The spring 2023 Ritt Lectures will take place on Monday, April 17, 2023 (4:30pm start time on Monday) and Friday, April 21, 2023 from 4:45 – 5:45pm in room 520. Professor Camillo De Lellis (Institute for Advanced Study), will deliver a two talk series titled:
Title: Area-minimizing integral currents: singularitiesand structure
Abstract: Area-minimizing integral currents are a naturalgeneralization of area-minimizing oriented surfaces. The concept was pioneeredby De Giorgi for hypersurfaces of the Euclidean space, and extended by Federerand Fleming to any codimension and general Riemannian ambients. These classicalworks of the fifties and sixties establish a general existence theory for theoriented Plateau problem of finding surfaces of least area spanning a givencontour.
Celebrated examples of singular 7-dimensional minimizersin R^8 and of singular 2-dimensional minimizers in R^4 are known since long andin fact in these cases there is no smooth oriented minimizer and any smoothminimizing sequence converges to the singular ones in an appropriate sense. A first theorem which summarizes the work of severalmathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, andFederer) and a second theorem of Almgren from 1980 give dimension bounds forthe singular set which match the one of the examples, in codimension 1 and ingeneral codimension respectively.
In these lectures I will focus on the case of generalcodimension and address the question of which structural results can be furtherproved for the singular set. A recent theorem by Liu proves that the latter canin fact be a fractal of any Hausdorff dimension \alpha \leq m-2. On the other hand it seems likely that it is an(m-2)-rectifiable set, i.e. that it can be covered by countably many C^1submanifolds leaving aside a set of zero (m-2)-dimensional Hausdorff measure.This conjecture is the counterpart, in general codimension, of a celebratedwork of Leon Simon in the nineties for the codimension 1 case. In these lectures I will explain why the problem is verychallenging, how it can be broken down into easier pieces, and present a lineof attack based on recent joint works with Anna Skorobogatova and Paul Minter.