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Talks

Translating Solitons in the Hyperbolic Einstein Space-time

Ankara University

We classify those rotationally invariant translators of the mean curvature flow in the Hyperbolic Einstein Space-time \(\mathbb{H}^n\times_{-1}\mathbb{R}\). Next, we consider a connected, compact space-like translator whose boundary is the boundary of a bounded open domain in a slice. If the domain is invariant by an isometry \(\sigma\) of \(\mathbb{H}^n\), then the traslator is invariant by \(\sigma\times id\). We then characterize one of the rotationally invariant examples.

Seminario 1 (IMAG)

Gromov's h-Principle and distributions

Universidad Complutense de Madrid

In this talk we will introduce Gromov's $h$-principle theory from a basic and accessible perspective. We will motivate it through visual examples with special emphasis on the method of Convex Integration. Many problems in Differential Topology involve differential relations (differential equations, inequalities, etc.). In many contexts, it can be proven that there exists an $h$-principle: this means that the resolution of certain geometric problems can be reduced to studying the underlying Algebraic Topology. We will show how these techniques can be applied to the study of maximal growth distributions on smooth manifolds. Prototypical examples of these objects are Contact and Engel structures.

Seminario 2 (IMAG)

Large conformal metrics with prescribed gauss and geodesic curvatures

Pontificia Universidad Católica de Chile

In this talk, our goal is to discuss the existence of at least two distinct conformal metrics with prescribed gaussian curvature and geodesic curvature respectively, $K_{g}= f + \lambda$ and $k_{g}= h + \mu$, where f and h are nonpositive functions and \lambda and \mu are positive constants. Utilizing Struwe's monotonicity trick, we investigate the blowup behavior of the solutions and establish a non-existence result for the limiting PDE, eliminating one of the potential blow-up profiles.

Seminario 2 (IMAG)

A Morse-theoretic glance at phase transitions approximations of mean curvature flows

Pontificia Universidad Católica de Chile

The Allen–Cahn equation is a semilinear parabolic partial differential equation that models phase-separation phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows. In this talk, we employ Morse-theoretical considerations to construct eternal solutions of the Allen–Cahn equation that connect unstable equilibria in compact manifolds. We describe the space of such solutions in a round 3-sphere under a low-energy assumption, and indicate how these solutions can be used to produce geometrically interesting eternal MCFs. This is joint work with Jingwen Chen (University of Pennsylvania).

Seminario 2 (IMAG)

On the topology of compact locally homogeneous plane waves

Universidad de Granada

A compact flat Lorentzian manifold is the quotient of the Minkowski space by a discrete subgroup \(\Gamma\) of the isometry group, acting properly, freely and cocompactly on it. A classical result by Goldman, Fried and Kamishima states that, up to finite index, \(\Gamma\) is a uniform lattice in some connected Lie subgroup of the isometry group, acting properly and cocompactly, generalizing Bieberbach theorem to the Lorentzian signature. Such compact quotients are called "standard". More generally, a compact quotient of a homogeneous space \(G/H\) of a Lie group \(G\) is standard if the fundamental group action extends to a proper cocompact action of a connected Lie subgroup of \(G\). It turns out that looking for standard quotients is an easier problem when studying the existence of compact quotients of homogeneous spaces. This talk is about compact locally homogeneous plane waves. Plane waves can be thought of as a deformation of Minkowski spacetime, they are of great mathematical and physical interests. In this talk, we describe the isometry group of a 1-connected homogeneous non-flat plane wave, and show that compact quotients are “essentially" standard. As an application, we obtain that the parallel flow of a compact plane wave is equicontinuous. This is a joint work with M. Hanounah, I. Kath and A. Zeghib.

Aula A14 (Facultad de Ciencias)

Uniqueness of semigraphical translators

Columbia University

Translators in \(\mathbb{R}^3\) are solitons of the mean curvature flow for embedded 2-surfaces. In the semigraphical case, where the translators are allowed graphical as well as vertical components, Hoffman-Martín-White classified the surfaces into six types. They conjectured the uniqueness of the objects within two families contained in slabs, the "helicoids" and the "pitchforks," for any given width. We present the proof of the conjecture by combining an arc-counting argument motivated by Morse-Radó theory for translators with a rotational application of the maximum principle. We then discuss applications of this result to the classification of semigraphical translators in \(\mathbb{R}^3\) and their limits, related to the work of Hoffman-Martín-White and Gama-Martín-Møller. This is joint work with F. Martín and M. Sáez.

Aula A20 (Facultad de Ciencias)

Events

XVIII International Young Researchers Workshop in Geometry, Dynamics and Field Theory

Warsaw (Poland)

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The 18th Young Researchers Workshop in Geometry, Dynamics and Field Theory is a yearly event to promote young researchers in the field of differential geometry and its relations to dynamics and field theory. The 18th edition will take place in the University of Warsaw. This event offers researchers in the field, especially to the younger participants, a platform to share their latest results to an international audience and discuss current topics. The workshop will contain three mini-courses in key topics in the field, selected talks proposed by the participants, and a poster session.

XI International Meeting on Lorentzian Geometry

Mérida (México)

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The Meeting is intended for all kind of researchers interested in Lorentz Geometry and its applications to General Relativity. It provides an excellent opportunity to exhibit their latest results and to create new ways of collaboration. For PhD students the meeting will represent an ideal way to have their first contact with current research topics on the subject. Furthermore, the schedule includes an advanced course given by an expert in the field.

Congreso Bienal de la Real Sociedad Matemática Española 2024

Pamplona (Spain)

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En este congreso Bienal RSME 2024 se darán a conocer los últimos avances en investigación en diferentes áreas de matemáticas y se facilitará el establecimiento de lazos de colaboración entre distintos grupos de investigación de nuestro país. Además de las habituales Conferencias Plenarias está prevista la celebración de Sesiones Especiales y exposición de pósteres. La asistencia al congreso permitirá disfrutar además de variadas actividades programadas en Pamplona y alrededores.

Iberian Strings 2024

Porto (Portugal)

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Iberian Strings 2024 is the 16th-installment of the annual meeting of the Spanish and Portuguese String Theory community, where recent developments in the field of supergravity, strings, branes and gauge theory are discussed.