IMPA - O Instituto de Matemática Pura e Aplicada

The Brauer group of an algebraic variety, first defined by Grothendieck in 1968, is a fundamental and poorly understood object in arithmetic geometry. In this talk I will survey recent techniques and results, joint with A.N. Skorobogatov, that determine Brauer groups of diagonal and related surfaces, namely those defined by an equality of two polynomials. In order to do so, one needs a sufficient understanding of the integral l-adic and singular cohomologies of such varieties.  Online talk: meet.google.com/hzq-vads-ntx

Recent works by M. Gromov, A. Petrunin, O. Chodosh, and C. Li deals with the question: Given a closed smooth manifold M, minimize the maximum normal curvatures among all immersions of M into (high-dimensional) Euclidean spaces, under the constraint that the image lies in a closed ball of radius one.

Replacing this constraint with the condition that the (extrinsic) diameter is at most two, one arrives at a different, but still natural, question.

The result I will present is a first step in the exploration of this new question, namely: For any immersed submanifold M with diameter at most two, the maximum normal curvature is at least one, and equality holds if and only if M is diffeomorphic to a sphere or (real/complex/quaternionic/octonionic) projective space, embedded either as an affine sphere, or as a "Veronese" variety.

Time allowing, I will discuss: (1) a generalization to submanifolds of spheres and hyperbolic spaces, (2) the proof, which uses A. Schur's "Bow" Lemma and K. Sakamoto's classification of submanifolds with planar geodesics, and (3) natural open questions.

I will describe how many Poisson varieties that have originated in Lie theory like Poisson-Lie groups, flag varieties and generalized Schubert cells may be viewed as decorated moduli spaces of flat connections on surfaces. This interpretation leads to a simple construction of their symplectic groupoids and corresponding Morita equivalences. This talk is based on https://arxiv.org/pdf/2504.09293

We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in Fourier-Lebesgue spaces. In particular, the admissible regularities go beyond the  $H^{-1}$-scale, which is a well-known threshold for KdV. As a byproduct, by reversing the gauge transform, we are able to improve on the known theory for KdV and derive sharp local well-posedness in Fourier-Lebesgue spaces with large integrability exponent. Our strategy is based on an infinite normal form reduction and Fourier restriction estimates, together with a thorough exploitation of algebraic cancellations. Additionally, our method is totally independent of the KdV completely integrable structure, and extends to other non-integrable models with quadratic nonlinearities.

Singular Riemannian foliations are certain partitions of Riemannian manifolds, and the traditional sources of examples are isometric group actions and isoparametric hypersurfaces. When the ambient manifold is a sphere, it has long been known that such examples are, in the appropriate sense, algebraic. In 2018, Lytchak and Radeschi have shown algebraicity for a general singular Riemannian foliation in a sphere. In joint work with Samuel Lin and Marco Radeschi, we extend the Lytchak-Radeschi theorem from spheres to any compact normal homogeneous space, a class that includes all compact symmetric spaces. Time-permitting, I'll comment on the ingredients of the proof(s).