MAU34308 Techniques in Geometry

Module Code	MAU34308
Module Title	Techniques in Geometry
Semester taught	Semester 2
ECTS Credits	5
Module Lecturer	Prof. Nikhil Savale
Module Prerequisites	MAU22206/MAU34214 Calculus on Maniforlds

This module is examined in a 2-hour examination at the end of Semester 2.
Continuous assessment contributes 20% towards the overall mark.
The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows:
1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session;
2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam;
3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental session.
Capping of reassessments applies to Theoretical Physics (TR035) students enrolled in this module. See full text at https://www.tcd.ie/teaching-learning/academic-affairs/ug-prog-award-regs/derogations/by-school.php Select the year and scroll to the School of Physics.

11 weeks of teaching with 3 lectures per week.

On successful completion of this module, students will be able to

explain notions of symplectic manifolds and geometric formulation of Hamiltonian mechanics.
demonstrate understanding of complex manifolds and some applications of Hodge thery.
use notions of Morse theory and some of its applications to differential topology.

Differential forms and de Rham cohomology
Symplectic manifolds; symplectomorphisms; Lagrangian submanifolds
Darboux and Moser theorems, Lagrangian neighborhood theorem
Complex vector bundles
Almost-complex structures, compatibility, integrability
Kähler manifolds, Dolbeault cohomology, Hodge theory, projective embeddings
Morse functions, Morse lemma, sublevel sets and attaching cells, Morse inequalities