Smooth maps between arbitrary subsets of Euclidean spaces. Diffeomorphisms. Boundaryless differentiable manifolds. Tangent space at a point to a manifold. Tangent bundle. Derivative of a map between manifolds. Sard's Lemma. Manifolds with boundary. Module 2 degree. Brouwer degree. Brouwer's fixed point theorem. Kakutani's example. Shauder's fixed point theorem. Continuation principle. Applications to differential and integral equations. Elements of bifurcation theory.
Furi M., Lectures registry (top-diff_14-15.pdf), http://www.dma.unifi.it/~furi/didattica/registri/top/
Guillemin V. - Pollak A., Differential Topology, Prentice-Hall, Inc., 1974.
Hirsch M.W., Differential Topology, Graduate Texts in Math., Vol. 33, Springer Verlag, 1976.
Lloyd N.G., Degree Theory, Cambridge Tracts in Mathematics, Vol. 73, Cambridge University Press, 1978.
Milnor J.W., Topology from the differentiable viewpoint, The Univ. Press of Virginia, 1965.
Spivak M., Calculus on Manifold, W.A. Benjamin, Inc., 1965.
Learning Objectives
Knowledge acquired: Being aware of mathematical concepts carried out during the lectures.
Competence acquired: Being able to join each other the concept carried out during the lectures.
Skills acquired (at the end of the course): Being able to solve the problems assigned during the lectures.
Prerequisites
The topics of the courses of Geometry and Calculus I and II. Elements of Functional Analysis. Basic knowledge of Lebesgue measure.
Teaching Methods
Lectures and tutorials. Development and discussion of homework.
Further information
Office hours: see the web page http://www.dma.unifi.it/~furi/
Type of Assessment
Oral exam.
Course program
Course Contents (detailed programme):
PRELIMINARIES
Smooth maps between arbitrary subsets of Euclidean spaces. Diffeomorphisms. Inverse function theorem. Implicit function theorem. Tangent cone at a point of a subset of an Euclidean space. Tangent space (i.e. the space spanned by the tangent cone). Differential (at a point) of a smooth map between arbitrary subsets of Euclidean spaces (as restriction to the tangent space of the differential of any smooth local extension). Tangent bundle of a subset of an Euclidean space. Singular subset of a set and its invariance under a diffeomorphism. Maxima and minima of a real function defined on a subset of an Euclidean space. Necessary conditions (of first and second order) for maxima and minima. Sufficient conditions (of first and second order) for maxima and minima.
DIFFERENTIABLE MANIFOLDS
Differentiable manifolds in Euclidean spaces. Manifolds with boundary. Charts and parametrizations. Properties of the tangent space at some point of a manifold. Theorem about regularity of solutions for boundaryless manifolds. Critical points and regular points of a map. Critical values and regular values of a map. Proper maps between metric spaces. A proof of the Fundamental Theorem of Algebra. Sard's Lemma. Theorem about regularity of solutions for manifolds with boundary. Maxima and minima for real functions on differentiable manifolds. Lagrange multipliers (necessary as well as sufficient conditions for maxima and minima of a constrained problem). Orientation of a differentiable manifold. Orientable and non-orientable manifolds.
DEGREE THEORY AND APPLICATIONS
Fixed point property. Retracts of a topological space (and relation with that fixed point property). Brouwer's Fixed Point Theorem. Kakutani's example. Schauder's Fixed Point Theorem. Peano's Existence Theorem for ordinary differential equations. Applications of the Schauder Fixed Point Theorem to some boundary value problems for ordinary differential equations. Continuation theorem in Banach spaces and some applications to boundary value problems for ordinary differential equations. Mod 2 degree for proper maps between (not necessarily orientable) differentiable manifolds. Brouwer degree for proper maps between oriented manifolds. Equivalence between the algebraic degree and the topological degree for complex polynomials. Tangents vector fields on a differentiable manifolds. Hairy ball theorem: you can't comb a hairy ball flat without creating a cowlick. Bifurcation theory in the finite dimensional context. Necessary as well as sufficient conditions for a point to be of bifurcation. Liking number of two closed curves in R^3. Axiomatic theory of Brouwer's degree and topological consequences.