The ATM workshop will consist of two components:

1. 3 sets of minicourses on geometry and topology. These will aim at preparing the students for working in one of the areas of the minicourses.

2. One day devoted to research talks on Ricci flow.

**The following will be the resource persons.**

Name |
Affiliation |
Duration | Topic |

Mahuya Datta (With local support from Dheeraj Kulkarni) |
ISI Kolkata | 6 lectures 1.5 hrs each | Introduction to Symplectic Topology |

Chitrabhanu Choudhuri |
IISER Pune | 6 lectures 1.5 hrs each | Introduction to Rational Homotopy Theory |

Anish Ghosh |
TIFR, Mumbai | 6 lectures 1.5 hrs each | Introduction to Hyperbolic Geometry |

Harish Seshadri |
IISc Bangalore | 3 lectures 1 hr each | Research talks on Ricci flow |

** Target Audience:**

PhD students working in Geometry and Topology.

**Syllabus and Time-table**

**Symplectic Topology**:

Mahuya Datta, (with local support from Dheeraj Kulkarni)– 6 lectures 1.5hrs each. Mon-Sat: 9:30 - 11 am.

1. Basic Theory of Symplectic manifolds, Symplectic vector spaces, Symplectic manifolds; examples Symplectomorphisms, Hamiltonian vector fields and flows

2. Tautological form on cotangent bundles

3. Stability theorem, Moser’s stability theorems, Moser-Weinstein theorem on lagrangian submanifolds, Lagrangian intersections and fixed points of symplectomorphisms

4. Symplectic reduction, Coisotropic reduction of a symplectic manifold, Hamiltonian action of a Lie group, The Mersden-Weinstein quotient, Hamiltonian circle action

5. Symplectic topology of Euclidean spaces: (at most 2 from below)

(a) Weinstein conjecture

(b) Non-squeezing theorem

(c) Poincare-Birkoff theorem. Billiard problem.

**Rational Homotopy Theory:**Chitrabhanu Choudhuri, (with local support from Samik Basu, Somnath Basu):– 6 lectures 1.5 hrs each. Mon-Sat (14-19 December 2015): 11:30 - 1pm.

Homtopy groups and Rational Homtopy groups of a space. Localising spaces with respect to Q. Cohomology of K(Q,n). The rational cochain algebra of a space - A(X). Sullivan algebras, minimal models. Computing rational homotopy groups of a space using minimal models. Postnikov towers, Hirsch extensions. Formal spaces. Formality of Kahler manifolds.

**Hyperbolic Geometry:**Anish Ghosh, (with local support from Kingshook Biswas, Mahan Mj): Introduction to Hyperbolic Geometry– 6 lectures 1.5 hrs each.Mon-Sat: 2:30 - 4 pm.

We shall start with the geometry of the hyperbolic plane and hyperbolic space (isometries, geodesics, etc.) and then move on to ergodic theory on the modular surface and (if time permits) on more general hyperbolic manifolds.

**Ricci Flow**Harish Seshadri, Soma Maity, Atreyee Bhattacharya:

3 research talks at 10-11, 11:30-12:30, 2-3, Monday 21 December 2015.