Course 2024-2025 a.y.

• Complex analysis:
  • Complex numbers and Euler's formula
  • Holomorphic functions and Cauchy's integral theorem for basic curves
  • Homotopy between loops and fundamental group
  • Cauchy's integral formula
  • Power series representation
  • Relation to Fourier series and harmonic functions
  • Special holomorphic functions such as exp and log
  • Poles and residues
  • Applications of residues to compute particular integrals
  • If time allows: covering spaces, ramified coverings, Riemann surfaces

• Basics of differential geometry:
  • Inverse and implicit function theorem
  • Equivalent definitions of embedded differentiable manifolds
  • Tangent space and differential of smooth maps between manifolds
  • Constrained optimization: Lagrange multipliers
  • Multilinear alternating forms and differential forms
  • Stokes' theorem (statement)
  • Riemannian metrics and geodesics
  • Exponential map and characterizations of failure of injectivity (statement)
  • Completeness and Hopf-Rinow theorem (statement)
  • If time allows: basic notions of curvature and classical results relating geodesics, curvature, and topology (statements)

• Degree theory:
  • Definition of degree of smooth maps between manifolds
  • Invariance under smooth homotopies and independence on the codomain point
  • Characterization using volume forms (statement)
  • Brouwer's fixed-point theorem
  • Applications to topology and Nash equilibria in game theory

• Know and apply fundamental notions in complex analysis: holomorphic functions, Cauchy's integral formula, power series, calculus of residues
• Know and apply fundamental notions in differential geometry: concrete examples of manifolds, parametrization, tangent space, differential of maps, metrics, geodesics, differential forms, Stokes' theorem
• Solve constrained optimization problems using Lagrange multipliers
• Understand degree theory and its topological applications such as Brouwer's fixed-point theorem

• Apply the calculus of residues in simple settings
• Solve constrained optimization problems in simple settings
• Operate with objects and tools from complex analysis and differential geometry
• Make use of the presented methodological tools in applied sciences such as computer science and physics

• Lectures
• Practical Exercises
• Individual works / Assignments

Online lectures have the same conceptual role as face-to-face lectures. The actual choice between offering face-to-face lectures, online lectures, or both will depend on the university's policies.

Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.

Students will be evaluated on the basis of written exams, which can be taken in one of the two following ways. 

The exam can be split into two partial exams. Each partial may contain multiple-choice questions and open-answer questions; each partial weighs for one-half of the final mark. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating: 

• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects. 
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

The exam can also be taken as a single general exam, which contains both multiple-choice questions and open-answer questions. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year. This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating: 

• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects. 
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

 
We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi.
 

• Lecture notes (main reference, updated regularly on Blackboard)
   
• Needham, Tristan. Visual complex analysis. Clarendon Press (new edition), 1999. ISBN: 978-0-1985-3446-4
   
• Giaquinta, Mariano;  Modica, Giuseppe. Mathematical analysis. An Introduction to functions of several variables.
  Birkhäuser Boston, Inc., Boston, MA,  2009. ISBN: 978-0-8176-4374-4