Formal group laws and Quillen's theorem.|
|Topic 6 (SNB)||6 (90 minute)||Free loop space fibrations, string topology. Loop spaces of surfaces and Birkhoff's curve shortening. Geodesics as critical points. Energy functional on the free loop space. Bumpy metrics and Morse-Bott functions. Gromoll-Meyer's result and Gromov's conjecture.|
|Lecture 1||Tea||Lecture 2||Lunch/ Library consultation||Tutorials||Tea||Tutorials||Snacks|
|Day||Date||9.30- 11.00||11.00 - 11.30||11.30 - 13.00||13.00-15.00||15.00 - 16.00||16.00- 16.30||16.30-17.30||17.30 – 18.00|
|Monday||13-06-2016||Topic 1||Topic 2||On topics 1 & 2||On topics 1 & 2|
|Tuesday||14-06-2016||Topic 1||Topic 2||On topics 1 & 2||On topics 1 & 2|
|Wednesday||15-06-2016||Topic 1||Topic 2||On topics 1 & 2||On topics 1 & 2|
|Thursday||16-06-2016||Topic 1||Topic 2||On topics 1 & 2||On topics 1 & 2|
|Friday||17-06-2016||Topic 1||Topic 2||On topics 1 & 2||On topics 1 & 2|
|Saturday||18-06-2016||Topic 1||Topic 2||On topics 1 & 2||On topics 1 & 2|
|Monday||20-06-2016||Topic 3||Topic 4||On topics 3 & 4||On topics 3 & 4|
|Tuesday||21-06-2016||Topic 3||Topic 4||On topics 3 & 4||On topics 3 & 4|
|Wednesday||22-06-2016||Topic 3||Topic 4||On topics 3 & 4||On topics 3 & 4|
|Thursday||23-06-2016||Topic 3||Topic 4||On topics 3 & 4||On topics 3 & 4|
|Friday||24-06-2016||Topic 3||Topic 4||On topics 3 & 4||On topics 3 & 4|
|Saturday||25-06-2016||Topic 3||Topic 4||On topics 3 & 4||On topics 3 & 4|
|Monday||27-06-2016||Topic 5||Topic 6||On topics 5 & 6||On topics 5 & 6|
|Tuesday||28-06-2016||Topic 5||Topic 6||On topics 5 & 6||On topics 5 & 6|
|Wednesday||29-06-2016||Topic 5||Topic 6||On topics 5 & 6||On topics 5 & 6|
|Thursday||30-06-2016||Topic 5||Topic 6||On topics 5 & 6||On topics 5 & 6|
|Friday||01-07-2016||Topic 5||Topic 6||On topics 5 & 6||On topics 5 & 6|
|Saturday||02-07-2016||Topic 5||Topic 6||On topics 5 & 6||On topics 5 & 6|
- Marvin J. Greenberg and John R. Harper - Algebraic Topology – a first course, Benjamin, 1981
- Edwin H. Spanier - Algebraic Topology, Tata-Mc Graw Hill
- Allen Hatcher - Algebraic Topology, 2001
- Robert M. Switzer - Algebraic Topology- Homotopy and Homology, Sringer-Verlag, 1976.
- R.E. Mosher and M.C. Tangora – Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, 1968.
- Adams, J.F. Stable Homotopy and Generalised Homology.
- Complex cobordism and stable homotopy groups of spheres, D. Ravenel. Academic Press.
- Milnor - Morse Theory.
- Gromoll, Meyer - Periodic geodesics on compact Riemannian manifolds.
- Bott - On the iteration of closed geodesics and Strum intersection theory.
- 11. Further reading:
- Gromoll, Meyer - On differentiable functions with isolated critical points.
- Abraham - Transversality in manifold of mappings.
- Anant R. Shastri: Basic Algebraic Topology, CRC Press, Taylor and Francis group, 2013.
- MILNOR J., ``Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007. Surgery theory was initiated by Milnor in the period 1956-1960 for the purpose of classifying smooth manifolds that are homotopy equivalent to a sphere of dimension \geq 5. In addition to published research articles, this volume contains four sets of notes for lectures he gave in this period; these lectures had remained in mimeographed form until their formal publication in this volume. They still provide excellent instruction in differential topology.
- MILNOR J., Morse theory, (based on lecture notes by M. Spivak and R. Wells) Annals of Mathematical Studies No. 51, Princeton University Press, 1963.
- MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
- MILNOR J. and STASHEFF J., Characteristic classes, Annals of Mathematics Studies No. 76, Princeton University Press, 1974. This monograph is based on more succinct mimeographed notes by Milnor and Stasheff for lectures of Milnor given in 1959 and still available at <http://www.maths.ed.ac.uk/~aar/papers/milnorcc.pdf>.
- Mukherjee, Amiya, Topics in Differential Topology, Hindistan Book Agency, 2005.
- Shastri, A.R., Elements of Differential Topology, CRC Press, 2011.
- STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.
There are many more new books and resources in the internet.
|Sr||SID||Full Name||Gender||Affiliation||Position in College/ University||University /Institute M.Sc./ M.A.||Year of Passing M.Sc./ M.A|
|1||7379||Mr. Lokenath Kundu||Male||Ramakrishna Mission Vivekananda University||PhD||Visva-Bharati||2013|
|2||7542||Mr. Aritra Bhowmick||Male||Indian Statistical Institute, Kolkata||PhD||Indian Institute of Technology, Kanpur||2015|
|3||7720||Mr. Ripan Saha||Male||IISER Mohali||PhD||IIT Kanpur||2013|
|4||7808||Mr Pravakar Paul||Male||Indian Statistical Institute||MSc Student||Indian Statistical Institute||Awaiting Result|
|5||7829||Mr. Satyendra Kumar Mishra||Male||IIT KANPUR||PhD||IIT KANPUR||2013|
|6||7831||Mr. Abhishek Sarkar||Male||IIT, KANPUR||PhD Student||VISVA-BHARATI||2013|
|7||7876||Mr Rishi Kumar||Male||IIT Kanpur||Project Associate||IIT Kanpur||2015|
|8||7941||Mr. Sourav Kanti Patra||Male||Ramakrishna Mission Vidyamandira||PhD Student||Chennai Mathematical Institute||2015|
|9||7945||Mr. Bhaskar Palit||Male||Ramakrishna Mission Vidyamandira||PhD||Ramakrishna Mission Vidyamandira||2015|
|10||8002||Ms Pinka Dey||Female||IISER MOHALI||PhD||JADAVPUR UNIVERSITY, WEST BENGAL,KOLKATA||2014|
|11||8023||Mr. Suraj Prakash Yadav||Male||IIT Kanpur||MSc Student||IIT Kanpur||Awaiting Result|
|12||8059||Mr. Abhra Abir Kundu||Male||Indian Statistical Institute||MMath Student||Indian Statistical Institute M.Math||Awaiting Result|
|13||8084||Mr. Arijit Mukherjee||Male||Hyderabad Central University||PhD||University of Calcutta||2014|
|14||8126||Mr. Anubhav Mukherjee||Male||Indian Statistical Institute||M-Math Student||Indian Statistical Institute||Awaiting Result|
|15||7648||Mr. Chinglensana Phanjoubam||Male||North Eastern Hill University||Asst. Prof.||Delhi University||2006|
|16||7649||Mr. Sainkupar Marwein Mawiong||Male||North Eastern Hill University Shillong||Asst. Prof.||North Eastern Hill University||2004|
|17||7672||Mr. Dipen Ranjan Panging||Male||North-Eastern Hill University||M.Phil||Tezpur University||2012|
|18||8066||Ms Swathi Krishna||Female||IISER Mohali||PhD student||University of Hyderabad||2013|
|19||8119||Mr. Ranendu Adhikary||Male||Jadavpur University||PhD||Jadavpur University||2013|
|20||7674||Mr. Sudhir Kumar||Male||IISER||Bs-Ms student||IISER Pune||Awaiting Result|
How to reach
- Come to Guwahati Airport/Railway Station by Air/Train.
- Accommodation is booked in the New NEHU Guest House, NEHU Campus, Shillong-793022.
- Enter the campus by Gate no. 2 and at the gate ask the security guards for the direction to the New NEHU Guest House.
- Take a shared Taxi from Airport to NEHU campus, or take a shared Taxi/ Sumo from the Railway station to NEHU Campus.
(Most taxis go to town and NEHU Campus is out of the way, so you may have to take a local taxi from town to the NEHU Campus, which will charge about Rs. 200/-. Sometimes the taxi from Guwahati agrees to drop you at New Guest House at the NEHU Campus for this they may charge some extra money say Rs. 150 to 200 in addition to usual fare)
- It may be raining some times, so carry umbrellas/raincoats. On a rainy day temperature drops so carry at least a half sweater.
- Important local telephone/mobile No.s :
- 0364-2725800 (New NEHU Guest House),
- 09863020684 (Angom Tiken Singh)
- 09863022257 (Sainkupar Mawiong)
- 09402951147 (Ph Chinglensana)