Ioannina Winter School 2024: 

Speaker: Athanasios Chatzikaleas (Ecole Polytechnique Fédérale de Lausanne, Switzerland)
Title: Isoperimetric inequalities on black holes and the final state of the universe
Abstract: In 1973, Sir Roger Penrose asked: "Is the future always predictable?" and rephrased this question to a beautiful geometric statement and that can be formulated as an isoperimetric inequality for black holes. In this series of lectures, we shall discuss the physical meaning of this inequality, set the necessary geometric background, as well as present a proof of the Riemannian version of it, following the pioneer work of Huisken-Ilmanen (2001) using the inverse mean curvature flow.

Speaker: Andreas Savas-Halilaj (University of Ioannina, Greece)
Title: Mean curvature flow and related equations
Abstract: This mini-course focuses on the deformation of submanifolds along their mean curvature vector in Euclidean space and Riemannian manifolds. We will introduce the fundamental principles of mean curvature flow and examine its connections  to harmonic heat flow and Hamilton's Ricci flow. Subsequently, we will demonstrate how these methods can be applied to derive results on the topology and rigidity of manifolds, as well as on mappings between them. In particular, we will explore  how mean curvature flow and Ricci flow provide insights into Smale's question of whether a smooth diffeomorphism of the  sphere can be deformed into an isometry. We will also review recent developments in this area, highlighting recent contributions of R. Bamler, B. Kleiner, K. Smoczyk, and M.-T. Wang.

Speaker: Konstantinos Zemas (University of Bonn, Germany)
Title: Rigidity vs Flexibility of Isometric Embeddings: From Weyl to the Nash-Kuiper theorem
Abstract: One of the most famous paradoxes in differential geometry concerns the isometric embeddings of S^2 into R^3: While the only C^2 isometric embeddings are rigid motions, there exists a plethora of C^1 isometric embeddings that wrinkle S^2 into any tiny region of R^3. This counter-intuitive result is a consequence of the celebrated Nash-Kuiper theorem, the proof of which uses the algorithmic technique of convex integration, a term that was established after Gromov’s contributions to the theory of partial differential relations. This technique has nowadays a broad spectrum of applications in nonlinear partial differential equations, in problems ranging from differential geometry to the equations of fluid mechanics. The goal is of this mini-course is to describe some classical rigidity theorems in that respect, and to present the main ideas of this suprising non-uniqueness phenomenon, known as the h-principle, indicating the two main mechanisms with which it arises: the high codimension of the ambient space, and the low-regularity of the desired solutions.

Speaker: Christos Saroglou (University of Ioannina, Greece)
Title: Mahler's conjecture, the Bourgain-Milman inequality and its connection to the local theory of Banach spaces
Abstract: For a finite dimensional normed space X, denote by B_X its unit ball, by X^* its dual space and by l_p^n the n-dimensional l^p space. A fundamental question in the local theory of Banach spaces is to estimate the volume product v.p.(X)=V(B_X)xV(B_{X^*}. While the upper bound v.p.(X)<=v.p.(l_2^n) is classical (n=dimX), a precise optimal lower bound in unknown. The famous conjecture of Mahler states that v.p.(X)>=v.p.(l_1^n). I will try to explain the current status of the conjecture and describe proofs of some recent developments  towards its solution. While Mahler's conjecture is in general open, a weaker - but asymptotically optimal - statement due to J. Bourgain and V. D. Milman is known to be true. If time permits, I will try to explain the proof of the Bourgain-Milman inequality and some of its applications.