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Summary.
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| The course is aimed to revisit some
elementary notions of multilinear algebra (such as
tensor algebras, exterior algebras, and
determinant theory) from an advanced point of
view, in order to study the elementary geometry of
Grassmannian manifolds parametrizing
fixed-dimensional vector subspaces in an ambient,
possibly infinite-dimensional, vector space. The
lectures will be interdisciplinary in nature.
Grassmannian manifolds are special and simple
algebraic varieties that nevertheless allow the
discovery of the prototypes of many important
concepts, still widely studied, such as moduli
spaces, classiying spaces, and intersection
theory. |
Programme.
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- Elementary linear
algebra from an advanced perspective: modules
over a ring, algebras, free modules.
Submodules. Ideals;
- Vector spaces as modules over a field.
Vector subspaces, bases, dimension,
homomorphisms. The dual vector space;
- Construction of real and complex projective
space and its projective Pluecker embedding;
- The Riemann sphere. Set-theoretical
definition of the complex Grassmann manifold,
its structure as an algebraic manifold, and
its Plücker embedding in projective space;
- Tautological sequences of vector bundles
over the Grassmannian. The Grassmannian as the
first example of a fine moduli space;
- Intersection theory of the Grassmannian as a
generalization of Bézout's theorem;
- Open problems session.
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Prerequisites.
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Basic on Linear
Algebra and Calculus 1 and 2. Basics of abstract
algebra. What is a group, a ring, a module over a
ring, an algebra over a ring R, a derivation of an
R-algebra. Basics in analytic geometry (linear
subvarieties: lines, planes, hyperplanes...)
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References.
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Lecture notes will be prepared
along the course delivering. The course will be a
proper subset of the content of the books
- J. Harris, Algebraic Geometry, A
First Course, Springer GTM 133,
1992;
- L. Gatto, P. Salehyan, Hasse-Schmidt
Derivations on Grassmann Algebras,
Springer International, 2018;
- S. Amukugo, E. Lazarus, J. Lichela, G.
Marelli, M. Mugochi, The LMS-MARM Lectures
on "Linear ODE, an algebraic
perspective", 2025, forthcoming.
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