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Renan Assimos (Leibniz Universität Hannover, Germany)

Title**: ***When discrete geometry meets a geometric flow.*

**Abstract:**We use ideas from discrete geometry to build new degree zero harmonic maps from surfaces of genus greater than one into two dimensional spheres. Those surfaces are then used as counter examples to a conjecture of Emery about harmonic map images and closed geodesics.

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**Panagiotis Batakidis (Aristotle University of Thessaloniki, Greece)****Title: ***Poisson geometry of Bott-Morse singularities.***Abstract: **The study of foliations on 3-manifolds is closely related to their topology and complications arise when one allows for singularities. A big class of them is named Bott-Morse singularities, and it has been shown that certain Poisson structures do exist on manifolds with such foliations. The talk is about work in progress on the cohomology of such structures aiming to provide invariants for the differentiable structure.**Slides:** [PDF]

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**Athanasios Chatzikaleas (Universität Münster, Germany)****Title: ***Non-linear periodic waves in the Anti-de Sitter spacetime and islands of stability.*

Abstract: In 2006, Dafermos-Holzegel conjectured that the Anti-de Sitter spacetime is an unstable solution to the Einstein equations under reflective boundary conditions for general initial data. Rostworowski-Maliborski enhanced this conjecture by proving numerical evidence that indicate the existence of "special" initial data leading to time-periodic solutions for the Einstein-Klein-Gordon system which are in fact stable. Motivated by these, we construct families of arbitrary small time-periodic solutions to several toy models on the fixed Anti-de Sitter background providing a rigorous proof of the numerical constructions above in a simpler setting. The models we consider include the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the fixed Anti-de Sitter spacetime and our proof relies on the modifications of a theorem of Bambursi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system.**Slides:** [PDF]

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**Athanasios Chatzikaleas (Universität Münster, Germany)Title of the minicourse: **

*The Hawking singularity theorem.*

**Abstract:**In 1965, the British physicist and mathematician Roger Penrose published a quintessentially mathematical paper [6] sketching the proof of a theorem in semi-Riemannian geometry. Roughly speaking, his theorem states that black hole formation is a robust prediction of General Relativity. One year later, in 1966, Stephen Hawking [3,4] generalized Penrose’s theorem in order to apply it to the universe as a whole. Both theorems, now called "Penrose-Hawking Singularity Theorems", spurred on many developments in General Relativity. In fact, in 2020, Roger Penrose was awarded the Nobel Prize in Physics for his discovery. In this note, we state, prove and discuss the Hawking’s Singularity Theorem according to which our universe must itself contain a singularity deep in its past, from which all matter and energy emanated in a Big Bang. To do so, we will follow the book of Godinho–Natario [2] as well as the lecture notes of Aretakis [1] and Holzegel [5].

**Bibliography:**

[1] S. Aretakis,

*General relativity*. Lecture Notes (2013).

[2] L. Godinho & J. Natario,

*An introduction to Riemannian geometry*, Universitext, Springer (2014).

[3] S.W. Hawking,

*The occurrence of singularities in cosmology I*, Proc. Roy. Soc. London Ser. A,

**294**, 511-521 (1996).

[4] S.W. Hawking,

*The occurrence of singularities in cosmology II*, Proc. Roy. Soc. London Ser. A,

**295**, 490-493 (1996).

[5] G. Holzegel,

*General relativity and the analysis of black hole spacetimes*, Lecture Notes, (2022).

[6] R. Penrose,

*Gravitational collapse and space-time singularities*, Phys. Rev. Lett.,

**14,**57-59, (1965).

**Lecture Notes:**[PDF]

**Slides:**[PDF]

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**Anestis Fotiadis (Aristotle University of Thessaloniki, Greece)**

**Title:**

*Area preserving maps to constant curvature surfaces.*

**Abstract:**We study area preserving maps between Riemann and Lorentz surfaces equipped with a conformal metric. Under the assumption that orthogonality is also preserved, we provide a classication of these diffeomorphisms. Finally, we construct a family of such maps when the target surface is of constant curvature. This is a joint work with Dr. Effie Papageorgiou.

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**Manousos Maridakis (Aristotle University of Thessaloniki, Greece)**

**Title:**

*Αφηρημένες ανισότητες Lojasiewicz-Simon και εφαρμογές.*

**Abstract:**Οι ανισότητες Lojasiewicz-Simon βρίσκουν πλήθος εφαρμογών όπως στις αποδείξεις μοναδικότητας εφαπτόμενων κώνων, αποδείξεις μοναδικότητας μοντέλων ιδιομορφίας στις γεωμετρικές ροές και αποδείξεις διακριτού φάσματος κρίσιμων τιμών συναρτησοειδών ενέργειας. Στην ομιλία θα παρουσιάσουμε μια αφηρημένη εκδοχή ανισότητας Lojasiewicz-Simon σε χώρους Banach, θα περιγράψουμε τις ιδέες πίσω απο την απόδειξη και χρόνου επιτρέποντος, θα μιλήσουμε για εφαρμογές σε ανισότητες Lojasiewicz-Simon με βέλτιστες νόρμες Sobolev για το συναρτησοειδές Dirichlet και για συναρτησοειδή ενέργειας τύπου Yang-Mills.

**Slides:**[PDF]

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**Francisco Martin (Universidad de Granada, Spain)**

**Title:**

*Finite entropy translating solitons in slabs.*

**Abstract:**This talk will be about self-translating solitons for the mean curvature flow of embedded 2-surfaces, under the natural geometric assumptions of finite topology, finite entropy and finite width. We first define "wing numbers", invariants of such solitons, which appear in a simple formula that computes the entropy, which is therefore in particular quantized into integer values. Asking further which examples of such solitons exist in the low entropy range, we use Morse theory a la Rado to prove a uniqueness theorem: The unique simply connected embedded translating solitons of finite width and entropy three are the so-called pitchforks, recently discovered by Hoffman-Martin-White. This is joint work with E.S. Gama and N.M. Moller.

**Slides:**[PDF]

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**Stavros Papadakis (University of Ioannina, Greece)**

**Title of the minicourse:**

*An introduction to K3 surfaces and Calabi-Yau manifolds.*

**Abstract:**An interesting class of compact complex surfaces is the class of K3 surfaces, which consists of the simply connected surfaces which have an everywhere nonzero holomorphic 2-form. Their higher dimensional generalizations are known as Calabi-Yau manifolds. The purpose of this mini-series is to give an introduction to the geometric properties of the K3 surfaces and the Calabi-Yau manifolds.

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**Effie Papageorgiou (University of Crete, Greece)**

**Title:**

*Asymptotic behavior of solutions to the heat equation on*

*noncompact symmetric spaces.*

**Abstract:**[PDF]

**Slides:**[PDF]

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**Nikolaos Roidos (University of Patras, Greece)Title:**

*The fractional porous medium equation on manifolds with conical singularities.*

**Abstract:**We will talk about existence, uniqueness and maximal regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. We will also discuss the space asymptotic behavior of the solutions close to the singularities and its relation to the local geometry.

**Slides:**[PDF]

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**Christos Saroglou (University of Ioannina, Greece)****Title:** *Non existence results for the *Lp*-Minkowski problem.*

**Abstract:**After giving some basic definitions, such as the support function and the surface area measure of a convex body, I will try to describe a major topic in Convex Geometry, the so-called L

_{p}-Minkowski problem. The main focus will be in the following result: Given a real number p<1, a positive integer

*n*and a proper subspace H of R

^{n}, the measure on the Euclidean sphere S

^{n-1}, which is concentrated in H and whose restriction to the class of Borel subsets of the intersection S

^{n-1}∩H equals the spherical Lebesgue measure on S

^{n-1}∩H, is not the L

_{p}-surface area measure of any convex body. This, in particular, disproves a conjecture from "Bianchi, Böröczky, Colesanti, Yang, The L

_{p}-Minkowski problem for -

*n*<p<1, Adv. Math. (2019)". If time permits, I will mention a new non-existence result and explain how this yields a conjectured

*L*-

*infinity*estimate for the support function in low dimensions. Joint work in part with Karoly Böröczky.

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**Polyxeni Spilioti (****Aarhus University, Denmark****)****Title:** *On the spectrum of twisted Laplacians and the Teichm**ü**ller representation.***Abstract:** In this talk, we will present some recent results concerning the spectrum of Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyperbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface with are of Teichmüller type, we investigate the high energy limit and give a precise description of the bulk of the spectrum where Weyl’s law is satisfied in terms of critical exponents of the representation which are completely determined by the Manhattan curve associated to the Teichmüller deformation. This is joint work with Frederic Naud.**Slides:** [PDF]

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**Knut Smoczyk (Leibniz Universität Hannover, Germany)**

**Title:**

*Mean curvature flow of area decreasing maps.*

**Abstract:**I will present recent progress on the deformation of area decreasing maps by their mean curvature, including new a priori estimates and convergence results.

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**Raphael Tsiamis (Harvard University, USA)**

**Title:**

*The hyperk*

*ä*

*hler structure on the cotangent bundle of a complex Lie group.*

**Abstract:**Hyperkähler manifolds have various analytic and algebraic properties that make them easy to describe, yet fairly rigid to construct. P. Kronheimer’s construction [1] exhibits an abstract hyperkähler structure on the total space T

^{*}G

_{C }of the cotangent bundle of the complexification of any Lie group G. This structure arises via infinite-dimensional hyperkähler reduction on the space of solutions to set of ODEs called Nahm’s equations, and the resulting hyperkähler structure is challenging to describe explicitly even for basic Lie groups. We present a partial description of the resulting hyperkähler structure and the family of Kähler structures G

_{C}arising as sections of the tangent bundle in the general case; this extends some results of Stenzel [3] on homogeneous spaces. We also perform concrete computations on the case of G=SU(2), for which G

_{C}=SL(2,C). These results represent work in progress within our broader study of the hyperkähler structure on T*G

_{C}

*.*This is joint work with Richard B. Melrose.

**Bibliography:**

[1] P.B. Kronheimer,

*The hyperk*

*ä*

*hler structure on the cotangent bundle of a complex Lie group*. MSRI preprint (1998).

[2] G. Patrizio & P.M. Wong,

*Stein manifolds with compact symmetric center.*Math. Ann.

**289**(3), 355-382 (1991).

[3] M. Stenzel,

*Ricci-flat metrics on the complexification of a compact rank one symmetric space.*Manuscripta Math.

**(80)**, 151-163 (1993).

**Slides:**[PDF]

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**Amalia-Sofia Tsouri (University of Ioannina, Greece)**

**Title:**

*Isometric deformations of pseudoholomorphic curves in*

**S**

^{6}.

**Abstract:**A fundamental problem in the theory of isometric immersions is the study of rigidity and deformability of a given isometric immersion. We provide important properties of the moduli space of all noncongruent minimal surfaces in the

*n*-sphere that are locally isometric to a pseudoholomorphic curve

*g*in the nearly Kähler 6

*-*sphere.

**Slides:**[PDF]

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**Artemis Vogiatzi (Queen Mary University of London, UK)**

**Title:**

*Convexity estimates for high codimension mean curvature flow in Riemannian manifolds and surgery.*

**Abstract:**Geometric Analysis is a study of geometric problems that can be formulated as problems of partial differential equations. In this talk, we will see that the blow-ups of compact solutions to the mean curvature flow in a Riemannian manifold, subject to the initial pinching condition

|A|^{2}-c(n)|H|^{2}+d(n)<0,

for a suitable constant c(n) must be of codimension one. This means that near a singularity the surface is quantitively cylindrical, which is important in order to implement surgery and continue the flow beyond the first singular time.

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**Konstantinos Zemas (****Universität Münster, Germany****)**

**Title:**

*Geometric rigidity estimates in fixed and variable domains and applications.*

**Abstract:**Quantitative rigidity results, besides their inherent geometric interest, have played a prominent role in the mathematical study of variational models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for functionals within the framework of nonlinear elasticity. After a quick review of the aforementioned results, we will present appropriate generalizations to the setting of variable domains, where the geometry of the domain comes into play in terms of a suitable surface energy of its boundary. If time permits, we will also discuss applications of this new rigidity estimates in questions related to models for elastic materials with free surfaces. This is joint work with Manuel Friedrich and Leonard Kreutz.

**Slides:**[PDF]

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