BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Neshan Wickramasekera (University of Cambridge) DTSTART:20260922T060000Z DTEND:20260922T080000Z DTSTAMP:20260827T132738Z UID:NCTS-GMT/38 DESCRIPTION:Title: Planar frequency\, asymptotic normal forms\, and local topology of area- minimizing currents.\nby Neshan Wickramasekera (University of Cambridg e) as part of NCTS international Geometric Measure Theory seminar\n\n\nAbs tract\nA fundamental problem in geometric measure theory is to understand the local structure of $n$-dimensional area-minimizing rectifiable current s $T$ of codimension at least 2. Almgren's 1983 theory provides a powerfu l general framework establishing the sharp Hausdorff dimension upper bound $n-2$ for the singular set (subsequently made more accessible by De Lelli s–Spadaro). The work of White\, Chang\, and Micallef–White gives a rem arkably complete structure theory when $T$ is 2-dimensional\, in which ca se the singularities are isolated. In higher dimensions\, however\, the lo cal structure of $T$ and the nature of its singularities remain much more subtle\, particularly at $branch$ $points$\, where one tangent cone is a plane.\n\nIn a series of papers with Brian Krummel\, we develop a new fram ework for this problem in arbitrary dimension $n$. \nIts geometric philoso phy differs from the classical theory: it unifies decay estimates at branc h points with the dimension and structure of the singular set\, as well as with the structure of $T$. \nA central conceptual novelty is the introduc tion of a new intrinsic frequency function\, called the $planar$ $frequenc y$. Unlike the frequency used in the classical theory\, planar frequency i s defined directly in terms of geometric quantities integrated over the cu rrent\, without first constructing auxiliary center manifolds at branch po ints. The approximate monotonicity of planar frequency provides quantitati ve control of the rate at which $T$ approaches planes and leads to a natur al decomposition of the singular set according to planar decay.\n\nI will describe this framework and some of its main consequences.\nAmong these ar e a more direct proof of Almgren's $n-2$ bound\, ${\\mathcal H}^{n-2}$-alm ost everywhere uniqueness of tangent cones\, and a detailed asymptotic des cription of $T$ at typical branch points. In particular\, at ${\\mathcal H }^{n-2}$-almost every branch point $Z$ there is a unique tangent plane\, a n intrinsic rational invariant—the $branching$ $order$ ${\\mathcal O}_{T }(Z) >1$—and a unique\, nonzero\, ${\\mathcal O}_{T}(Z)$-homogeneous cyl indrical multi-valued tangent function. Together\, these provide an $asymp totic$ $normal$ $form$ for $T$ at $Z$ with quantitative decay for the rema inder. Corollaries of this normal form include a locally finite decomposit ion of the singular set into disjoint\, locally compact\, locally $(n-2)$- rectifiable sets with locally finite ${\\mathcal H}^{n-2}$ measure\, and a sharp branching order criterion under which a branch point $Z$ is $classi cal$\; that is\, near $Z$\, the support of $T$ is homeomorphic to an \n$n$ -disk and admits a $C^{1\, \\mu}$ parameterization\, while the entire sing ular set is an $(n-2)$-dimensional $C^{1\, \\mu}$ submanifold consisting o nly of branch points with the same density and branching order as $Z$. Thi s is a natural higher-dimensional analogue of \nthe Chang–Micallef–Whi te structural description \nin dimension $2$. \n \n\nA central theme of th e talk will be how planar frequency avoids the need to construct center ma nifolds uniformly across all branch points as in the classical framework. Instead\, it identifies precisely the regime in which a center manifold be comes necessary and reduces its use to a canonical case. This reduction is crucial for our asymptotic normal form and also leads to substantial te chnical simplifications \nover the classical approach.\n\nI will also brie fly comment on related contemporaneous work of De Lellis\, Minter\, and S korobogatova.\n LOCATION:https://researchseminars.org/talk/NCTS-GMT/38/ END:VEVENT END:VCALENDAR