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Ph.D. Course Details

Elements of Geometric Analysis in finite and infinite dimensions

Lecturer

Valentino Magnani (Università di Pisa)
valentino.magnani@unipi.it

Description

In the first part of the course, we provide an overview of various results and problems arising from different synthetic formulations of Geometric Analysis. We start by introducing general tools from Analysis in Metric spaces, especially in relation to geodesics, Sobolev spaces, and differentiation of measures. Then we describe the main differences between the “commutative’’ Geometric Analysis of Euclidean spaces and the “noncommutative’’ Geometric Analysis of the so-called homogeneous Lie groups. A special attention will be devoted to the problem of computing the intrinsic area of submanifolds in a noncommutative homogeneous group, pointing out some recent open questions. In the second part of the course, we mainly study some classes of infinite dimensional Lie groups. We introduce weak and strong Riemannian metrics, the Levi-Civita covariant derivative, the concept of curvature and geodesic distances in infinite dimensions. We also present some counterexamples and problems that only appear in infinite dimensions. An astonishing phenomenon is the vanishing of the geodesic distance, which P. Michor and D. Mumford conjectured to be related to the blow-up of the sectional curvature. Finally, some rigidity results concerning biLipschitz embeddings of homogeneous groups into Banach spaces will be discussed in relation to general versions of Rademacher’s differentiation theorem. **Prerequisites** Basic facts about Banach spaces and basic notions of Differential Geometry, Measure Theory, and Multilinear Algebra.

Scheduled lessons

  • 2025-02-28 16:00 (60 minutes), Aula Seminari (Department of Mathematics).
  • 2025-03-11 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-03-14 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-03-18 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-03-21 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-03-25 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-03-28 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-04-01 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-04-04 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-04-08 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-04-11 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-04-15 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-04-18 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-04-22 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-04-29 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-05-02 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-05-06 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-05-09 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-05-13 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-05-16 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-05-20 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-05-23 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).
  • 2025-05-27 14:00 (120 minutes), Aula O1 (Polo Fibonacci, University of Pisa).
  • 2025-05-30 16:00 (120 minutes), Aula G (Polo Fibonacci, University of Pisa).

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