The 20th edition of the Geometry Colloquium at the Autonomous University of Yucatán, México, includes specialists from the Institute of Mathematics and the Faculty of Sciences of UNAM, CIMAT, and the Autonomous Metropolitan University (Iztapalapa Unit) among its invited speakers. The Colloquium activities include a mini-course, panoramic lectures (focused on a broad audience), conferences, and research reports.

On tuesday 16th of december I will deliver the talk Growing of geodesic balls on diagonal Heintze groups. Talk will be in spanish. Access is free. the talk will be delivered at the school of mathematics of the Autonomous University of Yucatan and online upon registration.

Abstract

Let M be a homogeneous, simply connected manifold with negative curvature. Heintze proves that M is isometric to a solvable Lie group with an invariant left metric of a special type. The metric is determined by an operator A in the Lie algebra that is compatible with the bracket of the algebra and with the metric. Such groups are well known from the standpoint of metric space theory. If A is diagonalizable on the complex numbers, explicit calculations can be made using dynamical systems techniques: the geodesic equations represent a physical system with a potential barrier, the curvature formulas are simplified, and the behavior of the Jacobi fields can be approximated.

By studying the stability of the Jacobi equation, an asymptotic formula for the growth of the volume of geodesic spheres and balls in diagonal Heintze groups can be found. This formula complements recent results in horosphere growth and relates to the growth of geodesic spheres in horospheric products, for which a formula for volume entropy is known. This talk intuitively presents the geometry of these spaces and examines the geometric details necessary for calculating the volume. Finally, I present the main obstacles to performing the calculation without assuming that the operator A is diagonal and the progress made in this area.

This talk is loosley based on the preprint Volume entropy of a family of rank one, split-solvable Lie groups of Abelian type

Venue

The Facultad de Matemáticas at the Universidad Autónoma de Yucatán (UADY) is a distinguished academic institution in Mexico dedicated to the study and advancement of mathematics, computer science, and applied sciences. Known for its strong focus on geometry research, education, and community engagement, the faculty offers undergraduate and graduate programs that prepare students for careers in academia, industry, and research. Through its collaborative projects and specialized courses, it contributes significantly to the scientific and technological development of the region.

Mérida, the vibrant capital of Yucatán, Mexico, offers Colloquium attendees a unique blend of rich cultural heritage and modern amenities. Known as the “White City,” Mérida is famed for its colonial architecture, lively plazas, and welcoming atmosphere. Visitors can explore historic sites, including the grand Paseo de Montejo, and savor authentic Yucatecan cuisine, such as cochinita pibil and panuchos. The city is also a gateway to stunning Mayan ruins, like Uxmal and Chichen Itza, and is close to beautiful cenotes and beaches. Mérida’s warmth and charm make it an ideal setting for both academic exchange and cultural immersion.

Registration

The event will take place at the Facultad de Matemáticas, conferences will be in spanish, details and the registration form can be found at the event site: https://sites.google.com/view/xx-coloquio-de-geometria/horario