On 24 and 27 February 2025, for the cycle “Crossings - Math Seminars”, Camillo de Lellis (IAS Princeton) will hold the following seminars:
#1
Technical Seminar: Area-minimizing integral currents: singularities and structure
Abstract: Area-minimizing integral currents were introduced by De Giorgi, Federer, and Fleming to build a successful existence theory for the {\em oriented} Plateau problem. While celebrated examples of singular minimizers were discovered soon after, a first theorem which summarizes the work of several mathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, and Federer) and a second theorem of Almgren from 1980 give general dimension bounds for the singular set which match the one of the examples, in codimension 1 and in general codimension respectively.
In joint works with Anna Skorobogatova and Paul Minter we prove that in higher codimension the singular set is (m-2)-rectifiable and the tangent cone is unique at a.e. point. Independently and at the same time, a proof of the same result has been discovered also by Krummel and Wickramasekera. This theorem is the counterpart, in general codimension, of a celebrated work of Leon Simon in the nineties for the codimension 1 case. Moreover, a recent theorem by Liu proves that the singular set can in fact be a fractal of any Hausdorff dimension <= m-2, indicating that the above structure theorem is indeed close to optimal.
Date and time: Monday 24 February, at 2:30 pm
Place: U1-06 Building, Quadrilatero della Scienza - Università di Milano-Bicocca
#2
Math Forum: When reason produces monsters while it is wide awake
Abstract: I turn with terror and horror from this lamentable scourge”. This sentence was uttered by a very famous mathematician towards the end of the XIX century, while referring to the work of another very famous colleague. The object which generated such virulent reaction is actually nowadays rather well accepted, in fact it is often mentioned in basic textbooks on differential calculus. In this lecture I will argue that it is just the first of a long series of counterintuitive mathematical constructions, which all share some common aspects. I will touch upon famous examples of the fifties, sixties, eighties and nineties of the last century, all looking like bizarre games of mathematicians made to defy common sense. However I will finally turn to some discoveries of the last few years, which confirm an old hypothesis of a theoretical physicist, recipient of the Nobel prize in chemistry!
Date and time: Thursday 27 February, at 2:30 pm
Place: U6-04 Building Room Martini, Piazza dell'Ateneo Nuovo, 1 - Università di Milano-Bicocca
A coffee break will follow
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