Title: Rigidity and volume growth of gradient solitons

Abstract: In this talk we consider a new program of studying solitons using a geometric flow for a general tensor q. Our approach is based in assumptions in order to extend results about Ricci solitons. In this direction, we identify an identity, first exploited in the pioneering work of Richard Hamilton in the case of Ricci solitons, which we call Hamilton's identity. We show that a version of this identity for an arbitrary geometric flow allows one to recover results about rigidity, the growth of the potential function, volume growth that have been proven for gradient Ricci solitons.

Title: Soon

Abstract: Soon.

Title: Classification of fractional Delaunay metrics on a twice-punctured sphere

Abstract: Fractional Delaunay metrics are the rotationally symmetric, periodic, complete metrics on the twice-punctured sphere with constant fractional Q_s-curvature, extending the classical Delaunay family from the scalar curvature case s = 1 to the nonlocal regime. In this talk, we present a full classification theorem: for every dimension n ≥ 3 and every fractional order s sufficiently close to one, any complete, conformally flat, constant Q_s-curvature metric on S^n \ {p, q} must be a member of the Delaunay family. The proof combines the Emden–Fowler reduction with a sharp comparison argument centered on the spherical bubble. The key input is that the bubble has a Morse index equal to one, which forces any solution whose maximum is normalized at t = 0 to coincide with a unique Delaunay profile. This collapses the moduli space to the necksize parameter and rules out all other possible behaviors. We further show that all fractional Delaunay metrics are nondegenerate for s close to one. These results provide a complete description of constant fractional Q_s-curvature metrics with two isolated singularities near the classical (s = 1) case.

Title: The Prékopa–Leindler inequality, stability and applications

Abstract: A cornerstone of convex analysis, the Prékopa–Leindler inequality is a fundamental tool in several related fields, such as partial differential equations and probability theory. In this talk, we explore how this classical inequality is connected to several important results in convex and geometric analysis, including log-Sobolev and Brascamp–Lieb inequalities.

As a by-product of our investigation, we show how recent results on the sharp stability of the Prékopa–Leindler inequality lead to sharpened versions of these inequalities. This, in turn, motivates the broader study of stability phenomena for Prékopa–Leindler and other geometric inequalities.

Based on joint work with Alessio Figalli, and with João Miguel Machado.

Title: Twisted Kähler–Einstein metrics on flag varieties

Abstract: In this talk, we present a description of invariant twisted Kähler–Einstein (tKE) metrics on flag varieties. In addition, we discuss applications of the methods used in the proof of our main result, particularly regarding the existence of invariant twisted constant scalar curvature Kähler metrics.

Moreover, we provide a precise description of the greatest Ricci lower bound for arbitrary Kähler classes on flag varieties. From this characterization, we establish a sequence of inequalities related to optimal upper bounds for the volume of Kähler metrics, relying exclusively on tools from Lie theory.

Title: Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

Abstract: Let (Σ², ds²) be a compact Riemannian surface, possibly with boundary, and consider Schrödinger-type operators of the form L = Δ + V − aK together with natural Robin and Steklov-type boundary conditions incorporating a boundary potential W and (in the curvature-corrected setting) the geodesic curvature κ_g of ∂Σ. Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of Σ and the integrals of V and W, obtained via a Hersch balancing argument on the capped surface. As a geometric application, we derive sharp topological restrictions for compact two-sided free boundary minimal surfaces of Morse index at most one inside geodesic balls of negatively curved pinched Cartan–Hadamard 3-manifolds under a mild radius condition.

Based on a joint work with Marcos P. Cavalcante and Vinicius Souza.