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<h2>cvgmt weekly bulletin</h2>
<p><i>Weekly bulletin for <a href="http://cvgmt.sns.it">http://cvgmt.sns.it</a></i></p>
<h2>Summary</h2>
<p><b>Seminars by:</b> <a href='/seminar/884/'>Krummel</a></p>
<p><b>New papers by:</b> <a href='/person/4738/' >Wozniak</a>, <a href='/person/1400/' >Scala</a>, <a href='/person/1318/' >Bate</a>, <a href='/person/3720/' >Carazzato</a>, <a href='/person/232/' >Alberti</a>, <a href='/person/1259/' >Trevisan</a>, <a href='/person/3371/' >Topaloglu</a>, <a href='/person/76/' >Marchese</a>, <a href='/person/1196/' >Pegon</a>, <a href='/person/47/' >Bellettini</a>, <a >Mariani</a>, <a href='/person/4658/' >Dipasquale</a>, <a href='/person/146/' >Scianna</a>, <a href='/person/1846/' >Nenna</a>, <a href='/person/151/' >Pratelli</a>, <a >Fleschler</a>, <a href='/person/58/' >De Lellis</a>, <a href='/person/188/' >Stroffolini</a></p>
<p><b>Modified papers by:</b> <a href='/person/2207/'>Pasqualetto</a>, <a href='/person/226/'>Gigli</a>, <a href='/person/2883/'>Antonelli</a>, <a >Mateu</a>, <a >Rondi</a>, <a href='/person/1517/'>Spolaor</a>, <a href='/person/174/'>Mora</a>, <a href='/person/2575/'>Pozzetta</a>, <a href='/person/336/'>Velichkov</a>, <a href='/person/4241/'>Gennaioli</a>, <a href='/person/22/'>De Philippis</a>, <a >Verdera</a>, <a href='/person/62/'>Scardia</a>, <a href='/person/2719/'>Semola</a></p>
<h2>Events next week</h2>
<ul>
<li><p>
<b><a href='/event/768/'>Hausdorff School "Analysis of PDEs: Variational and Geometric Perspectives"</a></b><br />
Mon 10 July 2023 - Fri 14 July 2023<br />
Bonn<br />
</p></li>
<li><p>
<b><a href='/event/783/'>RISM Summer School: Exotic solutions and well-posedness in PDEs and ODEs</a></b><br />
Mon 10 July 2023 - Thu 13 July 2023<br />
Villa Toeplitz, via G.B. Vico 46 , 21100 Varese - Italy<br />
</p></li>
<li><p>
<b><a href='/event/784/'>C.I.M.E. course 2023 - Variational and PDE Methods in Nonlinear Science</a></b><br />
Mon 10 July 2023 - Fri 14 July 2023<br />
Cetraro (CS), Italy<br />
</p></li>
<li><p>
<b><a href='/event/725/'>Summer School/Conference at Imperial College London</a></b><br />
Mon 10 July 2023 - Fri 14 July 2023<br />
Imperial College London<br />
</p></li>
</ul>
<h2>Seminars next week</h2>
<h3>Wed 12 July 2023</h3><ul>
<li><p>
Agenda: Get-together (30 min), presentation Brian Krummel (60 min), questions and discussions (30 min).<br />
Brian Krummel: <b><a href='/seminar/884/'>Analysis of singularities of area minimizing currents</a></b><br />
, 10:00<br />
</p>
<div style="font-size: smaller;"><p>The monumental work of Almgren in the early 1980s showed that the singular set of a locally area minimizing rectifiable current $T$ of dimension $n$ and codimension ≥ 2 has Hausdorff dimension at most $n − 2$. In contrast to codimension 1 area minimizers (for which it had been established a decade earlier that the singular set has Hausdorff dimension at most $n − 7$), the problem in higher codimension is substantially more complex because of the presence of branch point singularities, i.e. singular points where one tangent cone is a plane of multiplicity 2 or larger. Almgren’s lengthy proof (made more accessible and technically streamlined in the much more recent work of De Lellis-Spadaro) showed first that the non-branch-point singularities form a set of Hausdorff dimension at most $n − 2$ using an elementary argument based on the tangent cone type at such points, and developed a powerful array of ideas to obtain the same dimension bound for the branch set separately. In this strategy, the exceeding complexity of the argument to handle the branch set stems in large part from the lack of an estimate giving decay of $T$ towards a unique tangent plane at a branch point.
</p>
<p>We will discuss a new approach to this problem (joint work with Neshan Wickramasekera). In this approach, the set of singularities (of a fixed integer density $q$) is decomposed not as branch points and non-branch-points, but as a set $B$ of branch points where $T$ decays towards a (unique) plane faster than a fixed exponential rate, and the complementary set $S$. The set $S$ contains all (density $q$) non-branch-point singularities, but a priori it could also contain a large set of branch points. To analyze $S$, the work introduces a new, intrinsic frequency function for $T$ relative to a plane, called the planar frequency function. The planar frequency function satisfies an approximate monotonicity property, and takes correct values (i.e. ≤ 1) whenever T is a cone (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 minimizing multivalued functions and strong Lipschitz approximation) imply that $T$ satisfies a key approximation property along $S$: near each point of $S$ and at each sufficiently small scale, $T$ is significantly closer to some non-planar cone than to any plane. This property together with a new estimate for the distance of $T$ to a union of non-intersecting planes and the blow-up methods of Simon and Wickramasekera imply that $T$ has a unique non-planar tangent cone at $\mathcal H^{n-2}$-a.e. point of $S$ and that $S$ is $(n − 2)$-rectifiable with locally finite measure. Analysis of $B$ using the planar frequency function and the locally uniform decay estimate along $B$ recovers Almgren’s dimension bound for the singular set of $T$ in a simpler way, and (again via Simon and Wickramasekera blow-up methods) shows that $B$ (and hence the entire singular set of $T$ ) is countably $(n − 2)$-rectifiable with a unique, non-zero multi-valued harmonic blow-up at $\mathcal H^{n-2}$-a.e. point of $B$.</p>
</div>
</li>
</ul>
<h2>New Papers</h2>
<p><b> Mariani, Trevisan:</b> <a href='/paper/6107/'>Wasserstein Asymptotics for Brownian Motion on the Flat Torus and Brownian Interlacements</a></p>
<p><b> Wozniak:</b> <a href='/paper/6108/'>Sobolev regularity for linear growth functionals acting on $\mathbb{C}$-elliptic operators</a></p>
<p><b> Carazzato, Pratelli, Topaloglu:</b> <a href='/paper/6110/'>On the existence of minimizing sets for a weakly-repulsive non-local
energy</a></p>
<p><b> Dipasquale, Stroffolini:</b> <a href='/paper/6111/'>Manifold-constrained free discontinuity problems and Sobolev approximation</a></p>
<p><b> De Lellis, Fleschler:</b> <a href='/paper/6112/'>An elementary rectifiability lemma and some applications</a></p>
<p><b> Alberti, Bate, Marchese:</b> <a href='/paper/6113/'>On the closability of differential operators</a></p>
<p><b> Nenna, Pegon:</b> <a href='/paper/6114/'>Convergence rate of entropy-regularized multi-marginal optimal transport costs</a></p>
<p><b> Bellettini, Scala, Scianna:</b> <a href='/paper/6115/'>Upper bounds for the relaxed area of $S^1$-valued Sobolev maps and its countably subadditive interior envelope</a></p>
<h2>Modified Papers</h2>
<p><b> Velichkov:</b> <a href='/paper/4367/'>Regularity of the one-phase free boundaries</a></p>
<p><b> Antonelli, Pasqualetto, Pozzetta, Semola:</b> <a href='/paper/5685/'>Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds</a></p>
<p><b> Mateu, Mora, Rondi, Scardia, Verdera:</b> <a href='/paper/5752/'>Explicit minimisers for anisotropic Coulomb energies in 3D</a></p>
<p><b> Gennaioli, Gigli:</b> <a href='/paper/5760/'>A note about charts built by Eriksson-Bique and Soultanis on metric measure spaces</a></p>
<p><b> De Philippis, Spolaor, Velichkov:</b> <a href='/paper/5998/'>(Quasi-)conformal methods in two-dimensional free boundary problems</a></p>
<h2>Open Positions</h2>
<p><b>(new)</b> <a href="/position/794/">postdoc in Freiburg</a> (deadline: Tue 18 July 2023)</p>
<p><a href="/position/785/">PhD Position in Analysis at TU Dortmund</a> (deadline: Thu 20 July 2023)</p>
<p><b>(new)</b> <a href="/position/792/">3-year PDRA position</a> (deadline: Sun 23 July 2023)</p>
<p><a href="/position/786/">Ph.D. positions in Mathematics: Ferrara, Modena-Reggio Emilia, Parma</a> (deadline: Thu 27 July 2023)</p>
<p><a href="/position/789/">2 Posizioni di Ricercatore TD Tipo B - Mat/05 - Dipartimento di Matematica dell'Univ. di Napoli Federico II</a> (deadline: Thu 27 July 2023)</p>
<p><b>(new)</b> <a href="/position/790/">Post-doc Position in Mathematics in Padova</a> (deadline: Mon 31 July 2023)</p>
<p><b>(new)</b> <a href="/position/791/">PhD positions in Mathematics at the Università of Napoli Federico II</a> (deadline: Wed 2 August 2023)</p>
<p><a href="/position/780/">Professors of Mathematics at ETH Zurich</a> (deadline: Tue 15 August 2023)</p>
<p><a href="/position/787/">8 PhD positions in "Mathematical and Physical Sciences for Advanced Materials and Technologies" at the SSM in Napoli</a> (deadline: Fri 25 August 2023)</p>
<p><a href="/position/788/">Post Doc position in Analysis, University of Trento</a> (deadline: Mon 28 August 2023)</p>
<p><b>(new)</b> <a href="/position/793/">Post-doc position, CEREMADE (Paris Dauphine Université PSL)</a> (deadline: Thu 31 August 2023)</p>
<p><a href="/position/766/">PhD Position in Analysis/Geometry at the University of Fribourg</a> (deadline: Thu 31 August 2023)</p>
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