Prossimi seminari del Dipartimento di Matematica

Multiplex networks are used to model complex systems from myriad applications. They generalize classical complex networks by recording different types of relationships, different interactions, or changing interactions over time between the same entities in different layers. They possess natural linear algebraic representations in terms of structured matrices, which makes efficient numerical linear algebra techniques a valuable tool for their analysis. In this talk, we give an overview over several network science problems that can be formulated in terms of matrix function expressions, which we approximate by polynomial and rational Krylov methods. We discuss centrality measures, the solution of stiff systems of non-linear differential equations with exponential Runge--Kutta integrators, as well as un- and semi-supervised community detection. Additionally, we present a nonlinear spectral method for core-periphery detection in multiplex networks. All presented methods have a linear runtime scaling, which allows the treatment of large-scale multiplex networks and we present numerical experiments for all considered problems.

Moduli of stable sheaves are interesting objects which reflect several properties of underlying spaces. In the theory of algebraic varieties, the canonical bundle is an important invariant. In particular variety with trivial canonical bundle are particulary important. For surfaces, K3 surfaces and abelian surfaces are the examples, and there are many works on these surfaces. Mukai proved that the moduli of stable sheaves on these surfaces are also good by showing they are holomorphic symplectic monifolds. Mukai also invented a quite important tool called Fourier-Mukai transform. With these machinary, he conjectured many interesting phenomena on the moduli spaces in 1980's, and most of them are confirmed affirmatively by Bridgeland's works on Fourier-Mukai transforms and stability conditions. In this lecture, I will explain some results on the moduli of stable sheaves on abelian surfaces including some of Mukai's conjectures: 1. Semi-homogeneous vector bundles and Fourier-Mukai transforms. 2. Stability conditions on abelian surfaces. 3. Moduli of stable sheaves and the Bogomolov factor. 4. Birational invariants and birational automorphisms of moduli spaces.