# TYAN Thematic Workshops in Mathematics: Differential Geometry

This fifth edition will happen online on February 21, 2022 at 11h (UTC -3, Brasília, Brazil time) and the main topic will be “Differential Geometry”.  The speakers are reputed scientists working in Latin America: Emílio Lauret (Universidad Nacional del Sur, Argentina), Jimmy Petean (CIMAT, Mexico), Keti Tenenblat (Universidade de Brasília, Brazil) e Raquel Perales (Universidad Nacional Autónoma de Mexico, Mexico). The moderators are: Bárbara Nelli (Università Degli Studi Dell’Aquila, Italy) and Christina Sormani (CUNY Lehman College, USA).

The preliminar schedule is described below:

## Titles, Abstracts of Talks and Short Bio of the Speakers

1. Emilio Lauret (Universidad Nacional del Sur, Argentina)

Title: Isospectrality among homogeneous Riemannian manifolds

Abstract: We will review several results on inverse spectral geometry in the setting of homogeneous Riemannian manifolds. Some key points will be Schueth’s examples of continuous curves of isospectral left-invariant metrics on compact Lie groups, Sutton’s construction of isospectral compact homogeneous Riemannian manifolds (as a generalization of Sunada-Pesce’s and), method spectral isolation/uniqueness problems among $G$-invariant metrics on a fixed homogeneous space $G/K$.

Short bio: Ph.D. from Universidad Nacional de Córdoba, Argentina, in 2011, Alexander von Humboldt Postdoctoral Fellow at Humboldt Universität zu Berlin in 2017-2018. He is currently Assistant Professor at Universidad Nacional del Sur, Bahía Blanca, Argentina. His research area is “Inverse Spectral Geometry”.   He won Stimulus Award 2020, “Dr. Eduardo Zarantonello”, by Academia Nacional de Ciencias Exactas, Físicas y Naturales, Buenos Aires, Argentina. More information can be found in https://sites.google.com/prod/view/emiliolauret/short-cv

2. Raquel Perales (Universidad Nacional Autónoma de Mexico, Mexico)

Title: Upper bound on the revised first Betti number and torus stability for RCD spaces

Abstract: In this talk, we will generalize some classical results to the class of $RCD(K,N)$ spaces, which is the synthetic notion of the class of riemannian manifolds satisfying $Ric \geq K$ and $dim \leq N$. This new class of spaces include Ricci limit spaces and Alexandrov spaces.  In particular, we will generalize the following:  Given a fixed dimension $n$, there exists a positive number $\varepsilon(n)$ so that any $n$-dimensional riemannian manifold satisfying almost nonnegative Ricci curvature, that is, $Ric_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)$, must have first Betti number smaller than or equal to $n$ and when it equals $n$ the manifold is a flat torus. Joint work with I. Mondello and A. Mondino.

Short bio: Raquel estudió la licenciatura en matemáticas en la Universidad de Guanajuato. Del 2009 al 2015 fue estudiante de doctorado de matemáticas de la Universidad de Stony Brook. Hizo un postdoctorado en el IMATE en la Ciudad de México y otro en el MSRI. Desde finales del 2016 se encuentra en el IMATE Oaxaca como Científica por México (antes catedrática CONACyT).  Sus temas de investigación son la geometría Riemanniana, el análisis geométrico y la teoría geométrica de la medida. Le gusta estudiar variedades con curvatura de Ricci acotada inferiormente, espacios de Alexandrov, espacios RCD(K,N) y nociones de curvatura escalar acotada inferiormente.

3. Keti Tenenblat (Universidade de Brasília, Brazil)

Title: On the self-similar solutions to the curvature flow for curves.

Abstract: We will present results on the soliton solutions to the curve shortening flow on the 2-dimensional unit sphere and on the 2-dimensional hyperbolic space. We will also discuss recent results on the self-similar solutions to the curvature flow and to the inverse curvature flow, for curves on the 2-dimensional light cone. In each case, the geometry of the curves will be described. The talk is based on my recent papers in collaboration with Hiuri F. dos Reis and Fabio N. da Silva.

4. Jimmy Petean (CIMAT, Mexico)

Title: The Yamabe invariant and the Yamabe equation

Abstract: In this talk, I will survey results on the Yamabe invariant, and use this to motivate results on the multiplicity of solutions to the Yamabe equation. The Yamabe invariant appears naturally in the study of the Hilbert-Einstein functional. There has been progress in the study and computation of the invariant, but many important questions about it remains open. In particular computations for Riemannian products are not known and this motivates the study of the space of solutions of the Yamabe equation. I will comment on some recent results in this direction.

Short bio: Jimmy Petean obtained his PhD in Mathematics in 1997 from the State University of New York at Stony Brook, under the supervision of Claude LeBrun. He spent one year as a postdoctoral visiting professor at the Max Planck Institut fur Mathematics at Bonn and after that, in 1998, joined the Centro de Investigacion en Matematicas (CIMAT) in Guanajuato, Mexico, where he is currently working. He works on problems on differential geometry and geometric analysis. His PhD thesis dealt with applications of Seiberg-Witten invariants to the study of indefinite Kahler-Einstein metrics on compact complex surfaces. He later studied the behavior of the Yamabe invariant under surgery and applications using cobordism theory. He also obtained results on the minimal entropy problem on closed manifolds, and multiplicity results for solutions of the Yamabe equation.

For the registration at the conference, please access: https://forms.gle/ugYxMAAaoJMHT3o7A