Abstract: In this talk, we use the framed bordism class of the Seiberg–Witten moduli space to give a negative answer to a question of Donaldson: whether, for a closed simply connected symplectic 4–manifold, the symplectic Torelli group is generated by squared Dehn twists along Lagrangian spheres. This is joint work with Hokuto Konno, Jianfeng Lin, and Juan Muñoz-Echániz.
Abstract: Harnack inequalities are among the most fundamental tools in the theory of partial differential equations, providing quantitative control over the oscillation of nonnegative solutions on small sets. Such estimates play a central role in the study of regularity, compactness, and the qualitative behavior of solutions to elliptic and parabolic equations. They also have significant geometric applications: Harnack-type estimates played an important role in the Ricci-flow analysis underlying Perelman’s proof of the Poincar ́e Conjecture, particularly in controlling curvature and understanding singularity formation.
In this talk, I will discuss Harnack inequalities for equations involving the Normalized Infinity-Laplacian in the presence of nonlinear lower-order gradient terms. This operator is both highly degenerate and singular, while the additional gradient interactions introduce further analytical challenges. Despite these difficulties, the approach relies on comparatively elementary yet robust methods that avoid some of the more technical machinery traditionally associated with highly degenerate elliptic equations, yielding new local estimates and Harnack-type bounds for positive solutions.
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