Abstracts

Higher regularity results for a nonlinear Fokker-Planck equation

In this talk, we investigate local regularity properties for a class of possibly degenerate nonlinear ultraparabolic operators, for which the nonlinear Kolmogorov-Fokker-Planck equation is a prototype. After proving local boundedness estimates for weak subsolutions, we combine them with a suitable Gehring lemma to obtain higher integrability estimates for the gradient in the diffusion direction. This is a joint work with J. Guerand and T. Isernia

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Smoothing effect of third-order operators with variable coefficients

In this talk, we will show the smoothing effect of a class of partial differential operators of order two or three with variable coefficients, which contain, among others, Schrödinger and KdV-type operators.
As an application of the smoothing estimates, a local well-posedness result for the associated nonlinear initial value problem will be given.

Singed Curvature Detection in the Primary Visual Cortex

We present a geometric model for curvature-sensitive cells in the primary visual cortex, motivated by the use of SE(2) geometry in modeling orientation detection. The central observation is that there exists a canonical affine subbundle of the cotangent bundle of the manifold of oriented contact elements of the retina, whose sections measure signed geodesic curvature along lifted curves, and which carries a natural Engel structure related to that of the double Cartan prolongation. On an open dense submanifold of this prolongation, the iterated Lie brackets of a pair of Engel generators span sim(2), identifying SIM(2) as the natural symmetry group for curvature detection. This leads to a two-layer integral transform, defined by composing the quasi-regular representations of SIM(2) and SE(2), which under a SIM(2)-equivariance condition on the mother window collapses to a SIM(2)-transform. The receptive profiles of curvature-sensitive cells are modeled by the coherent state family of this transform and characterized by a SIM(2)-adapted uncertainty principle. Work in collaboration with G. Citti.

Nonlocal theory for fractional kinetic equations

We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations  naturally arising in kinetic theory, which combine a first-order skew-symmetric operator with an elliptic operator involving fractional derivatives along only part of the coordinates. Under sufficient integrability along the transport variables on the nonlocal tail, we prove the first local L^\infty estimate for this class of equations. Then, we establish the first full Harnack inequality for solutions to kinetic integral equations under the aforementioned tail summability assumption, which appears in clear accordance with the very recent counterexample by Kassmann and Weidner (Adv. in Math., 2024). This is based on series of papaers by F. Anceschi, M. Kassmann, A. Loher, G. Palatucci, M. Weidner and myself.

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Renormalization of contact velocity fields with horizontal Sobolev regularity in Heisenberg groups

The classical Cauchy-Lipschitz theory ensures well-posedness of the flow equation associated with Lipschitz vector fields. A major breakthrough in extending this theory to rough velocity fields was achieved by DiPerna-Lions in the Sobolev setting, and later by Ambrosio in the BV framework. Since then, the theory has been significantly developed under various structural and regularity assumptions, both in Euclidean and metric measure settings.

In this talk, after reviewing the existing theory, we present a new well-posedness result for a class of rough velocity fields in the genuinely sub-Riemannian setting of the Heisenberg group. We describe the main ideas of our approach, and we explain why our result cannot be deduced either from existing Euclidean techniques or from available results in the metric measure framework. Based on a joint work with L. Ambrosio, G. Somma and D. Vittone.

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Hölder Regularity for Mixed Singular–Degenerate Doubly Nonlinear Equations

We investigate the local Hölder continuity of nonnegative, locally bounded solutions to a class of doubly nonlinear equations. We introduce a new approach that covers singular-degenerate and degenerate-singular cases, up to the relevant Barenblatt numbers. A key ingredient of the proof is a new integral L^1-L^1 Harnack estimate.