Title: Nearly G2-structures and G2-Laplacian co-flows

Abstract:  Nearly G2-structures in dimension seven are, up to scaling, critical points of a geometric flow called (modified) Laplacian co-flow. Moreover, since nearly G2-structures define Einstein metrics, they can also be associated to critical points of the volume-normalised Ricci flow. In this talk, we will discuss a recent joint work with Jason Lotay, showing that many of these nearly G2 critical points are unstable for the modified co-flow, and giving a lower bound on the index.

Room: 3-102.

Title: Rotational Surfaces with Second Fundamental Form of Constant Length

Abstract:  I will present a classification of rotational surfaces in three-dimensional Euclidean space whose second fundamental form

has constant length equal to one at every point. This defines a class of non-linear Weingarten surfaces. The tools used to obtain the

classification involve classical differential geometry and ordinary differential equations.

Room: 3-102.

Title: Gradient Einstein-type warped products: Rigidity, existence and nonexistence results via a nonlinear PDE

Abstract: We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry–Émery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results.

Room: 3-012.

Title: The holonomy of the Obata connection on Joyce hypercomplex manifolds

Abstract: The Obata connection is the hypercomplex analogue of the Levi-Civita connection on Riemannian manifolds. While a lot is known about Riemannian holonomy groups (including a full classification), very little is known about the Obata holonomy group. In this talk, I will present some new results exploring the Obata holonomy on compact Lie groups endowed with left invariant hypercomplex structures. The talk should be fairly non-technical and is based on a recent joint work with Beatrice Brienza, Giovanni Gentili and Luigi Vezzoni.

Room: 3-102.

Title: Non-existence results for warped product shrinking Ricci solitons

Abstract: A warped product Ricci soliton is a gradient Ricci soliton which is isometric to a warped product $M^n\times_{h}F^m$, where $M$ is the base, $F$ is the fiber, and $h:M\rightarrow\mathbb{R}$ is the warping function. Through the work of Ivey, Dancer-Wang, and others, we have examples of complete gradient Ricci solitons with this geometry, which are either steady or expanding. However, no examples in the shrinking case have shown up until this moment.

This talk aims to present non-existence results concerning warped product shrinking gradient Ricci solitons, including contributions given by the speaker.

Room: 4-003.

Title: Machine Learning Ricci-Flat Metrics

Abstract: Artificial Intelligence has rapidly become a powerful tool in scientific discovery, with growing impact across many disciplines—including Pure Mathematics. In particular, Machine Learning offers promising approaches to tackling computationally intensive problems in fields such as Complex Differential and Algebraic Geometry.

In this talk, we will focus on recent advances in the use of supervised learning to approximate Ricci-flat metrics on Calabi–Yau manifolds, a central challenge in both Mathematics and Theoretical Physics. We will also explore how data-driven methods can reveal hidden structures and patterns in Differential and Algebraic Geometry.

Keywords: Ricci-flat metrics; Calabi–Yau manifolds; Machine Learning in Geometry

Room: 3-011

Title: G-equivariant atoms

Abstract: In this talk, we introduce the concept of atoms developed by Katzarkov-Kontsevich-Pantev-Yu while presenting some generalizations, particularly in the G-equivalent context. We will discuss applications related to questions in G-equivariant birational geometry. This is based on two joint works (in progress) with Lino Grama, Ludmil Katzarkov, and Maxim Kontsevich.

Room: 3-102

Title: Subvariedades isométricas com aplicações de Gauss isométricas.

Abstract: Apresentarei um resultado recente sobre subvariedades isométricas do espaço Euclidiano, em codimensão 2, cujas aplicações de Gauss são também isométricas. Este é um trabalho em conjunto com M. Dajczer y Th. Vlachos.

Room: 3-102

Title: Invariant G2- instantons on homogeneous manifolds.

Abstract: In this talk, we will discuss recent advances on invariant G2-instantons on some particular homogeneous spaces. In the first part, we will consider 7-dimensional 2-step nilpotent Lie groups endowed with an invariant coclosed. For this case, we characterize the corresponding Lie algebras where the characteristic connection is a G2-instanton.  

In the second part, we will consider the seven dimensional Stiefel manifold V^{5,2}=SO(5)/SO(3). According to the reductive decomposition of  V^{5,2}, we provide an explicit description of all invariant G_2 structures on the Stiefel 7-manifold. As a consequence, we classify the invariant connections on homogeneous principal bundles over  V^{5,2} with gauge group U(1) and SO(3), satisfying either the instanton condition. 

Finally, as an application of both cases, we will analyze the heterotic G2-system which is the 7 dimensional analogue of the Hull-Strominger system in complex manifolds.This talk features two joint works, one with Viviana del Barco (Unicamp) and Andrew Clarke (UFRJ) and one with Luis Portilla (UBO).

Room: 3-102.

Title: Equivariant construction of spheres with Zoll families of minimal hypersurfaces.

Abstract: We studied a generalized notion of Zoll metric for the context of minimal surfaces. We construct examples of Euclidean spheres with such property. 

We also showed that the real projective spaces RP^n are flexible with this extended notion of Zoll metrics. This contrasts with the classical theory of Zoll metrics where the real projective space only has the round metrics as Zoll metrics.

Our construction is equivariant with respect to the orthogonal group, and this gives us information even for classical Zoll metrics. In particular, given any discrete subgroup G of O(3) which does not contain the antipodal map, we show that there exists Euclidean Zoll 2-spheres with the group of isometries being precisely G.

Room: 5-003.

Title: Rigidity of Einstein metrics and symmetric spaces

Abstract: The moduli space of Einstein metrics on a given closed manifold is not well-understood, even locally. I review the deformation theory of Einstein metrics and discuss some recent progress on compact symmetric spaces. In particular, I give a new conceptual description of their infinitesimal Einstein deformations. This is a joint effort with Stuart J. Hall and Uwe Semmelmann.

Room: 5-101.