Sumários

Vector bundles - II

27 Março 2024, 15:30 • Giordano Cotti

In this lecture:

  • we introduced the notion of morphisms of vector bundles;
  • isomorphism classification problem for vector bundles: we prove that two vector bundles on a same smooth manifolds are isomorphic if and only if they admit systems of transition functions which are "cobordant";
  • we recalled basic examples of vector bundles (trivial, tangent, cotangent bundles);
  • we introduced several operations of vector bundles: dualization, Whitney direct sum, tensor product. These operations allow to construct new bundles from given ones.

Vector bundles - I

26 Março 2024, 10:00 • Giordano Cotti

In this lecture:

  • we introduced the notion of a real/complex vector bundle on a smooth manifold M;
  • we introduced the notion of transition functions of a vector bundle, and discussed their "cocycle relations";
  • we showed that it is possible to (re)construct a vector bundle from the datum of functions satisfying the cocycle relations.

Multi-linear algebra VI

20 Março 2024, 15:30 • Giordano Cotti

In this lecture:

  • we introduced the tensor algebra T(V) of a finite dimensional vector space;
  • we equipped T(V) of a graded, associative, non-commutative, unital algebra structure;
  • we introduced the notions of symmetric and anti-symmetric (or alternating) multi-linear functions;
  • we introduced the notions of symmetric and alternating tensors, spanning linear subspaces of T(V);
  • we introduced the notions of symmetrizing and anti-symmetrizing maps defined on the tensor algebra;
  • we formulated two universal problems for symmetric and anti-symmetric multi-linear functions, and explained how to solve them using the spaces of symmetric and anti-symmetric tensors, respectively;
  • we defined an algebra structure on the space of anti-symmetric tensors, called exterior algebra of V;
  • we discussed several properties of the exterior algebra of V (bases, dimension, properties of the wedge product);
  • we defined an algebra structure on the space of symmetric tensors, called symmetric tensor algebra of V;
  • we discussed several properties of the symmetric tensor algebra of V (bases, dimension, properties of the symmetric tensor product).
A problem was formulated as left as exercise for the students: given a linear map between two linear spaces E and F, show that there are well-defined induced linear maps from the tensors powers, symmetric tensor powers, and exterior tensors powers of E and F. Moreover, if the given linear map is represented by a matrix M (with respect two fixed bases of E and F), compute the matrices representing the induced linear maps (with respect the induced bases).

Multi-linear algebra - V

19 Março 2024, 10:00 • Giordano Cotti

During this lecture, we discussed and exemplified the identification of (1,1)-tensors on a vector space V with its endomorphisms. More generally, we exemplified the interpretation of tensors as "generalized matrices". Additionally, we explained how to interpret the trace of an endomorphism as the contraction of the corresponding (1,1)-tensor.

Multi-linear algebra - IV

13 Março 2024, 15:30 • Giordano Cotti

In this lecture:

  • we briefly recall the operations of product and direct sum of vector spaces (by mentioning the corresponding universal properties);
  • we discussed properties of the tensor product as operation on vector spaces (compatibility with the dual, commutativity/associativity up to canonical isomorphism, distributivity property w.r.t. direct sums);
  • we introduced the classical terminology about tensors (covariant, contravariant, and tensors of (r,s)-type), and briefly mentioned the difference with the conventions used in the physical literature;
  • we introduced the notion of "components (w.r.t. a base)" of a (r,s)-type tensor on the vector space V;
  • we derived the transformation rules for the components of a (r,s)-type tensor on V under base change;
  • we introduced Einstein convention on sums over repeated indices;
  • we made explicit how the tensor product operations read in components;
  • we introduced the notion of "contraction" of a tensor w.r.t. a covariant index and a contravariant one.