Mercoledì 23 settembre 2026, alle 17:00, in Aula Magna, avremo il piacere di ascoltare Andrea Merlo (Universidad del País Vasco),

che terrà un seminario dal titolo “Lipschitz functions and the geometry of rough sets".

Giovedì 24 settembre 2026, alle 16:00, in Aula Riunioni, avremo il piacere di ascoltare Malte Borken (Max Planck Institute for Mathematics in the Sciences)

che terrà un seminario dal titolo “A characterization of submanifolds of mxn matrices satisfying optimal rigidity estimates".

on Wednesday, 23 sept. 2026, 5pm, in Aula Magna, we will have the pleasure of listening to Andrea Merlo (Universidad del País Vasco);

the title of the talk is “Lipschitz functions and the geometry of rough sets”.

on Thursday, 24 sept. 2026, 4pm, in Aula Riunioni, we will have the pleasure of listening to Malte Borken (Max Planck Institute for Mathematics in the Sciences);

the title of the talk is “A characterization of submanifolds of mxn matrices satisfying optimal rigidity estimates”.

Abstract: Rademacher’s theorem tells us that Lipschitz functions on Euclidean space admit linear approximations at almost every point. Can the approximation properties of Lipschitz functions also tell us whether a set has a regular geometric structure?

For sets whose measure is comparable to a fixed power of the radius, David and Semmes proposed the following principle: if every Lipschitz function can be well approximated by affine functions, with quantitative control of the locations and scales where approximation fails, then the set must be uniformly rectifiable. This means that every ball contains a substantial portion of the set that admits a Lipschitz parametrization, with uniform bounds. This is the WALA conjecture.

In this talk, I present a proof of the conjecture and explain the geometric ideas connecting curves, differentiability and affine approximation. I also briefly discuss applications to elliptic partial differential equations, arising from joint work with Mihalis Mourgoglou, where estimates for solutions reveal geometric properties of the boundary.

Abstract: By a theorem of Friesecke, James and Müller from 2002, the set of rotations SO(n) satisfies the following rigidity estimate: If u is an H^1-vector field defined on some bounded Lipschitz domain, then the L^2-distance of Du to a single rotation can be controlled by the L^2-norm of the distance of Du to SO(n). We are interested in the following question: Exactly which structural properties of the set SO(n) enable it to satisfy the aforementioned rigidity estimate? I will discuss a new approach to the problem, which allows one to find necessary and sufficient conditions for a general compact C^1-submanifold of mxn matrices to satisfy such a rigidity estimate. Furthermore, this approach allows one to show that the estimate is stable under small graphical perturbations of the submanifold. This talk is based on my master's thesis, written under the supervision of Konstantinos Zemas and Sergio Conti.

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Luigi Forcella, Associate Professor
University of Pisa
Department of Mathematics
Largo Bruno Pontecorvo 5
57127 Pisa, Italy

https://pagine.dm.unipi.it/forcella/index.html