Lecture Schedule: Difference between revisions

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Lectures

Tentative lecture plan is as follows:

Week 1

Topics: Review of differential geometry

Topological and smooth manifolds,
Vector and tensor fields,
Algebra of differential forms,
De Rham Cohomology

Reading:

Taubes Ch 1
Frenkel Sections 1-3
Nakahara Sections 5.1, 5.2, 5.4. 6.
Sontz Chapters 2, 3, 5

Week 2

Topics: Review of differential geometry

vector fields
Lie brackets
diffeomorphisms
Lie derivatives
Lie groups and Lie algebras.
Exponential maps
adjoint representation
Basics of Symplectic Geometry
Hamiltonian vector fields
Hamiltonian group actions
Moment maps and co-moment maps.

Reading:

Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5

Week 3

Topics:

Further details on smooth actions,
Symplectic structure on cotangent bundles
Quotients of free, proper, smooth actions
Principal bundles.
Frame bundles.
Associated fibre bundles.

Reading:

Week 4

Topics:

Associated fibre bundles: vector bundles, adjoint bundle.
Reduction and extension of structure group.
Group of gauge transformations.
Linear connections.

Reading:

Week 5

Topics:

Connections: Koszul connections, principal connections, Ehresmann connections.
Equivalences between them.
Atiyah sequence and differential operators.

Reading:

@@ Line 1: / Line 1: @@
-====== Lectures =======
+== Lectures ==
 Tentative lecture plan is as follows:
@@ Line 6: / Line 5: @@
 Topics: Review of differential geometry
-  * Topological and smooth manifolds,
+* [[Topological and smooth manifolds]],
-  * Vector and tensor fields,
+* Vector and tensor fields,
-  * Algebra of differential forms,
+* Algebra of differential forms,
-  * De Rham Cohomology
+* De Rham Cohomology
 Reading:
-  * Taubes Ch 1
+* Taubes Ch 1
-  * Frenkel Sections 1-3
+* Frenkel Sections 1-3
-  * Nakahara Sections 5.1, 5.2, 5.4. 6.
+* Nakahara Sections 5.1, 5.2, 5.4. 6.
-  * Sontz Chapters 2, 3, 5
+* Sontz Chapters 2, 3, 5
 === Week 2 ===
 Topics: Review of differential geometry
-  * vector fields
+* vector fields
-  * Lie brackets
+* Lie brackets
-  * diffeomorphisms
+* diffeomorphisms
-  * Lie derivatives
+* Lie derivatives
-  * Lie groups and Lie algebras.
+* Lie groups and Lie algebras.
-  * Exponential maps
+* Exponential maps
-  * adjoint representation
+* adjoint representation
-  * Basics of Symplectic Geometry
+* Basics of Symplectic Geometry
-  * Hamiltonian vector fields
+* Hamiltonian vector fields
-  * Hamiltonian group actions
+* Hamiltonian group actions
-  * Moment maps and co-moment maps.
+* Moment maps and co-moment maps.
 Reading:
-  * Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5
+* Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5
 === Week 3 ===
 Topics:
-  * Further details on smooth actions,
+* Further details on smooth actions,
-  * Symplectic structure on cotangent bundles
+* Symplectic structure on cotangent bundles
-  * Quotients of free, proper, smooth actions
+* Quotients of free, proper, smooth actions
-  * Principal bundles.
+* Principal bundles.
-  * Frame bundles.
+* Frame bundles.
-  * Associated fibre bundles.
+* Associated fibre bundles.
@@ Line 53: / Line 52: @@
 Topics:
-  * Associated fibre bundles: vector bundles, adjoint bundle.
+* Associated fibre bundles: vector bundles, adjoint bundle.
-  * Reduction and extension of structure group.
+* Reduction and extension of structure group.
-  * Group of gauge transformations.
+* Group of gauge transformations.
-  * Linear connections.
+* Linear connections.
 Reading:
-=== Week 5 ====
+=== Week 5===
 Topics:
-  * Connections: Koszul connections, principal connections, Ehresmann connections.
+* Connections: Koszul connections, principal connections, Ehresmann connections.
-  * Equivalences between them.
+* Equivalences between them.
-  * Atiyah sequence and differential operators.
+* Atiyah sequence and differential operators.
 Reading: