BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Brian Krummel (University of Melbourne) DTSTART:20230712T080000Z DTEND:20230712T100000Z DTSTAMP:20260423T005830Z UID:NCTS-GMT/17 DESCRIPTION:Title: Analysis of singularities of area minimizing currents\nby Brian Krum mel (University of Melbourne) as part of NCTS international Geometric Meas ure Theory seminar\n\n\nAbstract\nThe monumental work of Almgren in the ea rly 1980s showed that the singular set of a locally area minimizing rectif iable current $T$ of dimension $n$ and codimension $\\geq 2$ has Hausdorff dimension at most $n-2$. In contrast to codimension 1 area minimizers (f or which it had been established a decade earlier that the singular set ha s Hausdorff dimension at most $n-7$)\, the problem in higher codimension i s substantially more complex because of the presence of branch point singu larities\, i.e. singular points where one tangent cone is a plane of multi plicity 2 or larger. Almgren's lengthy proof (made more accessible and tec hnically streamlined in the much more recent work of De Lellis--Spadaro) s howed first that the non-branch-point singularities form a set of Hausdorf f dimension at most $n-2$ using an elementary argument based on the tangen t cone type at such points\, and developed a powerful array of ideas to ob tain the same dimension bound for the branch set separately. In this strat egy\, the exceeding complexity of the argument to handle the branch set st ems in large part from the lack of an estimate giving decay of $T$ towards a unique tangent plane at a branch point. \n\nWe will discuss a new appr oach to this problem (joint work with Neshan Wickramasekera). In this appr oach\, the set of singularities (of a fixed integer density $q$) is decomp osed not as branch points and non-branch-points\, but as a set ${\\mathcal B}$ of branch points where $T$ decays towards a (unique) plane faster tha n a fixed exponential rate\, and the complementary set ${\\mathcal S}$. T he set ${\\mathcal S}$ contains all (density $q$) non-branch-point singula rities\, but a priori it could also contain a large set of branch points. To analyse ${\\mathcal S}$\, the work introduces a new\, intrinsic frequen cy function for $T$ relative to a plane\, called the planar frequency func tion. The planar frequency function satisfies an approximate monotonicity property\, and takes correct values (i.e. $\\leq 1$) whenever $T$ is a con e (for which planar frequency is defined) and the base point is the vertex of the cone. These properties of the planar frequency function together with relatively elementary parts of Almgren’s theory (Dirichlet energy m inimizing multivalued functions and strong Lipschitz approximation) imply that $T$ satisfies a key approximation property along $S$: near each point of ${\\mathcal S}$ and at each sufficiently small scale\, $T$ is signific antly closer to some non-planar cone than to any plane. This property toge ther with a new estimate for the distance of $T$ to a union of non-interse cting planes and the blow-up methods of Simon and Wickramasekera imply tha t $T$ has a unique non-planar tangent cone at $\\mathcal{H}^{n-2}$-a.e. po int of $\\mathcal{S}$ and that ${\\mathcal S}$ is $(n-2)$-rectifiable with locally finite measure. Analysis of ${\\mathcal B}$ using the planar freq uency function and the locally uniform decay estimate along ${\\mathcal B} $ recovers Almgren’s dimension bound for the singular set of $T$ in a si mpler way\, and (again via Simon and Wickramasekera blow-up methods) shows that ${\\mathcal B}$ (and hence the entire singular set of $T$) is counta bly $(n-2)$-rectifiable with a unique\, non-zero multi-valued harmonic blo w-up at $\\mathcal{H}^{n-2}$-a.e. point of ${\\mathcal B}$.\n LOCATION:https://researchseminars.org/talk/NCTS-GMT/17/ END:VEVENT END:VCALENDAR