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# On Legendrian dual surfaces of a spacelike curve in the $3$-dimensional lightcone

## Handan YildirimUniversity of Istanbul

In this talk which is based on the joint work with Kentaro Saji given in [3], taking into account the Legendrian dualities in [2] which are extensions of the Legendrian dualities in [1], we first introduce new extended Legendrian dualities for the 3-dimensional pseudo-spheres of various radii in Lorentz-Minkowski 4-space. Secondly, by connecting all of these Legendrian dualities continuously, we construct Legendrian dual surfaces (lying in these 3-dimensional pseudo-spheres) of a spacelike curve in the 3-dimensional lightcone. Finally, we investigate the singularities of these surfaces and show the dualities of the singularities of a certain class of such a surface in the 3-dimensional lightcone.
[1] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Moscow Mathematical Journal, 9 (2009), 325-357.
[2] S. Izumiya, H. Yildirim, Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space, Topology and its Applications, 159(2012), 509-518.
[3] K. Saji, H. Yildirim, Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone, Journal of Geometry and Physics, 104593, https://doi.org/10.1016/j.geomphys.2022.104593

# On the canonical contact structure of the space of null geodesics of a spacetime: the role of Engel geometry in dimension 3

The space of null geodesics of a spacetime (a Lorentzian manifold with a choice of future) sometimes has the structure of a smooth manifold. When this is the case, it comes equipped with a canonical contact structure. I will introduce the theory for a countable number of metrics on the product $S^2\times S^1$. Motivated by these examples, I will comment on how Engel geometry can be used to describe the manifold of null geodesics, by considering the Cartan deprolongation of the Lorentz prolongation of the spacetime. This allows us to characterize the 3-contact manifolds which are spaces of null geodesics, and to retrieve the spacetime they come from. This is joint work with R. Rubio.

# Critical embeddings for the first eigenvalue of the Laplacian

## Eduardo LongaUniversidade de Sao Paulo

The eigenvalues of the Laplace-Beltrami operator on a closed Riemannian manifold are very natural geometric invariants. Although in many problems the Riemannian structure is kept fixed, the eigenvalues can be seen as functionals in the space of metrics. This is the suitable setting for the calculus of variations. In this vein, El Soufi and Ilias have characterised the metrics which are critical for the first eigenvalue among all metrics of fixed volume and among all metrics of fixed volume in a conformal class. In the talk, I will prove a similar characterisation for some critical metrics which are induced by embeddings into a fixed Riemannian manifold.

# Minimal surfaces as an interdisciplinary topic

## Hao ChenUniversity of Göttingen

All my works on minimal surfaces has been motivated or inspired by natural sciences, including material sciences, bio-membranes, fluid dynamics, etc. I will give an informal talk (since I’m not natural scientist) about how minimal surface theory could benefit from interdisciplinary interactions.

## Hao ChenUniversity of Göttingen

Traizet’s node opening technique has been very powerful to construct minimal surfaces. In fact, it was first applied to glue saddle towers into minimal surfaces. But for technical reasons, the construction has much room to improve. I will talk about the ongoing project that addresses to various technical problems in the gluing construction. In particular, careful treatment of Dehn twist has revealed very subtle interactions between saddle towers.

# Triply periodic minimal surfaces

## Hao ChenUniversity of Göttingen

Triply periodic minimal surfaces (TPMSs) are minimal surfaces in flat 3-tori. I will review recent discoveries of new examples of TPMSs and outline future steps towards an eventual complete classification of TPMSs of genus 3.

# Geometric Modelling of nonlinear dynamics processes by fractal approximation methods

## Olha ZalevskaNational Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”

Small variations of nonlinear dynamics by means of geometric modeling are investigated. Visualization of the dynamic process is reproduced using three-dimensional cellular automata. The dependences of the transition positions of the dynamic system between the chaotic and stable states are established. Theoretical aspects of geometric fractal approximation of the dynamic process, criteria for establishing the stable state of the system, geometric fractal derivatives and integration are considered.

# Hyperbolic domains in real Euclidean spaces

## Franc ForstneričUniversity of Ljubljana

In a recent joint work with David Kalaj (2021), we introduced a new Finsler pseudometric on any domain in the real Euclidean space $\mathbb R^n$ for $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with the Kobayashi pseudometric on complex manifolds. This "minimal pseudometric" describes the maximal rate of growth of hyperbolic conformal minimal surfaces in a given domain. On the unit ball, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric. I will discuss sufficient geometric conditions for a domain to be (complete) hyperbolic, meaning that its minimal pseudometric is a (complete) metric.

# The mathematical theory of globally hyperbolic supersymmetric configurations in supergravity

I will give a pedagogical introduction to the incipient mathematical theory of globally hyperbolic supersymmetric configurations in four-dimensional Lorentzian supergravity. First, I will introduce the basics of supergravity in four dimensions as well as the notion of globally supersymmetric configuration as a solution to a system of first-order spinorial differential equations, that is, the supergravity spinorial equations. Then, I will introduce the theory of spinorial polyforms associated with bundles of irreducible real Clifford modules, which provides a convenient geometric framework to study the global geometric and topological properties of the solutions to the supergravity spinorial equations. Then, I will consider the evolution problem for globally hyperbolic supersymmetric configurations, focusing on the constraint equations, their moduli of solutions, and the construction of explicit evolution flows, which we reformulate as the supergravity flow equations for a coupled family of functions and global co-frames on a Cauchy hypersurface. This will lead us to explore in detail the case of (possibly Einstein) globally hyperbolic Lorentzian four-manifolds equipped with a parallel or Killing spinor, obtaining several results about the differentiable topology and geometry of such manifolds. Finally, I will mention several open problems and open directions for future research.

# Sharp differential inequalities for the isoperimetric profile in spaces with Ricci lower bounds

## Gioacchino AntonelliScuola Normale Superiore di Pisa

In this talk I will discuss sharp differential inequalities for the isoperimetric profile function in spaces with Ricci bounded from below, and with volumes of unit balls uniformly bounded from below. After that, I will highlight some of the consequences of such inequalities for the isoperimetric problem. After a short introduction about the notion of perimeter in the metric measure setting, I will pass to the motivation and statement of the sharp differential inequalities on Riemannian manifolds. Hence, I will discuss the proof, which builds on a non smooth generalized existence theorem for the isoperimetric problem (after Ritoré-Rosales, and Nardulli), and on a non smooth sharp Laplacian comparison theorem for the distance function from isoperimetric boundaries (after Mondino-Semola). At the end I will discuss how to use such differential inequalities to study the behaviour of the isoperimetric profile for small volumes. This talk is based on some results that recently appeared in a work in collaboration with E. Pasqualetto, M. Pozzetta, and D. Semola. Some of the tools and ideas exploited for the proofs come from other works in collaboration with E. Bruè, M. Fogagnolo, and S. Nardulli.

# Una demostración del teorema de Gauss-Bonnet evitando triangulaciones

El objetivo de la charla es explorar la geometría compleja de superficies (orientables y compactas) para obtener una demostración del teorema de Gauss-Bonnet de forma intrínseca. Este tipo de abordaje es una buena oportunidad para introducir estructuras geométricas que se aplican en una multitud de áreas fuera de la Geometría Diferencial Clásica (e.g. Geometrías Compleja y Simpléctica); además, no se necesita ninguna familiaridad con la topología de las variedades (algo esencial al utilizar triangulaciones), solamente hace falta una buena noción de cálculo con varias variables.

Sala de Conferencias (IMAG)

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# XXI Geometrical Seminar

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The Geometrical Seminar started its activities more than 30 years ago under the name Yugoslav Geometrical Seminar. The aim of these meetings is to bring together mathematicians and physicists interested in geometry and its applications, to give lectures on new results, exchange ideas, problems and conjectures.
The 21st will be held in June 2021.

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# Workshop on Surface Theory —UY60—

## Tokio (Japón) & Virtual

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Congreso en honor de Umehara y Yamada por su 60 cumpleaños