Lasse Rempe (University of Manchester)
A counterexample to Eremenko\'s conjecture abstract
Abstract:
Let f be an entire function (i.e., a holomorphic self-map of the complex plane), and suppose that f is transcendental, i.e., not a polynomial. The *escaping set* of f consists of those points that tend to infinity under repeated application of f. (For example, all real numbers belong to the escaping set of the exponential map, since they tend to infinity under repeated exponentiation.) In 1989, Eremenko conjectured that every connected component of the escaping set is unbounded. Eremenko\'s conjecture has been a central problem in transcendental dynamics. Prior to our most recent work, a number of stronger versions of the conjecture had been disproved, while weaker ones had been established, and the conjecture has also been shown to hold for a number of classes of functions. I will describe recent work with David Martí-Pete and James Waterman in which we construct a counterexample to the conjecture. The talk should be accessible to a general mathematical audience, including postgraduate students.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Michela Artebani (Universidad de Concepción)
Abstract:
The classification of Mori dream spaces - equivalently, normal projective varieties with finitely generated Cox ring - is closely tied to positivity properties of the anticanonical class. In particular, toric and Fano varieties provide fundamental classes of examples with finitely generated Cox rings. In contrast, Calabi-Yau varieties lie at the boundary of positivity: there is no general criterion deciding when they are Mori dream spaces, and their birational geometry (and birational automorphism groups) can be remarkably rich. In this talk we focus on Calabi--Yau varieties $X$ arising as general anticanonical hypersurfaces of smooth toric Fano varieties $Z$. Our results are formulated in terms of primitive pairs of the anticanonical polytope of $Z$, which is a smooth reflexive polytope. We present two complementary theorems. The first result provides sufficient combinatorial conditions on primitive pairs ensuring that $X$ is a Mori dream space, and it yields an explicit presentation of the Cox ring $R(X)$ in terms of $R(Z)$ and the defining equation of $X$. The second result goes in the opposite direction: the existence of certain relations among primitive pairs forces $\\mathrm{Bir}(X)$ to be infinite, and hence $X$ cannot be a Mori dream space. The proof of the first theorem builds on the approach of Herrera-Laface-Ugaglia on Cox rings of embedded varieties, while the second generalizes ideas of Kawamata and Ottem for anticanonical hypersurfaces in products of projective spaces. As an application, we obtain a complete classification of Mori dream Calabi-Yau hypersurfaces in dimensions $2$ and $3$. In particular, for these hypersurfaces there is a sharp dichotomy: either $R(X)$ is finitely generated or $\\mathrm{Bir}(X)$ is infinite. This is joint work with Antonio Laface and Luca Ugaglia.
10:30 • Universität Basel, Spiegelgasse 5, Seminarraum 05.002
Gabriel Wittum
Parallel Adaptive Simulation of Processes from Science and Engineering abstract
Abstract:
Numerical simulation has become one of the major topics in Computational Science. To promote modelling and simulation of complex problems new strategies are needed allowing for the solution of large, complex model systems. Crucial issues for such strategies are reliability, efficiency, robustness, usability, and versatility.After discussing the needs of large-scale simulation we point out basic simulation strategies such as adaptivity, parallelism and multi-grid solvers. To allow adaptive, parallel computations the load balancing problem for dynamically changing grids has to be solved efficiently by fast heuristics. These strategies are combined in the simulation system UG (“Unstructured Grids”) being presented in the following.In the second part of the seminar we show the performance and efficiency of this strategy in various applications. In particular, the application and benefit of parallel adaptive multi-grid methods to modelling drug permeation through human skin is shown in detail.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Matteo Longo (University of Padova)
p-adic families of Heegner points on Shimura curves and p-adic L-functions abstract
Abstract:
Heegner points (and their higher counterparts, Heegner cycles) are powerful tools in the study of the arithmetic of elliptic curves and modular forms. The question of their variation along p-adic families of modular forms (over modular curves) and the relation with BDP p-adic L-functions has been addressed through several different approaches by many authors: B. Howard in 2007, Castella in 2020, Jetchev-Loeffler-Zerbes and Büyükboduk-Lei in 2021. I will report on some progress and partial results on analogues of these theories in the case of modular forms over quaternionic Shimura curves (defined over the field of rational numbers). The main ingredient is the systematic use of Serre-Tate expansion on Shimura curves as a replacement of Fourier expansions of elliptic modular forms, following an old approach by A. Mori. This is a work in collaboration with P. Magrone and E. Walchek.