Speaker: Athanasios Chatzikaleas (National and Kapodistrian University of Athens, Greece)
Title: The Friedmann Equations and the Ultimate Fate of the Universe
Abstract: In the early 20th century, Alexander Friedmann developed dynamic solutions to Albert Einstein’s field equations of General Relativity, demonstrating how the universe can evolve over time based on its matter and energy content. His insights gained significant recognition after Edwin Hubble’s discovery that galaxies are receding, confirming the present-day expansion of the universe. This talk presents a step-by-step derivation of the Friedmann equations and explores how they form the foundation of modern cosmology. By analyzing the interplay between different cosmic components (matter, radiation, curvature and dark energy), we develop the mathematical framework to produce various scenarios for how the universe might end. Drawing on major observational milestones, including the detection of the cosmic microwave background in the mid-20th century, this talk brings together insights from cosmology, differential geometry, and differential equations to examine the ultimate fate of the universe.

Speaker: Andreas Savas-Halilaj (University of Ioannina, Greece)
Title: The Berstein Problem
Abstract: The Bernstein problem is a fundamental question in differential geometry and the theory of minimal surfaces. Posed originally by Sergei Bernstein in 1915, it asks whether every entire minimal graph over Rn (a solution to the minimal surface equation defined on the whole n-dimensional Euclidean space) must be an affine function. Bernstein proved that for 2-dimensional graphs, any such solution is necessarily linear, implying that the only global minimal surfaces in R3 that are graphs are planes. Subsequent work extended this rigidity result to higher dimensions, with De Giorgi (case n=3), Almgren (case n=4), Simons (case n=5,6,7), establishing that the Bernstein property holds for dimensions less than 8. Surprisingly, in dimensions greater than or equal to 8, Bombieri, De Giorgi, and Giusti constructed counterexamples, showing that nontrivial entire minimal graphs exist. This was the first example in the theory of PDEs where such a dimension-dependent dichotomy appears. Thus, the Bernstein problem revealed a striking dimension-dependent distinction in the behavior of minimal surfaces, linking geometric analysis, partial differential equations, and the calculus of variations. The problem and its resolution have had a profound influence on geometric measure theory and the study of elliptic partial differential equations.

In this minicourse, we will give an overview of the Bernstein problem, discuss its higher-codimension version, and present recent developments based on papers by Chodosh & Li (Acta Math. 2024) and Cationo, Mastrolia & Roncoroni (GAFA 2024).

Speaker: Konstantinos Zemas (Hausdorff Center for Mathematics, Bonn, Germany)
Title: The Willmore Conjecture
Abstract: One of the most fundamental quantities associated to a surface immersed in R^3 is its Willmore energy, defined as the integral of the square of its mean curvature. While this functional is inherently geometrically interesting, being invariant under conformal transformations of R^3, it is also very natural in terms of applications in physics and biology, as it appears for instance in the modelling of thin elastic objects and cell-membranes. From a variational viewpoint, the main question is to understand the minimizers of the Willmore energy among surfaces of fixed topological type. While it is relatively easy to see that the absolute minimum among all closed surfaces is precisely achieved by round spheres with value 4π, the problem of minimizing this energy among the class of immersed tori turned out to be an extremely challenging task.

According to Willmore’s initial conjecture (1965), the minimum in the class of genus 1 surfaces should be achieved (up to conformal transformations) for the stereographic projection of the Clifford torus with value 2π^2. The goal of this mini-course is to describe the main steps of the full proof of the conjecture, given in the pioneering work of F. C. Marques and A. Neves (Ann. Math. 2014), ingeniously using the min-max theory for minimal hypersurfaces in S3 due to Almgren and Pitts from the 1970’s. 

Speaker: Panagiotis Polymerakis (University of Thessaly, Greece)
Title: Small Eigenvalues of Geometrically Finite Manifolds
Abstract:  In this mini-course, we will survey some results on small eigenvalues of geometrically finite manifolds in the sense of Bowditch. This is a class of complete Riemannian manifolds of pinched negative sectional curvature, which includes manifolds of finite volume. Initially, we will study hyperbolic surfaces of finite area, focusing on the result of Otal-Rosas asserting that the number of small eigenvalues of such a surface is bounded in terms of its Euler characteristic. We will then discuss the higher dimensional analogue for hyperbolic manifolds of finite volume, established by Buser-Colbois-Dodziuk. Finally, we will address recent progress in generalizing these results to the broader class of geometrically finite manifolds.