PHD COURSE – ACADEMIC YEAR 2023-24
| PROFESSOR | COURSE | HOURS/ CREDITS | ABSTRACT |
| Prof. Niels Benedikter, Unimi | Mathematical Methods for Many-Body Quantum Systems |
November 2023 15 hours (3 credits) |
The first goal of the course will be the introduction or review of functional analytic methods for the study of quantum mechanical systems. The next goal will be the familiarization with classical rigorous results in the analysis of Schrödinger operators. Ultimately the students will learn be introduced to many-body quantum systems and learn modern analytical methods for their analysis. POSTER |
| Prof. Luigi Lombardi, Unimi | Geometric and numerical invariants of derived categories of sheaves |
December 2023 15 hours (3 credits) |
In algebraic Geometry derived categories of sheaves on projective varieties are objects of homological nature that encode several properties of the underlying variety. One of the main conjectures relating derived categories to birational geometry is the so-called DK-hypothesis. This asks whether K-equivalent varieties have the same derived category. In this course we will survey this problem. We will first introduce the general background regarding triangulaterd categories, derived functors, Fourier-Mukai transforms and semiortogonal decompositions, and then discuss Bridgeland’s result for the 3-dimensional case. Time permitting we will also discuss the problem of the invariance of the number of linearly independent holomorphic one-forms. POSTER |
| Prof. Alberto Vezzani, Unimi | Adic Spaces |
March 2024 15 hours (3 credits) |
Learning the basics of non-archimedean analytic geometry, focusing on recent developments and applications. They will include Huber’s theory of adic spaces, Scholze’s theory of perfectoid spaces, Fargues-Fontaine curves and, if time permits, Clausen-Scholze’s notion of analytic rings. POSTER |
| Prof. Anna Laura Mazzucato, Penn State University | Boundary layers and applications to the vanishing viscosity limit for incompressible fluids |
January 2024 16 hours (3 credits) |
The main goal of the course is to introduce the theory of boundary layers with emphasis on viscous boundary layers for incompressible flows, discussing some recent developments in the rigorous analysis of viscous boundary layers and the vanishing viscosity limit. We will first discuss the case of boundary layers for linear equations (ordinary differential equations and the Laplace equation), in order to present the topics in a context more familiar to the students, and then for fluids leading to Prandtl equation. We will provide the necessary tools to analyze more complex models and to let interested students study open problems from an analytic standpoint. POSTER |
| Prof. Adrian Muntean, Karlstad University, Sweden | Two-scale Convergence Homogenization Techniques and Multiscale Modeling |
February 2024 16 hours (3 credits) |
1.recognise relevant separated scales and types of multiscale problems; 2. select appropriate small parameters required for asymptotic developments; 3. formally scale up microscopic systems using arguments from asymptotic two-scale homogenisation; 4.rigorously scale up microscopic systems with compactness and arguments from two-scale convergence; 5. determine the quality of certain homogenisation strategies by using information from corrector estimates; 6. numerically illustrate the significance of homogenisation of partial differential equations formulated in perforated domains. POSTER |
| Prof. Amnon Neeman, Unimi | Analytic techniques applied to derived and triangulated categories |
January 2024 15 hours (3 credits) |
In algebraic geometry (more specifically in the three exposes by Illusie in SGA6), there was a careful study of several derived categories of coherent/quasicoherent sheaves that one can usefully associate to a scheme X. And the course will be about the recent developments in this old field, progress that came about by adapting some methods from analysis. POSTER |
| Dr. Riccardo Tione, MPI MiS – Leipzig, Germany | Differential inclusions and convex integration |
March 2024 15 hours (3 credits) |
The aim of this course is to overview classical and recent results about differential inclusions. After introducing some general objects and notions, such as Young Measures and elliptic differential inclusions, the course will focus on methods for constructing solutions via convex integration. A variety of examples will be considered, such as the eikonal equation, critical points of polyconvex functionals and very weak solutions to the p-Laplace equation. POSTER |
| Prof. Federico Binda, Unimi | Derived geometry |
February 2025 15 hours (3 credits)
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Learning the basic tools used in derived geometry, with applications ranging from deformation theory to recent breakthroughs in K-theory and (topological) Hochschild homology due to Antieu, Bhatt, Mathew, Morrow, Nikolaus, Scholze and others . We will build the theory from scratch, starting from Toen-Vezzosi’s “classical” approach. |