BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andrew Sageman-Furnas (University Göttingen) DTSTART:20200421T170000Z DTEND:20200421T180000Z DTSTAMP:20260422T225637Z UID:OSGA/1 DESCRIPTION:Title: Nav igating the space of Chebyshev nets\nby Andrew Sageman-Furnas (Univers ity Göttingen) as part of Online Seminar "Geometric Analysis"\n\n\nAbstra ct\nMany materials are built from a grid of flexible but nearly inextensib le rods that behaves as a shell-like structure. Everyday examples range fr om fabrics made of 1000s of interwoven yarns\; to kitchen strainers made o f 100s of plastically deforming wires\; to architectural gridshells or med ical stents made of 10s of elastically deforming rods. In this talk\, I em phasize the geometric constraints common to these different physical syste ms. We build from a differential geometric model for woven fabric\, initia lly introduced by Pafnuty Chebyshev in 1878\, that directly encodes the in extensibility of the two families of rods.\n\nWe discuss the theory of Che byshev nets through a series of applied\, collaborative efforts in computa tional fabrication and inverse design. Theoretical obstructions expose the challenges in finding Chebyshev nets on surfaces with large amounts of cu rvature\, suggesting a limited shape space. However\, we show that a caref ul reformulation of the problem\, combined with a discrete analog of Cheby shev nets\, leads to computational tools that reveal a vibrant design spac e.\n LOCATION:https://researchseminars.org/talk/OSGA/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Siran Li (Rice University) DTSTART:20200421T180000Z DTEND:20200421T190000Z DTSTAMP:20260422T225637Z UID:OSGA/2 DESCRIPTION:Title: Iso metric Immersions of Riemannian Manifolds into Euclidean Spaces\, Revisite d\nby Siran Li (Rice University) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe existence of isometric immersions of Riemanni an\nmanifolds into ambient Euclidean spaces has been a classical problem\n in geometric analysis and nonlinear PDEs. Seminal works by Darboux\,\nWeyl \, Nirenberg\, Nash\, Gromov\, etc. etc. have addressed this problem\nfrom different perspectives. In this talk we discuss three approaches\,\nsome are probably less known\, to the isometric immersions problem.\nThese incl ude (1)\, pseudo-holomorphic curve formulation of the Weyl\nproblem due to F. Labourie\; (2)\, Uhlenbeck gauge formulation for the\nPfaff system\; a nd (3)\, the fluid mechanical formulation for negatively\ncurved surfaces. \n LOCATION:https://researchseminars.org/talk/OSGA/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Bastian Käfer (RWTH Aachen) DTSTART:20200428T170000Z DTEND:20200428T180000Z DTSTAMP:20260422T225637Z UID:OSGA/3 DESCRIPTION:Title: A M öbius invariant energy for sets of arbitrary dimension and codimension\nby Bastian Käfer (RWTH Aachen) as part of Online Seminar "Geometric An alysis"\n\n\nAbstract\nWe consider the family of Möbius invariant energie s for m-dimensional submanifolds of $\\mathbb R^n$\, introduced by R. Kusn er and J. Sullivan\, defined on a class of sets\, which are given by the u nion of Lipschitz graphs and satisfy an additional condition of "nice" sel f-intersection.\nWe show for these sets that finite energy implies Reifenb erg-flatness through estimating the energy of certain subsets.\nThis final ly leads to a local representation given by a single graph and prevents an y kind of self-intersection.\nAs an immediate implication\, we obtain that every immersed $C^1$ manifold with finite energy is embedded.\nThis is jo int work with Heiko von der Mosel.\n LOCATION:https://researchseminars.org/talk/OSGA/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Huy The Nguyen (Queen Mary University London) DTSTART:20200505T170000Z DTEND:20200505T180000Z DTSTAMP:20260422T225637Z UID:OSGA/4 DESCRIPTION:Title: Hig h codimension mean curvature flow and surgery\nby Huy The Nguyen (Quee n Mary University London) as part of Online Seminar "Geometric Analysis"\n \nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/OSGA/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Jesse Ratzkin (University Würzburg) DTSTART:20200512T170000Z DTEND:20200512T180000Z DTSTAMP:20260422T225637Z UID:OSGA/5 DESCRIPTION:Title: On constant Q-curvature metrics with isolated singularities and a related fou rth order conformal invariant\nby Jesse Ratzkin (University Würzburg) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe Q-curva ture of a Riemannian manifold is a higher order analog of its scalar curva ture\, and so many people have over the last two decades proven results ab out Q-curvature mirroring theorems about scalar curvature. I will present two such results. First\, I will describe a refined asymptotic expansion o f isolated singularities in the conformally flat case\, similar to work of Caffarelli\, Gidas and Spruck in the scalar curvature setting. Then I wil l describe a conformal invariant and prove a convergence result similar to a theorem of Schoen.\n LOCATION:https://researchseminars.org/talk/OSGA/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Volker Branding (University Vienna) DTSTART:20200519T170000Z DTEND:20200519T180000Z DTSTAMP:20260422T225637Z UID:OSGA/6 DESCRIPTION:Title: Hig her order generalizations of harmonic maps\nby Volker Branding (Univer sity Vienna) as part of Online Seminar "Geometric Analysis"\n\nAbstract: T BA\n LOCATION:https://researchseminars.org/talk/OSGA/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Katharina Brazda (University Vienna) DTSTART:20200526T170000Z DTEND:20200526T180000Z DTSTAMP:20260422T225637Z UID:OSGA/7 DESCRIPTION:Title: The Canham-Helfrich model for multiphase biomembranes\nby Katharina Brazd a (University Vienna) as part of Online Seminar "Geometric Analysis"\n\n\n Abstract\nBiological membranes adopt a fascinating variety of shapes. The Canham-Helfrich variational model describes their equilibrium configuratio ns as surfaces of minimal elastic bending energy under area and volume con straints. In case of heterogeneous membranes with multiple phases\, latera l fluidity gives rise to an additional coupling between composition and cu rvature. We present an existence result for multiphase Canham-Helfrich min imizers with sharp phase interfaces obtained in the framework of oriented curvature varifolds with boundary. This is joint work with Luca Lussardi a nd Ulisse Stefanelli.\n LOCATION:https://researchseminars.org/talk/OSGA/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Miles Simon (University Magdeburg) DTSTART:20200623T170000Z DTEND:20200623T180000Z DTSTAMP:20260422T225637Z UID:OSGA/8 DESCRIPTION:Title: On the regularity of Ricci flows coming out of metric spaces.\nby Miles S imon (University Magdeburg) as part of Online Seminar "Geometric Analysis" \n\n\nAbstract\nJoint work with Alix Deruelle\, Felix Schulze\n\nWe consid er solutions to Ricci flow defined on manifolds M for a time interval $(0\ ,T)$ whose Ricci curvature is bounded uniformly in time from below\, and f or which the norm of the full curvature tensor at time $t$ is bounded by $c/t$ for some fixed constant $c>1$ for all $t \\in (0\,T)$.\nFrom previo us works\, it is known that if the solution is complete for all times $t>0 $\, then there is a limit\nmetric space $(M\,d_0)$\, as time t approaches zero. We show : if there is a open region $V$ on which $(V\,d_0)$ is *smoo th*\, then the\nsolution can be extended smoothly to time zero on $V$.\n LOCATION:https://researchseminars.org/talk/OSGA/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Elena Mäder-Baumdicker (University Darmstadt) DTSTART:20200505T180000Z DTEND:20200505T190000Z DTSTAMP:20260422T225637Z UID:OSGA/9 DESCRIPTION:Title: The Morse index of Willmore spheres and its relation to the geometry of minim al surfaces\nby Elena Mäder-Baumdicker (University Darmstadt) as part of Online Seminar "Geometric Analysis"\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/OSGA/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Remy Rodiac (University Paris-Saclay) DTSTART:20200407T170000Z DTEND:20200407T180000Z DTSTAMP:20260422T225637Z UID:OSGA/11 DESCRIPTION:Title: In ner variations and limiting vorticities for the Ginzburg-Landau equations< /a>\nby Remy Rodiac (University Paris-Saclay) as part of Online Seminar "G eometric Analysis"\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/OSGA/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Steenebrügge (RWTH Aachen) DTSTART:20200414T170000Z DTEND:20200414T180000Z DTSTAMP:20260422T225637Z UID:OSGA/12 DESCRIPTION:Title: A speed preserving Hilbert gradient flow for generalized integral Menger cur vature\nby Daniel Steenebrügge (RWTH Aachen) as part of Online Semina r "Geometric Analysis"\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/OSGA/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Marc Pegon (Université de Paris) DTSTART:20200616T170000Z DTEND:20200616T180000Z DTSTAMP:20260422T225637Z UID:OSGA/13 DESCRIPTION:Title: Pa rtial regularity for fractional harmonic maps into spheres\nby Marc Pe gon (Université de Paris) as part of Online Seminar "Geometric Analysis"\ n\n\nAbstract\nSimilarly to “classical” harmonic maps\, which are crit ical points of the Dirichlet energy\, fractional harmonic maps are defined as critical points of a fractional Dirichlet energy associated with the $ s$-power of the Laplacian\, for $s \\in (0\,1)$.\nIn this talk\, after a b rief reminder on classical harmonic maps\, I will present the fractional s etting and the partial regularity results we have obtained for maps valued into a sphere. In the case of half harmonic maps ($s=\\frac{1}{2}$)\, I w ill also recall the connection with minimal surfaces with free boundary\, which allowed us to improve known regularity results for energy minimizing maps into spheres.\n LOCATION:https://researchseminars.org/talk/OSGA/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Peter Topping (University of Warwick) DTSTART:20200630T170000Z DTEND:20200630T180000Z DTSTAMP:20260422T225637Z UID:OSGA/14 DESCRIPTION:Title: Un iqueness of limits in geometric flows\nby Peter Topping (University of Warwick) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nQu ite often when considering long-time behaviour of geometric flows\, or con sidering blow-ups of singularities in geometric PDE\, we extract limits us ing soft compactness arguments. For example\, a flow might easily be seen to converge to a limit at a *sequence* of times converging to infinity.\nT he more subtle question is then whether the flow converges as time converg es to infinity\, without having to restrict to a sequence of times.\n\nI w ill outline some of the issues that arise in this subject\, focussing on g radient flows for the harmonic map energy\, and sketch some recent work wi th M.Rupflin and J.Kohout.\n LOCATION:https://researchseminars.org/talk/OSGA/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Ruben Jakob (Technion) DTSTART:20200707T170000Z DTEND:20200707T180000Z DTSTAMP:20260422T225637Z UID:OSGA/15 DESCRIPTION:Title: Ge neric full smooth convergence of the elastic energy flow in the 2-sphere\nby Ruben Jakob (Technion) as part of Online Seminar "Geometric Analysi s"\n\n\nAbstract\nThe speaker is going to present his recent investigation of the ``Moebius\ninvariant Willmore flow'' (MIWF) in the 3-sphere and of some particular version of the\n``elastic energy flow'' (EEF) in the 2-sp here. We will discuss the\ninteraction between these two geometric flows v ia the Hopf fibration and the\nresulting possibility to transfer particula r insights about the ``EEF'' to\nthe ``MIWF''\, and vice versa special ins ights about the ``MIWF'' back\nto the ``EEF''. A big motivation for this p arallel investigation is the\nannounced proof (by the speaker) of the ''ge neric full smooth convergence''\nof the ``EEF'' in the 2-sphere.\n LOCATION:https://researchseminars.org/talk/OSGA/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Myfanwy Evans (University of Potsdam) DTSTART:20200616T180000Z DTEND:20200616T190000Z DTSTAMP:20260422T225637Z UID:OSGA/16 DESCRIPTION:Title: Ge ometric modelling of tangled structures\nby Myfanwy Evans (University of Potsdam) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\n This talk will introduce the use of geometric ideas in the characterisatio n and analysis of tangled biophysical systems. It will introduce the const ruction of idealised tangled structures using ideas of both symmetry and h omotopy of tangled lines on surfaces. These structures provide an extensiv e set of tangling motifs for the exploration of the behaviour of tangled m icrostructures in liquids\, and I will show preliminary results working to wards this goal\, including an example of the geometry-driven swelling of human skin cells.\n LOCATION:https://researchseminars.org/talk/OSGA/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Lynn Heller (University of Hannover) DTSTART:20200602T170000Z DTEND:20200602T180000Z DTSTAMP:20260422T225637Z UID:OSGA/17 DESCRIPTION:Title: Ar ea Estimates for High genus Lawson surfaces via DPW\nby Lynn Heller (U niversity of Hannover) as part of Online Seminar "Geometric Analysis"\n\n\ nAbstract\nStarting at a saddle tower surface\, we give a new existence pr oof of the Lawson surfaces\n$\\xi_{m\,k}$ of high genus by dropping some c losing conditions of the surface and then\ndeforming the corresponding DPW potential. As a byproduct\, we obtain for fixed mestimates\non the area o f $\\xi_{m\,k}$ in terms of their genus $g= mk \\gg 1$. This is joint work with\nSebastian Heller and Martin Traizet.\n LOCATION:https://researchseminars.org/talk/OSGA/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Sven Pistre (RWTH Aachen University) DTSTART:20200609T170000Z DTEND:20200609T180000Z DTSTAMP:20260422T225637Z UID:OSGA/18 DESCRIPTION:Title: Th e Radon transform and higher regularity of surfaces minimising a Finsler a rea\nby Sven Pistre (RWTH Aachen University) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nA Finsler metric is a smooth family o f smooth norms on the tangent bundle of a manifold. One possible generalis ation of the usual Riemannian notion of area in Finsler geometry is the Bu semann-Hausdorff area functional. In this talk I will consider high-codime nsional disk-type surfaces which minimise this area with respect to Platea u boundary conditions. $\\\\$\nI will show that the Busemann-Hausdorff are a functional fits into the Hildebrandt-von der Mosel framework on Cartan f unctionals. Existence of minimisers is then guaranteed under mild growth c onditions of the Finsler metric. Higher regularity ($W^{2\,2}_{\\textrm{lo c}} \\cap C^{1\,\\mu}$) of minimisers can be achieved by using functional analytic properties of the Radon transform. \n$\\\\$\nThe latter is an ope rator which assigns a function on the $(n−1)$-sphere its mean by integra tion over $(m-1)$-dimensional subspheres. One crucial property of this ope rator is its equivariance with respect to a Lie group action on the sphere and the $m$-Grassmannian. An infinitesimal version of this equivariance y ields the regularity results about area minimisers.\n LOCATION:https://researchseminars.org/talk/OSGA/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Christian Bär (University of Potsdam) DTSTART:20200714T170000Z DTEND:20200714T180000Z DTSTAMP:20260422T225637Z UID:OSGA/19 DESCRIPTION:Title: Co unter-intuitive approximations\nby Christian Bär (University of Potsd am) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe Nash -Kuiper embedding theorem is a prototypical example of a counter-intuitive approximation result: any short embedding of a Riemannian manifold into E uclidean space can be approximated by *isometric* ones. As a consequence\, any surface can be isometrically $C^1$-embedded into an arbitrarily small ball in $\\mathbb{R}^3$. For $C^2$-embeddings this is impossible due to c urvature restrictions.\n\nWe will present a general result which will allo w for approximations by functions satisfying strongly overdetermined equat ions on open dense subsets. This will be illustrated by three examples: re al functions\, embeddings of surfaces\, and abstract Riemannian metrics on manifolds.\n\nOur method is based on "weak flexibility"\, a concept intro duced by Gromov in 1986. This is joint work with Bernhard Hanke (Augsburg) .\n LOCATION:https://researchseminars.org/talk/OSGA/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Carla Cederbaum (University of Tübingen) DTSTART:20200721T170000Z DTEND:20200721T180000Z DTSTAMP:20260422T225637Z UID:OSGA/20 DESCRIPTION:Title: On CMC-foliations of asymptotically flat manifolds\nby Carla Cederbaum ( University of Tübingen) as part of Online Seminar "Geometric Analysis"\n\ n\nAbstract\nIn 1996\, Huisken and Yau proved existence of foliations by c onstant mean curvature (CMC) surfaces in the asymptotic end of an asymptot ically Euclidean Riemannian manifold. Their work has inspired the study of various other foliations in asymptotic ends\, most notably the foliations by Willmore surfaces (Lamm\, Metzger\, Schulze) and by constant expansion /null mean curvature surfaces in the context of asymptotically Euclidean i nitial data sets in General Relativity (Metzger). I will present a new fol iation by constant spacetime mean curvature surfaces (STCMC)\, also in the context of asymptotically Euclidean initial data sets in General Relativi ty (joint work with Sakovich). The STCMC-foliation is well-suited to defin e a notion of total center of mass in General Relativity.\n LOCATION:https://researchseminars.org/talk/OSGA/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Nadine Große DTSTART:20200728T170000Z DTEND:20200728T180000Z DTSTAMP:20260422T225637Z UID:OSGA/21 DESCRIPTION:Title: Bo undary value problems on singular domains: an approach via bounded geometr ies\nby Nadine Große as part of Online Seminar "Geometric Analysis"\n \n\nAbstract\nIn this talk\, we consider boundary value problems on domain s \nwith non smooth boundaries. We approach this problem by transferring i t\nto non-compact manifolds with a suffiently nice geometry -- the bounded geometry.\nThis gives a more general framework that allows to handle Diri chlet (or\nDirichlet-Neumann mixed) boundary value problems for domains wi th a\nlarger class of singularities on the boundary and gives a nice geome tric\ninterpretation. This is joint work with Bernd Ammann\n(Regensburg) a nd Victor Nistor (Metz).\n LOCATION:https://researchseminars.org/talk/OSGA/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Melanie Rupflin (University of Oxford) DTSTART:20200804T170000Z DTEND:20200804T180000Z DTSTAMP:20260422T225637Z UID:OSGA/22 DESCRIPTION:Title: Ł ojasiewicz inequalities near simple bubble trees for the $H$ surface equat ion\nby Melanie Rupflin (University of Oxford) as part of Online Semin ar "Geometric Analysis"\n\n\nAbstract\nIn this talk we discuss a gap pheno menon for critical points of\nthe $H$-functional on closed non-spherical s urfaces when $H$ is constant\, and in\nthis setting furthermore prove that sequences of almost critical points\nsatisfy Łojasiewicz inequalities as they approach the first non-trivial\nbubble tree.\n\nTo prove these resul ts we derive sufficient conditions for Łojasiewicz\ninequalities to hold near a finite-dimensional submanifold of\nalmost-critical points for suita ble functionals on a Hilbert space.\n\nThe presented results are joint wor k with Andrea Malchiodi and Ben Sharp.\n LOCATION:https://researchseminars.org/talk/OSGA/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Julian Scheuer (Cardiff University) DTSTART:20200811T170000Z DTEND:20200811T180000Z DTSTAMP:20260422T225637Z UID:OSGA/23 DESCRIPTION:Title: Co ncavity of solutions to elliptic equations on the sphere\nby Julian Sc heuer (Cardiff University) as part of Online Seminar "Geometric Analysis"\ n\n\nAbstract\nAn important question in PDE is when a solution to an ellip tic\nequation is concave. This has been of interest with respect to the sp ectrum of\nlinear equations as well as in nonlinear problems. An old techn ique going back\nto works of Korevaar\, Kennington and Kawohl is to study a certain two-point\nfunction on a Euclidean domain to prove a so-called c oncavity maximum principle\nwith the help of a first and second derivative test. To our knowledge\, so far\nthis technique has never been transferre d to other ambient spaces\, as the\nnonlinearity of a general ambient spac e introduces geometric terms into the\nclassical calculation\, which in ge neral do not carry a sign. In this talk we\nhave a look at this situation on the unit sphere. We prove a concavity maximum\nprinciple for a broad cl ass of degenerate elliptic equations via a careful\nanalysis of the spheri cal Jacobi fields and their derivatives. In turn we obtain\nconcavity of s olutions to this class of equations. This is joint work with Mat\nLangford \, University of Tennessee Knoxville.\n LOCATION:https://researchseminars.org/talk/OSGA/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Richard Bamler (UC Berkeley) DTSTART:20200818T170000Z DTEND:20200818T180000Z DTSTAMP:20260422T225637Z UID:OSGA/24 DESCRIPTION:Title: Ri cci flow in higher dimensions\nby Richard Bamler (UC Berkeley) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nI will present new results concerning Ricci flows in higher dimensions.\n LOCATION:https://researchseminars.org/talk/OSGA/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Max Engelstein DTSTART:20200825T170000Z DTEND:20200825T180000Z DTSTAMP:20260422T225637Z UID:OSGA/25 DESCRIPTION:Title: Wi nding for Wave Maps\nby Max Engelstein as part of Online Seminar "Geom etric Analysis"\n\n\nAbstract\nWave maps are harmonic maps from a Lorentzi an domain to a\nRiemannian target. Like solutions to many energy critical PDE\, wave maps can\ndevelop singularities where the energy concentrates o n arbitrary small\nscales but the norm stays bounded. Zooming in on these singularities yields\na harmonic map (called a soliton or bubble) in the w eak limit. One\nfundamental question is whether this weak limit is unique\ , that is to say\,\nwhether different bubbles may appear as the limit of d ifferent sequences of\nrescalings.\n\nWe show by example that uniqueness m ay not hold if the target manifold is\nnot analytic.  Our construction is heavily inspired by Peter Topping's\nanalogous example of a ``winding" bu bble in harmonic map heat flow. However\,\nthe Hamiltonian nature of the w ave maps will occasionally necessitate\ndifferent arguments.  This is joi nt work with Dana Mendelson (U Chicago).\n LOCATION:https://researchseminars.org/talk/OSGA/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Peter McGrath (North Carolina State University) DTSTART:20200901T170000Z DTEND:20200901T180000Z DTSTAMP:20260422T225637Z UID:OSGA/26 DESCRIPTION:Title: Qu antitative Isoperimetric Inequalities on Riemannian Surfaces\nby Peter McGrath (North Carolina State University) as part of Online Seminar "Geom etric Analysis"\n\n\nAbstract\nTalk Abstract: In this talk\, we introduce a scattering asymmetry which measures the asymmetry of a domain by quanti fying its incompatibility with an isometric circle action. We prove a qua ntitative isoperimetric inequality involving the scattering asymmetry and characterize the domains with vanishing scattering asymmetry by their rota tional symmetry. We also give a new proof of the sharp Sobolev inequality for Riemannian surfaces which is independent of the isoperimetric inequal ity. This is joint work with J. Hoisington.\n LOCATION:https://researchseminars.org/talk/OSGA/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Ryan Alvarado (Amherst College) DTSTART:20200908T170000Z DTEND:20200908T180000Z DTSTAMP:20260422T225637Z UID:OSGA/27 DESCRIPTION:Title: A characterization of the Sobolev embedding theorem in metric-measure spaces .\nby Ryan Alvarado (Amherst College) as part of Online Seminar "Geome tric Analysis"\n\n\nAbstract\nHistorically\, the Sobolev embedding theorem on domains has played a key role in establishing many fundamental results in the area of analysis and it is well known that the geometry of the und erlying domain is intimately linked to the availability of these embedding s. In fact\, certain geometrical characterizations of domains which suppor t Sobolev embeddings have been obtained in the Euclidean setting. In this talk\, we will revisit these embedding theorems in the more general contex t of metric-measure spaces and discuss some recent work which identifies a measure theoretic condition that is both necessary and sufficient to ensu re their veracity. A measure characterization of Sobolev extension domains in the metric setting as well as applications of our methods to spaces su pporting $p$-Poincaré inequalities will also be discussed. This talk is b ased on joint work with Przemysław Górka (Warsaw University of Technolog y)\, Piotr Hajłasz (University of Pittsburgh).\n LOCATION:https://researchseminars.org/talk/OSGA/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Fritz Hiesmayr (University College London) DTSTART:20200922T170000Z DTEND:20200922T180000Z DTSTAMP:20260422T225637Z UID:OSGA/29 DESCRIPTION:Title: A rigidity theorem for the Allen-Cahn equation in $S^3$\nby Fritz Hiesma yr (University College London) as part of Online Seminar "Geometric Analys is"\n\n\nAbstract\nWe present a recent rigidity theorem for the Allen-Cahn equation in the three-sphere: critical points with Morse index are symmet ric and vanish on a Clifford torus. One key ingredient is a novel Frankel- type property we establish for the nodal sets of any two distinct solution s: they intersect if they are connected. This in fact holds in all manifol ds with positive Ricci curvature. Time permitting we will discuss addition al rigidity results in higher-dimensional spheres.\n LOCATION:https://researchseminars.org/talk/OSGA/29/ END:VEVENT BEGIN:VEVENT SUMMARY:John Maddocks (EPF Lausanne) DTSTART:20201124T180000Z DTEND:20201124T190000Z DTSTAMP:20260422T225637Z UID:OSGA/31 DESCRIPTION:Title: Id eal knots: The trefoil\, analysis and numerics to experiment\nby John Maddocks (EPF Lausanne) as part of Online Seminar "Geometric Analysis"\n\n \nAbstract\nGeometrical knot theory is an area of mathematics that has bee n growing in\nactivity over the last few decades. It involves the study of specific shapes\nof knotted curves\, rather than their topology\, where t he specific knot shape\nis fixed by some criterion\, typically minimizing some form of knot energy.\nIn this talk I will introduce some older work o f both my collaborators and\nI\, as well as others\, on  the specific cas e of ideal\, or tightest\, knot\nshapes. I will start by explaining the an alytical difficulties\, along with\nsome associated theorems. Then I will describe some numerical results\nconcentrating on the specific case of the ideal trefoil. And finally I will\ndescribe some very recent experimental results for the ideal trefoil\nobtained by the group of Pedro Reis at the EPFL.\n LOCATION:https://researchseminars.org/talk/OSGA/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Marius Müller (Albert-Ludwigs-Universität Freiburg) DTSTART:20201201T180000Z DTEND:20201201T190000Z DTSTAMP:20260422T225637Z UID:OSGA/32 DESCRIPTION:Title: Th e Willmore Flow of Tori of Revolution\nby Marius Müller (Albert-Ludwi gs-Universität Freiburg) as part of Online Seminar "Geometric Analysis"\n \n\nAbstract\nThis is a joint work with Anna Dall'Acqua\, Adrian Spener an d Reiner Schätzle. \n\nWe study the $\\textcolor{red}{\\textbf{Willmore f low}}$ of tori that have a revolution symmetry - so-called tori of revolut ion. Luckily\, the Willmore flow preserves this symmetry. Because of that we can look at the flow as an evolution of the "profile curves" - a reduct ion of the dimension!\n\nWe will examine the geometry of this curve evolut ion and understand why it is somewhat natural to look at those curves in $ \\textcolor{red}{\\textbf{hyperbolic geometry}}$. We prove: \n\n$\\textcol or{green}{ \\textbf{If the hyperbolic length of the profile curves remains bounded\, then the Willmore flow converges.}}$\n\nThe remaining question: How can the hyperbolic length of those curves be controlled? We use varia tional methods to $\\textcolor{red}{\\textbf{control the hyperbolic length }}$ by the Willmore energy - but this control is only available below an e nergy level of $\\textcolor{red}{\\mathbf{8\\pi}}$. We obtain:\n\n$\\textc olor{green}{\\textbf{If we start the Willmore flow with a torus of revolut ion of Willmore energy below $8\\pi$\, then the flow converges}.}$ \n\nIf time allows: The threshold of $8\\pi$ is also sharp and plays an important role in the context of the Willmore functional. It is also the same thres hold that was already found by E. Kuwert and R. Schätzle for the Willmore flow of spheres.\n LOCATION:https://researchseminars.org/talk/OSGA/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Siffert (Universität Münster) DTSTART:20200929T170000Z DTEND:20200929T180000Z DTSTAMP:20260422T225637Z UID:OSGA/33 DESCRIPTION:Title: Co nstructing explicit p-harmonic functions\nby Anna Siffert (Universitä t Münster) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\n The study of $p$-harmonic functions on Riemannian manifolds has invoked th e interest of mathematicians and physicists for nearly two centuries.\nApp lications within physics can for example be found\nin continuum mechanics\ , elasticity theory\, as well as two-dimensional hydrodynamics problems in volving Stokes flows of incompressible Newtonian fluids.\n\nIn my talk I w ill focus on the construction of explicit $p$-harmonic functions on rank- one Lie groups of Iwasawa type.\nThis joint work with Sigmundur Gudmundsso n and Marko Sobak.\n LOCATION:https://researchseminars.org/talk/OSGA/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Renan Assimos (Leibniz Universitaet Hannover) DTSTART:20201013T170000Z DTEND:20201013T180000Z DTSTAMP:20260422T225637Z UID:OSGA/34 DESCRIPTION:Title: On a spherical Bernstein theorem by B. Solomon\nby Renan Assimos (Leibni z Universitaet Hannover) as part of Online Seminar "Geometric Analysis"\n\ n\nAbstract\nJoint work with J. Jost: A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geom etry\, 19(1)\, 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$\, whose Gauss map omits a neighborho od of an $S^{k−1}$ equator\, is totally geodesic in $S^{k+1}$. In this t alk\, I will present a new proof strategy for Solomon's theorem which allo ws us to obtain analogous results for higher codimensions. If time permits \, we sketch the proof for codimension 2 compact minimal submanifolds of $ S^{k+1}$.\n LOCATION:https://researchseminars.org/talk/OSGA/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Julia Menzel (Universität Regensburg) DTSTART:20201110T180000Z DTEND:20201110T190000Z DTSTAMP:20260422T225637Z UID:OSGA/35 DESCRIPTION:Title: Bo undary Value Problems for Evolutions of Willmore Type\nby Julia Menzel (Universität Regensburg) as part of Online Seminar "Geometric Analysis"\ n\n\nAbstract\nThe Willmore flow arises as the $L^2$-gradient flow of the Willmore energy which is itself given by the integrated squared mean curva ture of the considered surface. \n\n\n\nAfter a short introduction and rev iew of known results on the Willmore flow of curves and closed surfaces\, we discuss the existence of solutions to the Willmore flow of compact open surfaces immersed in Euclidean space subject to Navier boundary condition s.\n\n\n\nWe further study the elastic flow of planar networks composed of curves meeting in triple junctions. As a main result we obtain that start ing from a suitable initial network the flow exists globally in time if th e length of each curve remains uniformly bounded away from zero and if at least one angle at each triple junction stays uniformly bounded away from zero\, $\\pi$ and $2\\pi$.\n\n\n\nThis talk is based on my recently submit ted PhD thesis and includes joint work with H. Abels\, H. Garcke and A. Pl uda.\n LOCATION:https://researchseminars.org/talk/OSGA/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonas Hirsch (University of Leipzig) DTSTART:20201006T170000Z DTEND:20201006T180000Z DTSTAMP:20260422T225637Z UID:OSGA/36 DESCRIPTION:Title: On the regularity of area minimzing currents mod(p)\nby Jonas Hirsch (Un iversity of Leipzig) as part of Online Seminar "Geometric Analysis"\n\n\nA bstract\njoint work with C. De Lellis\, A Marches and S. Stuvard\n\nIn thi s talk I would like to give a glimpse on the regularity of area minimzing currents mod(p).\n\nMotivation: If one considers real soap films on e notice that from time to time one can find configurations where differen t soap films join on a common piece. One possibility to allow this kind of phenomenon is to consider flat chains with coefficients in $\\mathbb Z_p$ . For instance for $p = 2$ one can deal with unoriented surfaces\, for $p = 3$ one allows triple junctions.\n\nConsidering area minimzing currents w ithin this class the aim is to give a bound on the Hausdorff dimension of the singular set sing(T) in the interior. These are alle points where the precise representative of the minimiser T is not even locally supported on a piece of a $C^{1\,\\alpha}$ regular surface.
\nAfter a short introdu ction into general theory of currents mod(p)\, I will give you glimpse on the previously known results and on our new bound on the Hausdorff dimensi on of the set. If time permits I will give a short outlook of what we woul d be the expected result.\n LOCATION:https://researchseminars.org/talk/OSGA/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Brendle (Columbia University) DTSTART:20201027T180000Z DTEND:20201027T190000Z DTSTAMP:20260422T225637Z UID:OSGA/37 DESCRIPTION:Title: Th e isoperimetric inequality for minimal surfaces\nby Simon Brendle (Col umbia University) as part of Online Seminar "Geometric Analysis"\n\nAbstra ct: TBA\n LOCATION:https://researchseminars.org/talk/OSGA/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Campbell (University of Hradec Kralove) DTSTART:20201020T170000Z DTEND:20201020T180000Z DTSTAMP:20260422T225637Z UID:OSGA/38 DESCRIPTION:Title: Pa thological Sobolev homeomorphisms in GFT and NE\nby Daniel Campbell (U niversity of Hradec Kralove) as part of Online Seminar "Geometric Analysis "\n\n\nAbstract\nSobolev homeomorphisms are the natural choice for minimiz ation problems in non-linear elasticity. For the regularity of these probl ems it would be useful to be able to approximate these maps by smooth home omorphisms in their corresponding Sobolev space (the so-called Ball-Evans problem). We construct a pair of homeomorphisms for which is impossible si multaneously solving the Hajlasz problem. That is we construct a Sobolev h omeomorphism equalling identity on the boundary of a cube but with negativ e Jacobian almost everywhere.\n LOCATION:https://researchseminars.org/talk/OSGA/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Ursula Ludwig (University of Duisburg-Essen) DTSTART:20201208T180000Z DTEND:20201208T190000Z DTSTAMP:20260422T225637Z UID:OSGA/39 DESCRIPTION:Title: An Extension of a Theorem by Cheeger and Müller to Spaces with Isolated Con ical Singularities\nby Ursula Ludwig (University of Duisburg-Essen) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nAn important c omparison theorem in global analysis is the comparison of analytic and top ological torsion for smooth compact manifolds equipped with a unitary flat vector bundle. It has been conjectured by Ray and Singer and has been ind ependently proved by Cheeger and Mu ̈ller in the 70ies. Bismut and Zhang combined the Witten deformation and local index techniques to generalise t he result of Cheeger and Mu ̈ller to arbitrary flat vector bundles with a rbitrary Hermitian metrics. The aim of this talk is to present an extensio n of the Cheeger-Mu ̈ller theorem to spaces with isolated conical singula rities by generalising the proof of Bismut and Zhang to the singular setti ng.\n LOCATION:https://researchseminars.org/talk/OSGA/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Behnam Esmayli (Uni of Pittsburgh) DTSTART:20201117T180000Z DTEND:20201117T190000Z DTSTAMP:20260422T225637Z UID:OSGA/40 DESCRIPTION:Title: Co -area formula for maps into metric spaces\nby Behnam Esmayli (Uni of P ittsburgh) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nC o-area formula for maps between Euclidean spaces contains\, as its very sp ecial cases\, both Fubini's theorem and integration in polar coordinates f ormula. In 2009\, L. Reichel proved the coarea formula for maps from Eucli dean spaces to general metric spaces. I will discuss a new proof of the la tter by the way of an implicit function theorem for such maps. An importan t tool is an improved version of the coarea inequality (a.k.a Eilenberg in equality) that was the subject of a recent joint work with Piotr Hajlasz. Our proof of the coarea formula does not use the Euclidean version of it a nd can thus be viewed as a new (and arguably more geometric) proof in that case as well.\n LOCATION:https://researchseminars.org/talk/OSGA/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Hermann Karcher (University of Bonn) DTSTART:20201215T180000Z DTEND:20201215T190000Z DTSTAMP:20260422T225637Z UID:OSGA/41 DESCRIPTION:Title: Nu merical experiments with closed constant curvature space curves\nby He rmann Karcher (University of Bonn) as part of Online Seminar "Geometric An alysis"\n\n\nAbstract\nThe discovery story will be told with pictures illu strating all steps\, including the observation\nbelow which we us ed but could not prove. Since the shape of space curves is difficult to be \ncorrectly deduced from planar images\, most images will be red-green an aglyphs. They\ncan be looked at without red-green glasses\, but without gi ving the 3D impression.\n\nDec 15\,2020: H. Karcher: Closed consta nt curvature space curves\n\nThe only closed constant curvature s pace curves which I knew in 2004\nwere made from pieces of circles and hel ices.\nThe Frenet equations allow to construct space curves of constant cu rvature $\\kappa$\nby prescribing a torsion function $\\tau(s)$. For close d curves one needs periodic\ntorsion functions\, for example Fourier polyn omials. If one chooses these\nfunctions so that they are skew symmetric wi th respect to their zeros\, $\\tau(a-s)=-\\tau(a+s)$\, then\nthe resulting curves have the normal planes at these points as planes of\nmirror symmet ry. If adjacent symmetry planes have angles such as $\\pi/3$\,\nthen the c urves are forced by their symmetries to be closed. This gives the \nfirst collection of new examples.\n\nIf the torsion functions are even with resp ect to their extremal points\, i.e.\n$\\tau(a-s) = \\tau(a+s)$\, then the resulting curves have their principal normals\nat these points as symmetry axes for $180^\\circ$ rotations. If two such adjacent\nsymmetry normals a re coplanar and intersect under rational angles
($\\pi p/q$)\,\nthen the curves are again forced by their symmetries to close up. Therefore\non e can hope to get examples by solving a 2-parameter problem.\n\nThis is ma de simple by an observation which I cannot prove: The distance of \nadjacent symmetry normals depends in a surprisingly monotone way on the constant\nterm in the Fourier polynomial $\\tau(s) = b_0 + b_1\\\,\\sin(s ) + b_3\\\, \\sin(3s)$. This\nallows to consider $b_0 = b_0(\\kappa\, b_1\ , b_3)$\, such that the symmetry normals of the\nresulting curves are copl anar and hence intersect all in one point. Therefore we\nhave again to sol ve a 1-parameter problem by choosing $\\kappa\, b_1\, b_3$ in such a way\n that adjacent symmetry normals intersect with a rational angle. This gives a wealth\nof new examples.\n\nThe evolutes of such curves have also const ant curvature $\\kappa$\, but they have \nsingularities at the zeros of $\ \tau(s)$. This led to a search for closed examples with $\\tau(s) > 0$.\nA (2-11)-torus knot showed up and suggested to look for examples on tori. E asy ones\nare again found by symmetries and more complicated ones as solut ions of intermediate\nvalue problems. The formulas work also on cylinders and revealed easier examples than\nall the previous ones!\n\nThen E. Tjade n suggested to look for examples which are congruent to t heir evolutes.\nThey were found by modifying the Frenet equations. The $(2 \, 2n+1)$- torus knots among\nthem are in fact their own evolutes.\n LOCATION:https://researchseminars.org/talk/OSGA/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Tobias Weth (Goethe University Frankfurt) DTSTART:20210126T180000Z DTEND:20210126T190000Z DTSTAMP:20260422T225637Z UID:OSGA/42 DESCRIPTION:Title: Cr itical domains for the first nonzero Neumann eigenvalue in Riemannian mani folds\nby Tobias Weth (Goethe University Frankfurt) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe talk is concerned with geo metric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains of a Riemannian manif old. More precisely\, we analyze locally extremal domains for the first no ntrivial eigenvalue with respect to volume preserving domain perturbation s\, and we show that corresponding notions of criticality arise in the for m of overdetermined boundary value problems. Our results rely on an extens ion of Zanger's shape derivative formula which covers the case where the f irst nonzero Neumann eigenvalue is not simple. In the second part of the t alk\, we focus on product manifolds with euclidean factors\, and we classi fy the subdomains where the associated overdetermined boundary value probl em has a solution. If time permits\, I will also briefly discuss the first nontrivial Stekloff eigenvalue. \nThis is joint work with Moustapha Fall (AIMS Senegal).\n LOCATION:https://researchseminars.org/talk/OSGA/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Wilderich Tuschmann (Karlsruhe Institute of Technology) DTSTART:20210119T180000Z DTEND:20210119T190000Z DTSTAMP:20260422T225637Z UID:OSGA/43 DESCRIPTION:Title: Sp aces and Moduli Spaces of Riemannian Metrics\nby Wilderich Tuschmann ( Karlsruhe Institute of Technology) as part of Online Seminar "Geometric An alysis"\n\n\nAbstract\nConsider a smooth manifold with a Riemannian metric satisfying some sort of curvature constraint like\, for example\, positiv e scalar curvature\, non-negative Ricci or negative sectional curvature\, being Einstein\, Kähler\, Sasaki\, etc. A natural question to study is th en what the space of all such metrics does look like. Moreover\, one can a lso pose this question for corresponding moduli spaces of metrics\, i.e.\, quotients of the former by (suitable subgroups of) the diffeomorphism gro up of the manifold\, acting by pulling back metrics. \n\nThese spaces are customarily equipped with the topology of smooth convergence on compact su bsets and the quotient topology\, respectively\, and their topological pro perties then provide the right means to measure 'how many' different metri cs and geometries the given manifold actually does exhibit\; but one can t opologize and view those also in very different manners.\n\nIn my talk\, I will report on some general results and open questions about spaces and m oduli spaces of metrics with non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds\, and\, in particular\, also discuss broader non-traditional approaches from metri c geometry and analysis to these objects and topics.\n LOCATION:https://researchseminars.org/talk/OSGA/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Henrik Matthiesen (University of Chicago) DTSTART:20201103T181500Z DTEND:20201103T191500Z DTSTAMP:20260422T225637Z UID:OSGA/44 DESCRIPTION:Title: Ne w minimal surfaces from shape optimization\nby Henrik Matthiesen (Univ ersity of Chicago) as part of Online Seminar "Geometric Analysis"\n\n\nAbs tract\nI will discuss the connection between sharp eigenvalue bounds and m inimal surfaces in two cases:\nThe first eigenvalue of the Laplacian on a closed surface among unit area metrics\, and\nthe first Steklov eigenvalue on a compact surface with non empty boundary among metrics with unit leng th boundary.\nIn both cases maximizing metrics - if they exist - are induc ed by certain minimal immersions.\nMore precisely\, minimal immersions int o round spheres for the closed case and free boundary minimal immersions i nto Euclidean balls in the bordered case.\nI will discuss the solution of the existence problem for maximizers in both these cases\, which provides many new examples of minimal surfaces of the aforementioned types.\nThis i s based on joint work with Anna Siffert in the closed case and Romain Petr ides in the bordered case.\n LOCATION:https://researchseminars.org/talk/OSGA/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Rupert Frank DTSTART:20210112T180000Z DTEND:20210112T190000Z DTSTAMP:20260422T225637Z UID:OSGA/45 DESCRIPTION:Title: Wh ich magnetic fields support a zero mode?\nby Rupert Frank as part of O nline Seminar "Geometric Analysis"\n\n\nAbstract\nMotivated by the questio n from mathematical physics about the size of magnetic fields that support zero modes for the three dimensional Dirac equation\, we study a certain conformally invariant spinor equation. We state some conjectures and prese nt some results supporting them. Those concern\, in particular\, two novel Sobolev inequalities for spinors and vector fields.\n\nThe talk is based on joint work with Michael Loss.\n LOCATION:https://researchseminars.org/talk/OSGA/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Paul Creutz (University of Cologne) DTSTART:20210202T180000Z DTEND:20210202T190000Z DTSTAMP:20260422T225637Z UID:OSGA/46 DESCRIPTION:Title: Ar ea minimizing surfaces for singular boundary values\nby Paul Creutz (U niversity of Cologne) as part of Online Seminar "Geometric Analysis"\n\n\n Abstract\nFix a nonnegative integer g and a finite configuration of disjoi nt Jordan curves in Euclidean space. Then\, by a classical result of Dougl as\, there is an area minimizer among all surfaces of genus at most g whic h span the given curve configuration. In the talk I will discuss a general ization of this theorem to singular configurations of possibly non-disjoin t or self-intersecting curves. The proof relies on an existence result for minimal surfaces in singular metric spaces and does not seem amenable wit hin classical smooth techniques.\n\nThis is joint work with M. Fitzi.\n LOCATION:https://researchseminars.org/talk/OSGA/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Florian Litzinger (Queen Mary College) DTSTART:20210209T180000Z DTEND:20210209T190000Z DTSTAMP:20260422T225637Z UID:OSGA/47 DESCRIPTION:Title: Op timal regularity for Pfaffian systems and the fundamental theorem of surfa ce theory\nby Florian Litzinger (Queen Mary College) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe fundamental theorem of su rface theory asserts the existence of a surface immersion with prescribed first and second fundamental forms that satisfy the Gauss–Codazzi–Main ardi equations. Its proof is based on the solution of a Pfaffian system an d an application of the Poincaré lemma. Consequently\, the regularity of the resulting immersion crucially depends on the regularity of the solutio n of the corresponding Pfaffian system. This talk shall briefly review bot h the classical smooth case and the regularity theory and then introduce a n extension to the optimal regularity.\n LOCATION:https://researchseminars.org/talk/OSGA/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Gerhard Huisken (University of Tübingen) DTSTART:20210216T180000Z DTEND:20210216T190000Z DTSTAMP:20260422T225637Z UID:OSGA/48 DESCRIPTION:Title: Me an curvature flow with surgery\nby Gerhard Huisken (University of Tüb ingen) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe e volution of hypersurfaces in a Riemannian manifold along its mean curvatur e vector is governed by a quasilinear parabolic system that exhibits smoot hing behavior and singularity formation at the same time since the evoluti on of the geometry is governed by a non-linear reaction diffusion system. The lecture explains how for embedded 2-surfaces of positive mean curvatur e in general ambient manifolds long-time solutions can be constructed that contain finitely many surgeries near singular regions. Finally we discuss applications in Geometry and General Relativity.\n LOCATION:https://researchseminars.org/talk/OSGA/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephan Wojtowytsch (Princeton University) DTSTART:20210223T180000Z DTEND:20210223T190000Z DTSTAMP:20260422T225637Z UID:OSGA/49 DESCRIPTION:Title: Op timal transport for non-convex optimization in machine learning\nby St ephan Wojtowytsch (Princeton University) as part of Online Seminar "Geomet ric Analysis"\n\n\nAbstract\nFunction approximation is a classical task in both classical numerical analysis and machine learning. Elements of the r ecently popular class of neural networks depend nonlinearly on a finite se t of parameters. This nonlinearity gives the function class immense approx imation power\, but causes parameter optimization problems to be non-conve x. In fact\, generically the set of global minimizers is a (curved) manifo ld of positive dimension. Despite this non-convexity\, gradient descent ba sed algorithms empirically find good minimizers in many applications. We d iscuss this surprising success of simple optimization algorithms from the perspective of Wasserstein gradient flows in the case of shallow neural ne tworks in the infinite parameter limit.\n LOCATION:https://researchseminars.org/talk/OSGA/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Ruijun Wu (SISSA\, Trieste) DTSTART:20210302T180000Z DTEND:20210302T190000Z DTSTAMP:20260422T225637Z UID:OSGA/50 DESCRIPTION:Title: Su per Liouville equations on the 2-sphere\nby Ruijun Wu (SISSA\, Trieste ) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe 2D sup er Liouville equations\, from the super Liouville field theory\, is a conf ormally invariant system which couples the classical Liouville equation wi th a Dirac equation. We are interested in the existence of nontrivial solu tions. Aside from those known solutions induced from prescribing curvature equations and those from Killing spinors\, we introduced an additional (b ut natural) parameter and obtained new solutions via bifurcation theory. T his is a joint work with A. Malchiodi and A. Jevnikar.\n LOCATION:https://researchseminars.org/talk/OSGA/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Christian Ketterer (University of Toronto) DTSTART:20210309T180000Z DTEND:20210309T190000Z DTSTAMP:20260422T225637Z UID:OSGA/51 DESCRIPTION:Title: In scribed radius bounds for metric measure spaces with mean-H-convex boundar y\nby Christian Ketterer (University of Toronto) as part of Online Sem inar "Geometric Analysis"\n\n\nAbstract\nWe introduce a synthetic lower me an curvature bound for the\ntopological boundary of a subset in a metric m easure space that satisfies a\nlower Ricci curvature bound in the sense of Lott\, Sturm and Villani.  This \nlower mean curvature bound coincides with the classical notion in smooth\ncontext. As application I present a t heorem about sharp comparison estimates\nfor the inscribed radius of such subsets.  Moreover\, in the context of\nRCD(0\,N) metric measure spaces ( Riemannian curvature-dimension condition)\nequality holds if and only if t he subset is isometric to a geodesic ball\ncentered at the tip of an Eucli dean cone. This generalizes theorems in\nsmooth context by Kasue and Sakur ai to a singular framework. This is a joint\nwork with Annegret Burtscher\ , Robert McCann and Eric Woolgar.\n LOCATION:https://researchseminars.org/talk/OSGA/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Verena Bögelein (University Salzburg\, Austria) DTSTART:20210316T180000Z DTEND:20210316T190000Z DTSTAMP:20260422T225637Z UID:OSGA/52 DESCRIPTION:Title: Hi gher regularity in congested traffic dynamics\nby Verena Bögelein (Un iversity Salzburg\, Austria) as part of Online Seminar "Geometric Analysis "\n\n\nAbstract\nWe consider an elliptic system that is motivated by a con gested traffic dynamics problem. It has the form\n$$\n \\mathrm{div}\\bigg ((|Du|-1)_+^{p-1}\\frac{Du}{|Du|}\\bigg)=f\,\n$$\nand falls into the conte xt of very degenerate problems. Continuity properties of the gradient have been investigated in the scalar case by Santambrogio & Vespri and Colomb o & Figalli. \nIn this talk we establish the optimal regularity of weak so lutions in the vectorial case for any $p>1$. This is joint work with F. Du zaar\, R. Giova and A. Passarelli di Napoli.\n LOCATION:https://researchseminars.org/talk/OSGA/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Sonja Hohloch (University of Antwerp) DTSTART:20210323T180000Z DTEND:20210323T190000Z DTSTAMP:20260422T225637Z UID:OSGA/53 DESCRIPTION:Title: On recent advances in semitoric integrable systems\nby Sonja Hohloch (Un iversity of Antwerp) as part of Online Seminar "Geometric Analysis"\n\n\nA bstract\nRoughly speaking\, a semitoric system is a completely integrable Hamiltonian system on a 4-dimensional symplectic manifold that admits only nondegenerate singularities without hyperbolic components and whose flow gives rise to an $(\\mathbb S^1 \\times \\mathbb R)$-action. Coupled spin oscillators and coupled angular momenta are examples of such semitoric sys tems.\n\nSemitoric systems have been symplectically classified about a dec ade ago by Pelayo $\\&$ Vu Ngoc by means of five invariants. Recently\, th ere has been made considerable progress by various authors concerning the computation of these invariants.\n\nIn this talk\, we will give an introdu ction to semitoric systems before considering a recent\, intuitive family of semitoric systems that allows for explicit observation of bifurcation b ehaviour such as bifurcations between focus-focus and elliptic-elliptic si ngularities and other interesting geometric-topological features related t o singularities and bifurcations. The latter part is based on a joint work with A.\\ De Meulenaere.\n LOCATION:https://researchseminars.org/talk/OSGA/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Jimmy Scott (University of Pittsburgh) DTSTART:20210330T170000Z DTEND:20210330T180000Z DTSTAMP:20260422T225637Z UID:OSGA/54 DESCRIPTION:Title: Fr actional Korn-Type Inequalities and Applications\nby Jimmy Scott (Univ ersity of Pittsburgh) as part of Online Seminar "Geometric Analysis"\n\n\n Abstract\nWe show that a class of spaces of vector fields whose semi-norms involve the magnitude of ``directional" difference quotients is in fact e quivalent to the class of fractional Sobolev-Slobodeckij spaces. The equiv alence can be considered a Korn-type characterization of said Sobolev spac es. For vector fields defined on various classes of domains\, we obtain a relevant form of the inequality. As an application\, we consider variation al problems associated to strongly coupled systems of nonlocal equations m otivated by a continuum mechanics model known as peridynamics. We use the fractional Korn-type inequalities to characterize vector fields in associa ted energy spaces and obtain existence and uniqueness of solutions in frac tional Sobolev spaces.\n LOCATION:https://researchseminars.org/talk/OSGA/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Oded Stein (MIT) DTSTART:20210406T170000Z DTEND:20210406T180000Z DTSTAMP:20260422T225637Z UID:OSGA/55 DESCRIPTION:Title: Th e Biharmonic Equation in Geometry Processing\nby Oded Stein (MIT) as p art of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe Laplacian ha s been an extensively used tool of geometry processing and computer graphi cs for a long time.\nIn this talk we will take a look at a close relative of the Laplacian\, the Bilaplacian\, as well as its partial differential e quation\, the biharmonic equation.\nThe Bilaplacian can be used in applica tions such as smoothing\, interpolation\, character animation\, distance c omputation\, and more.\nWe will examine the biharmonic equation and its us e in geometry processing\, we will look at ways to discretize it for curve d surfaces\, and we will discuss different boundary conditions of the biha rmonic equation.\n LOCATION:https://researchseminars.org/talk/OSGA/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Amy Novick-Cohen (Technion-IIT\, Haifa) DTSTART:20210413T170000Z DTEND:20210413T180000Z DTSTAMP:20260422T225637Z UID:OSGA/56 DESCRIPTION:Title: Su rface diffusion\, and surface diffusion coupled with mean curvature motion \nby Amy Novick-Cohen (Technion-IIT\, Haifa) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nSurface diffusion as well as mean cur vature motion constitute geometric motions relevant to the modelling vario us phenomena arising in modeling thin poly-crystalline films. We first rev iew some special grooving solutions and traveling wave solutions. Afterwar ds we focus on certain composite axi-symmetric geometries\; here the stead y states may be described by piecing together Delaunay surfaces\, and rela ted evolutionary questions are pertinent to solid state wetting and dewett ing.\n LOCATION:https://researchseminars.org/talk/OSGA/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrea Malchiodi (Scuola Normale Superiore) DTSTART:20210420T170000Z DTEND:20210420T180000Z DTSTAMP:20260422T225637Z UID:OSGA/57 DESCRIPTION:Title: On critical points of the Moser-Trudinger functional\nby Andrea Malchiod i (Scuola Normale Superiore) as part of Online Seminar "Geometric Analysis "\n\n\nAbstract\nIt is known that in two dimensions Sobolev functions in $ W^{1\,2}$ satisfy critical embedding properties of exponential type. In 19 71 Moser obtained a sharp form of the embedding\, controlling the integrab ility of $F(u) := \\int \\exp(u^2)$ in terms of the Sobolev norm of $u$.\n On a closed Riemannian surface\, $F(u)$ is unbounded above for $\\|u\\|_{W ^{1\,2}} > 4 \\pi$. \nWe are however able to find critical points of $F$ c onstrained to any sphere \n$\\{ \\|u\\|_{W^{1\,2}} = \\beta \\}$\, with $\ \beta > 0$ arbitrary. The proof combines min-max theory\, a monotonicity a rgument by Struwe\, blow-up analysis and compactness estimates. This is jo int work with F. De Marchis\, O. Druet\, L. Martinazzi and P. D. Thizy.\n LOCATION:https://researchseminars.org/talk/OSGA/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Francesco Palmurella (ETH Zürich) DTSTART:20210427T170000Z DTEND:20210427T180000Z DTSTAMP:20260422T225637Z UID:OSGA/58 DESCRIPTION:Title: Th e parametric approach to the Willmore flow\nby Francesco Palmurella (E TH Zürich) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\n We introduce a parametric framework for the study of Willmore gradient flo ws\nwhich enables to consider a general class of weak\, energy-level solut ions and opens\nthe possibility to study energy quantization and finite-ti me singularities.\nIn this first work we restricted to a small-energy regi me and proved that\, for small-energy weak\nimmersions\, the Cauchy proble m in this class admits a unique solution.\n LOCATION:https://researchseminars.org/talk/OSGA/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Duzaar (University Erlangen-Nürnberg) DTSTART:20210504T170000Z DTEND:20210504T180000Z DTSTAMP:20260422T225637Z UID:OSGA/59 DESCRIPTION:Title: Hi gher integrability for porous medium type systems\nby Frank Duzaar (Un iversity Erlangen-Nürnberg) as part of Online Seminar "Geometric Analysis "\n\n\nAbstract\nIn this talk we report on recent developments concerning the higher integrability\nof the spatial gradient to porous medium type sy stems of the form\n$$\n \\partial_ t u- \\Delta(|u|^{m-1}u) = \\rm{div}\\\ , F.\n$$\n LOCATION:https://researchseminars.org/talk/OSGA/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Hans Knüpfer (University of Heidelberg) DTSTART:20210511T170000Z DTEND:20210511T180000Z DTSTAMP:20260422T225637Z UID:OSGA/60 DESCRIPTION:Title: Ga mma-limit for zigzag walls in thin ferromagnetic films\nby Hans Knüpf er (University of Heidelberg) as part of Online Seminar "Geometric Analysi s"\n\n\nAbstract\nIn the continuum theory\, the magnetization of a ferroma gnetic sample $\\Omega \\subset \\R^3$ is described by a unit vector field $m \\in H^1(\\Omega\,S^2)$. The minimization of the underlying micromagne tic energy leads to the formation of extended magnetic domains with unifo rm magnetization\, separated by thin transition layers. One type of such t ransition layers\, observed in thin ferromagnetic films are the so called zigzag walls. We consider the family of energies\n$$E_\\varepsilon[m] \\ = \\ \\frac{\\epsilon}{2}\\|\\nabla m\\|_{L^2(\\Omega)}^2 + \\frac 1{2\\var epsilon} \\|m \\cdot e_2\\|_{L^2(\\Omega)}^2 %\n + \\frac{\\pi\\lambda} {2|\\ln \\varepsilon|} \\|\\nabla \\cdot (m-M)\\|_{\\dot H^{-\\frac 12}}^2 \, \n$$\nvalid for thin ferromagnetic films. We consider a material in the form a thin strip and\n enforce a charged domain wall by suitable bo undary conditions on $m$. Here\, $M$ is an arbitrary fixed background fie ld to ensure global neutrality of magnetic charges. In the\n limit $\\var epsilon \\to 0$ and for fixed $\\lambda > 0$\, corresponding to the macros copic\n limit\, we show that the energy $\\Gamma$--converges to a limit e nergy where jump\n discontinuities of the magnetization are penalized ani sotropically. In\n particular\, in the subcritical regime $\\lambda \\leq 1$ one--dimensional charged\n domain walls are favorable\, in the superc ritical regime $\\lambda > 1$ the limit\n model allows for zigzaging two- -dimensional domain walls.\n LOCATION:https://researchseminars.org/talk/OSGA/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Catherine Bandle (University Basel) DTSTART:20210518T170000Z DTEND:20210518T180000Z DTSTAMP:20260422T225637Z UID:OSGA/61 DESCRIPTION:Title: Do main variations for boundary value problems.\nby Catherine Bandle (Uni versity Basel) as part of Online Seminar "Geometric Analysis"\n\n\nAbstrac t\nWe consider boundary value problems which are Euler-Lagrange equations of certain energy-functionals. Important questions in this context are: Ho w do they depend on the geometry of the domain on which they are defined? For instance\, does the energy assume a minimum among all domains of given volume? How does the optimal region\, if it exists\, look like? \n\nThe t echnique of domain variations studies the changes of functionals under inf initesimal deformations. It is a differential calculus that allows to deri ve necessary conditions for the geometry of an optimal domain. Its beginni ngs go back to Hadamard in 1908\, who calculated the first variation of Gr een's functions with Dirichlet boundary conditions. In this talk\, the fir st and second variations of the energy of torsion problem with Robin bound ary conditions will be discussed.\n LOCATION:https://researchseminars.org/talk/OSGA/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Sun-Yung Alice Chang (Princeton University) DTSTART:20210525T170000Z DTEND:20210525T180000Z DTSTAMP:20260422T225637Z UID:OSGA/62 DESCRIPTION:Title: On bi-Lipschitz equivalence of a class of non-conformally flat spheres\n by Sun-Yung Alice Chang (Princeton University) as part of Online Seminar " Geometric Analysis"\n\n\nAbstract\nThis is a report of some recent joint w ork with Eden Prywes and Paul Yang. The main\nresult is a bi-Lipschitz equ ivalence of a class of metrics on 4-shpere under curvature constraints. Th e proof involves two steps: first a construction of quasiconformal maps be tween\ntwo conformally related metrics in a positive Yamabe class\, follow ed by the step of applying\nthe Ricci flow to establish the bi-Lipschitz e quivalence from such a conformal class to the\nstandard conformal class on 4-spheres.\n LOCATION:https://researchseminars.org/talk/OSGA/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Shankar Venkataramani (University of Arizona) DTSTART:20210601T170000Z DTEND:20210601T180000Z DTSTAMP:20260422T225637Z UID:OSGA/63 DESCRIPTION:Title: On branch points and C^{1\,1} pseudospherical immersions\nby Shankar Ven kataramani (University of Arizona) as part of Online Seminar "Geometric An alysis"\n\n\nAbstract\nThis is a report of joint work with Toby Shearman. The key result is that one can define a (local) winding number of the Gaus s Map for $C^{1\,1}$ hyperbolic surfaces in $R^3$ and this degree is an ob struction for approximation by smooth immersions in $W^{2\,2}_{loc}$. I wi ll discuss the ideas behind the proof\, as well as the motivation for stud ying this question\, which comes from the mechanics of non-Euclidean plate s.\n LOCATION:https://researchseminars.org/talk/OSGA/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Sigurd Angenent (University of Wisconsin–Madison) DTSTART:20210608T170000Z DTEND:20210608T180000Z DTSTAMP:20260422T225637Z UID:OSGA/64 DESCRIPTION:Title: No nuniqueness in mean curvature flow and Ricci flow\nby Sigurd Angenent (University of Wisconsin–Madison) as part of Online Seminar "Geometric A nalysis"\n\n\nAbstract\nReporting on joint work with Ilmanen and Velazquez \, I will present examples of smooth solutions to MCF in $\\mathbb R^{d}$ with $d\\in\\{4\, 5\, 6\, 7\, 8\\}$ that form a conical singularity after which they allow many different forward smooth continuations. I will also show similar results obtained with Knopf concerning the Ricci flow in dim ensions $5\, \\dots\, 9$.\n LOCATION:https://researchseminars.org/talk/OSGA/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Martin Rumpf (University of Bonn) DTSTART:20210615T170000Z DTEND:20210615T180000Z DTSTAMP:20260422T225637Z UID:OSGA/65 DESCRIPTION:Title: Ri emannian calculus in shape spaces\nby Martin Rumpf (University of Bonn ) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nSpaces of curves and surfaces or spaces of images are considered as Riemannian manif olds.\nThe talk will develop a calculus on such spaces\, which enables\n\n
  • the computation of distances via minimizing a Riemannian path ener gy\,
    • \n
  • interpolation of shapes along geodesic paths\,
    • \n
  • ex trapolation via the Riemannian exponential map\,
    • \n
  • detail transfer via parallel transport\,
    • \n
  • key pose interpolation via Riemannian splines\, and
    • \n
  • statistical analysis via Riemannian PCA.
    • \n    \nTo this end a time discrete calculus is introduced and its\nconvergenc e is discussed.\n LOCATION:https://researchseminars.org/talk/OSGA/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Hans-Joachim Hein (University of Münster) DTSTART:20210622T170000Z DTEND:20210622T180000Z DTSTAMP:20260422T225637Z UID:OSGA/66 DESCRIPTION:Title: Sm ooth asymptotics for collapsing Calabi-Yau metrics\nby Hans-Joachim He in (University of Münster) as part of Online Seminar "Geometric Analysis" \n\n\nAbstract\nYau's solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds . Attempts to understand the degeneration patterns of these metrics in fam ilies have revealed many remarkable phenomena over the years. I will revie w some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a h olomorphic fibration of a fixed compact Calabi-Yau manifold. This relies o n a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivativ e of the remainder.\n LOCATION:https://researchseminars.org/talk/OSGA/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Andreas Bernig (Goethe University Frankfurt) DTSTART:20210706T170000Z DTEND:20210706T180000Z DTSTAMP:20260422T225637Z UID:OSGA/67 DESCRIPTION:Title: In trinsic volumes on pseudo-Riemannian manifolds\nby Andreas Bernig (Goe the University Frankfurt) as part of Online Seminar "Geometric Analysis"\n \n\nAbstract\nThe intrinsic volumes in Euclidean space can be defined via Steiner's tube formula and were characterized by Hadwiger as the unique co ntinuous\, translation and rotation invariant valuations. By the Weyl prin ciple\, their extension to Riemannian manifolds behaves naturally under is ometric embeddings.\n\nIn a series of papers with Dmitry Faifman and Gil S olanes\, we developed a theory of intrinsic volumes in pseudo-Euclidean sp aces and on pseudo-Riemannian manifolds. Fundamental results like Hadwiger 's theorem\, Weyl's principle and Crofton formulas on spheres have their n atural analogues in the pseudo-Riemannian setting.\n LOCATION:https://researchseminars.org/talk/OSGA/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Jerome Wettstein (ETH Zurich) DTSTART:20211123T180000Z DTEND:20211123T190000Z DTSTAMP:20260422T225637Z UID:OSGA/68 DESCRIPTION:Title: Pr operties of the Half-Harmonic Gradient Flow\nby Jerome Wettstein (ETH Zurich) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nIn t his talk\, we discuss properties of the fractional harmonic gradient flow with values in $S^{n-1}$ and its generalisation to arbitrary target manifo lds\, as investigated by the speaker in . Particular attention is spent on comparing the non-local case with the local one\, i.e. the harmonic map f low.\n LOCATION:https://researchseminars.org/talk/OSGA/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Haslhofer (University of Toronto) DTSTART:20210803T170000Z DTEND:20210803T180000Z DTSTAMP:20260422T225637Z UID:OSGA/69 DESCRIPTION:Title: Me an curvature flow through neck-singularities\nby Robert Haslhofer (Uni versity of Toronto) as part of Online Seminar "Geometric Analysis"\n\n\nAb stract\nIn this talk\, I will explain our recent work showing that mean cu rvature flow through neck-singularities is unique. The key is a classifica tion result for ancient asymptotically cylindrical flows that describes al l possible blowup limits near a neck-singularity. In particular\, this con firms Ilmanen’s mean-convex neighborhood conjecture\, and more precisely gives a canonical neighborhood theorem for neck-singularities. Furthermor e\, assuming the multiplicity-one conjecture\, we conclude that for embedd ed two-spheres mean curvature flow through singularities is well-posed. Th e two-dimensional case is joint work with Choi and Hershkovits\, and the h igher-dimensional case is joint with Choi\, Hershkovits and White.\n LOCATION:https://researchseminars.org/talk/OSGA/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Hans-Christoph Grunau (Otto-von-Guericke-Universität Magdeburg) DTSTART:20210713T170000Z DTEND:20210713T180000Z DTSTAMP:20260422T225637Z UID:OSGA/70 DESCRIPTION:Title: Bo undary value problems for the Willmore and the Helfrich functional for sur faces of revolution\nby Hans-Christoph Grunau (Otto-von-Guericke-Unive rsität Magdeburg) as part of Online Seminar "Geometric Analysis"\n\n\nAbs tract\nThis talk concerns joint works with A. Dall'Acqua\, K. Deckelnick\, M. Doemeland\, S. Eichmann\, and S. Okabe.\n\nA special form of the Helf rich energy for a sufficiently smooth (two dimensional) surface $ S \\sub set \\mathbb{R} ^3 $ (with or without boundary) is defined by\n $$\n {\\mathscr H}_\\varepsilon(S) := \\int_{S} H^2 \\\, d S + \\varepsilon \\int_{S} \\\, d S \,\n $$\n where $H$ denotes the mean curvature of $S$.\n The first integral may be considered as a bending energy and th e second as\n surface (stretching) energy. ${\\mathscr W} (S):={\\maths cr H}_0 (S)$ is\n called the Willmore functional.\n We consider surf aces of revolution $ S $\n $$\n (x\,\\varphi)\\mapsto \\big(x\, u(x)\\cos \\varphi\, u(x)\\sin \\varphi \\big) \\\, \,\n \\quad x\ \in[-1\,1]\,~\\varphi\\in[0\,2\\pi]\,\n $$\n with smooth strictly po sitive profile curve $u$ subject to Dirichlet\n boundary conditions\n $$\n u(-1)=\\alpha\,\\quad u(1)=\\beta\,\\quad u'(\\pm1)=0\n $$\n and aim at minimising ${\\mathscr H}_\\varepsilon$. Thanks to these bou ndary conditions the Gauss curvature integral $\\int_{S} K\\\, d S $ beco mes a constant and needs not to be considered.\n\nIn the first part of th e lecture I shall consider the Willmore case\, i.e.\n $\\varepsilon=0$. After briefly recalling minimisation in the symmetric case\n $\\alpha= \\beta$ (see [1\,4]) I shall show how much more complicated the problem\n becomes for $\\alpha\\not=\\beta$. Only when $\\alpha$ and $\\beta$ do not differ\n too much\, the profile curve will remain a graph while in general it will\n become a nonprojectable curve\, see [3].\n\nIn the se cond part\, ${\\mathscr H}_\\varepsilon$ is considered for\n $\\varepsi lon\\in[0\,\\infty)$\, but again in the symmetric setting $\\alpha=\\beta $. For $\\alpha \\ge \\alpha_m = c_m \\cosh(\\frac{1}{c_m})\\approx 1.895$ with $c_m\\approx 1.564$ the unique solution of the equation\n$\n\\frac{ 2}{c} = 1 + e ^ {-2/ c}\n$\, when one has a catenoid $v_\\alpha$ which globally minimises the surface\nenergy\, we find minimisers $u_\\varepsil on$ for any $\\varepsilon\\ge 0$\nand show uniform and locally smooth conv ergence $u_\\varepsilon \\to v_\\alpha$ under the singular limit\n$\\varep silon \\to \\infty$. These results are collected in [2].\n\nAt the end I s hall briefly mention recent work on obstacle problems [5].\n \n\n \n \n[1] A. Dall'Acqua\, K. Deckelnick\, and H.-Ch. Grunau\,\n Cl assical solutions to the Dirichlet problem for Willmore\n surfaces of r evolution\, Adv. Calc. Var. 1 (2008)\, 379-397.\n\n[2] K. Deckelnick\, H.-Ch. Grunau\, and M. Doemeland\, Boundary value problems fo r the Helfrich functional for surfaces of revolution\n - Existence and asymptotic behaviour\, Calc. Var. Partial Differ. Equ. 60< /b> (2021)\, Article number 32.\n\n[3] S. Eichmann and H.-Ch. Grunau\,\n Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions\,\n Adv. Calc. Var. 12 (20 19)\, 333–361.\n\n[4] H.-Ch. Grunau\, The asymptotic shape of a boundary layer of symmetric\n Willmore surfaces of revolution.\n In: C. Band le et al. (eds.)\, Inequalities and Applications 2010.\n Internatio nal Series of Numerical Mathematics 161 (2012)\, 19-29.\n\n[5] H.-Ch. Grunau and S. Okabe\,\n Willmore obstacle problems under Dirich let boundary conditions\, submitted.\n LOCATION:https://researchseminars.org/talk/OSGA/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Xavier Cabre (ICREA and Universitat Politecnica de Catalunya) DTSTART:20210720T170000Z DTEND:20210720T180000Z DTSTAMP:20260422T225637Z UID:OSGA/71 DESCRIPTION:Title: St able solutions to semilinear elliptic equations are smooth up to dimension 9\nby Xavier Cabre (ICREA and Universitat Politecnica de Catalunya) a s part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe regularit y of stable solutions to semilinear elliptic PDEs has been studied since t he 1970's. In dimensions 10 and higher\, there exist singular stable energ y solutions. In this talk I will describe a recent work in collaboration w ith Figalli\, Ros-Oton\, and Serra\, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.\n LOCATION:https://researchseminars.org/talk/OSGA/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Tobias Lamm (KIT) DTSTART:20210914T170000Z DTEND:20210914T180000Z DTSTAMP:20260422T225637Z UID:OSGA/73 DESCRIPTION:Title: Di ffusive stability results for the harmonic map flow and related equations< /a>\nby Tobias Lamm (KIT) as part of Online Seminar "Geometric Analysis"\n \n\nAbstract\nThe goal of this talk is to introduce the audience to the th eory of diffusive stability in the context of the harmonic map flow. This theory is useful when studying stability results for parabolic equations a nd we will illustrate its use for geometric equations such as the harmonic map flow.\nAdditionally\, we use this theory in order improve various uni queness results for solutions with rough initial data.\n LOCATION:https://researchseminars.org/talk/OSGA/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Jürgen Jost (MPI Leipzig) DTSTART:20210727T170000Z DTEND:20210727T180000Z DTSTAMP:20260422T225637Z UID:OSGA/74 DESCRIPTION:Title: No npositive curvature: Geometric and analytic aspects\nby Jürgen Jost ( MPI Leipzig) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\ nMotivated by questions from data analysis\, we develop a new approach to curvature of metric spaces. The approach works also for discrete metric sp aces and links curvature to deviations from hyperconvexity.\n LOCATION:https://researchseminars.org/talk/OSGA/74/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Waldron (University of Wisconsin - Madison) DTSTART:20210810T170000Z DTEND:20210810T180000Z DTSTAMP:20260422T225637Z UID:OSGA/75 DESCRIPTION:Title: Ha rmonic map flow for almost-holomorphic maps\nby Alex Waldron (Universi ty of Wisconsin - Madison) as part of Online Seminar "Geometric Analysis"\ n\n\nAbstract\nI'll describe some history\, recent results\, and open prob lems about harmonic map flow in dimension two.\n\nThe main result is as fo llows: let $\\Sigma$ be a compact oriented surface and $N$ a compact Kähl er manifold with nonnegative holomorphic bisectional curvature (e.g. $\\ma thbb{CP}^n$). For harmonic map flow starting from an almost-holomorphic ma p $\\Sigma \\to N$ (in the energy sense)\, the ``body map'' at each singul ar time is continuous\, and no ``neck'' appears between the body map and t he bubble tree.\n\nThis is joint work with Chong Song.\n LOCATION:https://researchseminars.org/talk/OSGA/75/ END:VEVENT BEGIN:VEVENT SUMMARY:Antonio De Rosa (University of Maryland) DTSTART:20210907T170000Z DTEND:20210907T180000Z DTSTAMP:20260422T225637Z UID:OSGA/76 DESCRIPTION:Title: Re gularity of anisotropic minimal surfaces\nby Antonio De Rosa (Universi ty of Maryland) as part of Online Seminar "Geometric Analysis"\n\n\nAbstra ct\nI will present regularity theorems for weak minimal surfaces with resp ect to anisotropic surface energies\, extending the celebrated isotropic c ounterparts proved by Allard.\n LOCATION:https://researchseminars.org/talk/OSGA/76/ END:VEVENT BEGIN:VEVENT SUMMARY:Rob Kusner (University of Massachusetts at Amherst) DTSTART:20210629T170000Z DTEND:20210629T180000Z DTSTAMP:20260422T225637Z UID:OSGA/77 DESCRIPTION:Title: ON THE CANHAM PROBLEM: BENDING ENERGY MINIMIZING SURFACES OF ANY GENUS AND ISOPERIMETRIC RATIO\nby Rob Kusner (University of Massachusetts at Amh erst) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nIn 197 0\, the biophysicist Peter CANHAM proposed that the shapes of red blood ce lls could be described variationally\, leading to the Canham problem: find the surfaces of genus $g$ embedded in $\\R^3$ that minimize their Willmor e bending energy $W=\\frac14 \\int H^2$ with given area and enclosed volum e\, or equivalently (since $W$ is scale invariant) with given isoperimetri c ratio $v \\in (0\, 1)$. Building on very recent work of Andrea MONDINO & Christian SCHARRER\, we solve the “existence” part of the problem\; i t suffices to find a comparison surface of genus $g$ with arbitrarily smal l isoperimetric ratio $v$ and $W < 8π$\, which we construct by gluing $g+ 1$ small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation $∆(∆+2)φ = 0$ on the $(g+1)$-punctured s phere. (If time permits\, we may discuss our ongoing work to understand t he “small $v$” limit\, as well as “uniqueness” aspects of the Canh am problem.)\n\n— Rob KUSNER\, UMassAmherst & CoronavirusU\n\n[joint wor k with Peter MCGRATH\, NorthCarolinaStateU]\n LOCATION:https://researchseminars.org/talk/OSGA/77/ END:VEVENT BEGIN:VEVENT SUMMARY:Fabian Rupp (Ulm University) DTSTART:20210824T170000Z DTEND:20210824T180000Z DTSTAMP:20260422T225637Z UID:OSGA/78 DESCRIPTION:Title: A dynamic approach to the Canham problem\nby Fabian Rupp (Ulm University ) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nMotivated by the Canham-Helfrich model for lipid bilayers\, the minimization of the Willmore energy among surfaces of given topological type subject to the co nstraint of fixed isoperimetric ratio has been extensively studied through out the last decade. In this talk\, we consider a dynamical approach by in troducing a non-local $L^2$-gradient flow for the Willmore energy\, which preserves the isoperimetric ratio. For topological spheres with initial en ergy below an explicit threshold\, we show global existence and convergenc e to a Helfrich immersion as $t\\to\\infty$. Our proof relies on a blow-up procedure and a constrained version of the \nŁojasiewicz--Simon gradient inequality.\n LOCATION:https://researchseminars.org/talk/OSGA/78/ END:VEVENT BEGIN:VEVENT SUMMARY:Theodora Bourni (University of Tennessee\, Knoxville) DTSTART:20210831T170000Z DTEND:20210831T180000Z DTSTAMP:20260422T225637Z UID:OSGA/79 DESCRIPTION:Title: An cient polygonal pancakes.\nby Theodora Bourni (University of Tennessee \, Knoxville) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract \nMean curvature flow (MCF) is the gradient flow of the area functional\; it moves the surface in the direction of steepest decrease of area. An im portant motivation for the study of MCF comes from its potential geometric applications\, such as classification theorems and geometric inequalities . MCF develops ``singularities'' (curvature blow-up)\, which obstruct the flow from existing for all times and therefore understanding these high cu rvature regions is of great interest. This is done by studying ancient so lutions\, solutions that have existed for all times in the past\, and whic h model singularities. In this talk we will discuss their importance and w ays of constructing and classifying such solutions. In particular\, we wil l focus on ``collapsed'' solutions and construct\, in all dimensions $n\\g e 2$\, a large family of new examples\, including both symmetric and asymm etric examples\, as well as many eternal examples that do not evolve by tr anslation. Moreover\, we will show that collapsed solutions decompose ``b ackwards in time'' into a canonical configuration of Grim hyperplanes whic h satisfies certain necessary conditions. This is joint work with Mat Lang ford and Giuseppe Tinaglia.\n LOCATION:https://researchseminars.org/talk/OSGA/79/ END:VEVENT BEGIN:VEVENT SUMMARY:Felix Schulze (University of Warwick) DTSTART:20210921T170000Z DTEND:20210921T180000Z DTSTAMP:20260422T225637Z UID:OSGA/80 DESCRIPTION:Title: Me an curvature flow with generic initial data\nby Felix Schulze (Univers ity of Warwick) as part of Online Seminar "Geometric Analysis"\n\n\nAbstra ct\nMean curvature flow is the gradient flow of the area functional and co nstitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed\, similar to Ricci Flow in the intrinsic setting\, to have the potential to serve as a tool to app roach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities\, which one in genera l might not be able to classify completely. Nevertheless\, a well-known co njecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a first step in this dire ction Colding-Minicozzi have shown in fundamental work that spheres and cy linders are the only linearly stable singularity models. As a second step toward Huisken's conjecture we show that mean curvature flow of generic in itial closed surfaces in $\\mathbb{R}^3$ avoids asymptotically conical and non-spherical compact singularities. The main technical ingredient is a l ong-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact self-similarly s hrinking solutions. This is joint work with Otis Chodosh\, Kyeongsu Choi a nd Christos Mantoulidis.\n LOCATION:https://researchseminars.org/talk/OSGA/80/ END:VEVENT BEGIN:VEVENT SUMMARY:Louis Dupaigne (Université Claude Bernard Lyon 1) DTSTART:20210928T170000Z DTEND:20210928T180000Z DTSTAMP:20260422T225637Z UID:OSGA/81 DESCRIPTION:Title: Th e best constant in Sobolev's inequality\, joint work with Ivan Gentil (Lyo n 1) and Simon Zugmeyer (Paris 5)\nby Louis Dupaigne (Université Clau de Bernard Lyon 1) as part of Online Seminar "Geometric Analysis"\n\n\nAbs tract\nDue to its conformal invariance\, the sharp Sobolev inequality take s\nequivalent forms on the three standard model spaces i.e. the Euclidean\ nspace\, the round sphere and the hyperbolic space. By analogy\, we introd uce\nthree weighted manifolds named after Caffarelli\, Kohn and Nirenberg (CKN)\nfor the following reason: the sharp Caffarelli-Kohn-Nirenberg inequ ality in\nthe standard Euclidean space can be reformulated as a (sharp) So bolev\ninequality written on the CKN Euclidean space. It is equivalent to similar\n(but new) Sobolev inequalities on the CKN sphere and the CKN hype rbolic\nspace. In addition\, the Felli-Schneider condition\, that is\, the region of\nparameters for which symmetry breaking occurs in the study of extremals\,\nturns out to have a purely geometric interpretation as an (in tegrated)\ncurvature-dimension condition. To prove these results\, we shal l use Bakry's\ngeneralization of the notion of scalar curvature\, (a weigh ted version of)\nOtto's calculus\, the reformulation of all the inequaliti es (and many more)\nas entropy-entropy production inequalities along appro priate gradient flows\nin Wasserstein space\, and eventually elliptic PDE methods as our best tool\nfor building rigorous and concise proofs.\n LOCATION:https://researchseminars.org/talk/OSGA/81/ END:VEVENT BEGIN:VEVENT SUMMARY:Filip Rindler (University of Warwick) DTSTART:20211005T170000Z DTEND:20211005T180000Z DTSTAMP:20260422T225637Z UID:OSGA/82 DESCRIPTION:Title: Sp ace-time integral currents of bounded variation\nby Filip Rindler (Uni versity of Warwick) as part of Online Seminar "Geometric Analysis"\n\n\nAb stract\nI will present aspects of a theory of space-time integral currents with bounded variation in time. This is motivated by a recent model for e lasto-plastic evolutions that are driven by the flow of dislocations (this model is joint work with T. Hudson). The classical scalar BV-theory can b e recovered as the 0-dimensional limit case of this BV space-time theory. However\, the emphasis is on evolutions of higher-dimensional objects\, mo st notably 1D loops moving within 3D domains (i.e.\, the codimension 2 cas e)\, which corresponds to dislocation dynamics in a material specimen. Bas ed on this\, I will discuss the notion of Lipschitz deformation distance b etween integral currents\, which arises physically as a (simplified) measu re of dissipation. In particular\, I will explain its relation to the boun daryless Whitney flat metric.\n LOCATION:https://researchseminars.org/talk/OSGA/82/ END:VEVENT BEGIN:VEVENT SUMMARY:Joan Verdera (Universitat Autònoma de Barcelona) DTSTART:20211012T170000Z DTEND:20211012T180000Z DTSTAMP:20260422T225637Z UID:OSGA/83 DESCRIPTION:Title: Th e regularity of the boundary of a vortex patch and commutators of singular integrals\nby Joan Verdera (Universitat Autònoma de Barcelona) as pa rt of Online Seminar "Geometric Analysis"\n\n\nAbstract\nI will introduce briefly the vorticity form of the Euler equation in the plane and show how singular integrals appear immediately. Then I will introduce vortex patch es and the problem of regularity of the boundary. I will describe some ele ments of a short proof I have found recently\, which also solves the regul arity problem for other transport equations. Commutators of singular integ rals play a key role\, as it is well-known.\n LOCATION:https://researchseminars.org/talk/OSGA/83/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrea Mondino (University of Oxford) DTSTART:20211019T170000Z DTEND:20211019T180000Z DTSTAMP:20260422T225637Z UID:OSGA/84 DESCRIPTION:Title: Op timal Transport\, weak Laplacian bounds and minimal boundaries in non-smoo th spaces with Lower Ricci Curvature bounds\nby Andrea Mondino (Univer sity of Oxford) as part of Online Seminar "Geometric Analysis"\n\n\nAbstra ct\nThe goal of the seminar is to report on recent joint work with\nDaniel e Semola\, motivated by a question of Gromov to establish a “synthetic\ nregularity theory" for minimal surfaces in non-smooth ambient spaces.\n\n In the setting of non-smooth spaces with lower Ricci Curvature bounds:\n\n
  • We establish a new principle relating lower Ricci Curvature bounds to the\npreservation of Laplacian bounds under the evolution via the Hopf -Lax\nsemigroup\;
    • \n
  • We develop an intrinsic viscosity theory of La placian bounds and prove\nequivalence with other weak notions of Laplacian bounds\;
    • \n
  • We prove sharp Laplacian bounds on the distance functi on from a set\n(locally) minimizing the perimeter: this corresponds to van ishing mean\ncurvature in the smooth setting\;
    • \n
  • We study the regu larity of boundaries of sets (locally) minimizing the\nperimeter\, obtaini ng sharp bounds on the Hausdorff co-dimension of the\nsingular set plus co ntent estimates and topological regularity of the\nregular set.
    • \n\nOptimal transport plays the role of underlying technical tool for addre ssing\nvarious points.\n LOCATION:https://researchseminars.org/talk/OSGA/84/ END:VEVENT BEGIN:VEVENT SUMMARY:Henrik Schumacher (Chemnitz University of Technology) DTSTART:20211026T170000Z DTEND:20211026T180000Z DTSTAMP:20260422T225637Z UID:OSGA/85 DESCRIPTION:Title: Re pulsive Curves and Surfaces\nby Henrik Schumacher (Chemnitz University of Technology) as part of Online Seminar "Geometric Analysis"\n\n\nAbstra ct\nRepulsive Curves and Surfaces\n\nI am going to report on recent work o n the numerical optimization of tangent-point energies of curves and surfa ces. After a motivation and brief introduction to the central computationa l tools (construction of suitable Riemannian metrics on the space of embed ded manifolds\, a polyhedral discretization of the energies\, and fast mul tipole techniques)\, I am going to show a couple of numerical results. Not much about the shape of minimizers has been know so far. So\, for the fir st time\, we will be able to admire the beauty of the energies' minimizers and gradient flows.\n\nThis is based on joint work with Philipp Reiter (C hemnitz University of Technology) and Caleb Brakensiek\, Keenan Crane\, an d Chris Yu (Carnegie Mellon University\, Pittsburgh).\n LOCATION:https://researchseminars.org/talk/OSGA/85/ END:VEVENT BEGIN:VEVENT SUMMARY:Asaf Shachar (The Hebrew University of Jerusalem) DTSTART:20211102T180000Z DTEND:20211102T190000Z DTSTAMP:20260422T225637Z UID:OSGA/86 DESCRIPTION:Title: No n-Euclidean elasticity: Embedding surfaces with minimal distortion\nby Asaf Shachar (The Hebrew University of Jerusalem) as part of Online Semin ar "Geometric Analysis"\n\n\nAbstract\nGiven two dimensional Riemannian ma nifolds $M\,N$\, I will present a sharp lower bound on the elastic energy (distortion) of embeddings $f:M \\to N$\, in terms of the areas' discrepan cy of $M\,N$.\n\nThe minimizing maps attaining this bound go through a pha se transition when the ratio of areas is $1/4$: The homotheties are the un ique energy minimizers when the ratio $\\frac{\\operatorname{Vol}(N)}{\\op eratorname{Vol}(M)} \\ge 1/4$\, and they cease being minimizers when $\\fr ac{\\operatorname{Vol}(N)}{\\operatorname{Vol}(M)} $ gets below $1/4$.\n\n I will describe explicit minimizers in the non-trivial regime $\\frac{\\op eratorname{Vol}(N)}{\\operatorname{Vol}(M)} < 1/4$ when $M\,N$ are disks\, and give a proof sketch of the lower bound. If time permits\, I will disc uss the stability of minimizers.\n LOCATION:https://researchseminars.org/talk/OSGA/86/ END:VEVENT BEGIN:VEVENT SUMMARY:Juncheng Wei (University of British Columbia) DTSTART:20211109T180000Z DTEND:20211109T190000Z DTSTAMP:20260422T225637Z UID:OSGA/87 DESCRIPTION:Title: Si ngularity formations in some geometric flows\nby Juncheng Wei (Univers ity of British Columbia) as part of Online Seminar "Geometric Analysis"\n\ n\nAbstract\nI will discuss constructions of finite or infinite time blow- ups for several geometric flows\, including harmonic maps flows\, half-har monic map flows and Yang-Mills flows. Phenomenon include forward bubblin g\, reverse bubbling\, bubbling continuations\, bubbling towers.\n LOCATION:https://researchseminars.org/talk/OSGA/87/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephanie Wang (UC San Diego) DTSTART:20211116T180000Z DTEND:20211116T190000Z DTSTAMP:20260422T225637Z UID:OSGA/88 DESCRIPTION:Title: Ca pturing surfaces with differential forms\nby Stephanie Wang (UC San Di ego) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nThe ext erior calculus of differential forms has been an important tool in solving PDEs in geometry processing. In this talk we expand the usage of differe ntial forms to a whole new way of representing curves and surfaces. By do ing so we reformulate the classical nonconvex Plateau minimal surface prob lem into a convex optimization problem.\n LOCATION:https://researchseminars.org/talk/OSGA/88/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Schmidt DTSTART:20220201T170000Z DTEND:20220201T180000Z DTSTAMP:20260422T225637Z UID:OSGA/89 DESCRIPTION:Title: Pe rimeter functionals with measure datum\nby Thomas Schmidt as part of O nline Seminar "Geometric Analysis"\n\n\nAbstract\nThe talk is concerned wi th perimeter functionals $\\mathscr{P}_\\mu$ given by\n\\[\n \\mathscr{P} _\\mu[A]:=\\mathrm{P}(A)-\\mu(A^+)\n\\]\non sets $A\\subset{\\mathbb R}^n$ of finite volume and finite perimeter\n$\\mathrm{P}(A)$\, where the fixed non-negative Radon measure $\\mu$ may be\nsingular and is (necessarily) e valuated on a suitable closure $A^+$ of\n$A$. It will be explained that se micontinuity and existence results for\n$\\mathscr{P}_\\mu$ crucially depe nd on a new type of isoperimetric condition\,\nwhich also admits some ($n{ -}1$)-dimensional measures $\\mu$\, and exemplary\nconfigurations will be discussed. The long-term goal of these considerations is\nto extend the va riational approach to prescribed mean curvature hypersurfaces in\nthe spir it of Caccioppoli\, De Giorgi\, Miranda\, Massari from $\\mathrm{L}^1$ mea n\ncurvature to mean curvature given by a possibly lower-dimensional measu re.\n LOCATION:https://researchseminars.org/talk/OSGA/89/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Körber (University of Vienna) DTSTART:20211207T180000Z DTEND:20211207T190000Z DTSTAMP:20260422T225637Z UID:OSGA/90 DESCRIPTION:Title: Fo liations of asymptotically flat 3-manifolds by stable constant mean curvat ure spheres\nby Thomas Körber (University of Vienna) as part of Onlin e Seminar "Geometric Analysis"\n\n\nAbstract\nStable constant mean curvatu re spheres encode important information on the asymptotic geometry of init ial data sets for isolated gravitational systems. In this talk\, I will pr esent a short new proof (joint with M. Eichmair) based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of such an initial data set by stable constant mean curvature spheres. In the case where the scalar curvature is non-negative\, our method also shows that the leaves o f this foliation are the only large stable constant mean curvature spheres that enclose the center of the initial data set.\n LOCATION:https://researchseminars.org/talk/OSGA/90/ END:VEVENT BEGIN:VEVENT SUMMARY:Josef Bemelmans (RWTH Aachen University) DTSTART:20211214T180000Z DTEND:20211214T190000Z DTSTAMP:20260422T225637Z UID:OSGA/91 DESCRIPTION:Title: A Central Result from Newton's Principia Mathematica: The Body of Least Resi stance\nby Josef Bemelmans (RWTH Aachen University) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nIn Newton's Principia Mathemat ica fundamental theorems\, e.g about the motion of planets around the sun\ , are proven by methods of ancient geometry rather than infinitesimal anal ysis\, as one might expect. There are however problems in the Principia th at are treated using techniques from calculus\; we present one that in tod ay's terminology belongs to the calculus of variations: to determine the s hape of a rotationally symmetric body of prescribed base and height such t hat its resistance in a uniform fluid flow becomes minimal.\n LOCATION:https://researchseminars.org/talk/OSGA/91/ END:VEVENT BEGIN:VEVENT SUMMARY:Gianmichele Di Matteo (Karlsruhe Institute of Technology) DTSTART:20220111T170000Z DTEND:20220111T180000Z DTSTAMP:20260422T225637Z UID:OSGA/92 DESCRIPTION:Title: A Local Singularity Analysis for the Ricci flow\nby Gianmichele Di Matte o (Karlsruhe Institute of Technology) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nIn this talk\, I will describe a refined local s ingularity analysis for the Ricci flow developed jointly with R. Buzano. T he key idea is to investigate blow-up rates of the curvature tensor locall y\, near a singular point. Then I will show applications of this theory to Ricci flows with scalar curvature bounded up to the singular time.\n LOCATION:https://researchseminars.org/talk/OSGA/92/ END:VEVENT BEGIN:VEVENT SUMMARY:Leah Schätzler (University Salzburg) DTSTART:20220118T170000Z DTEND:20220118T180000Z DTSTAMP:20260422T225637Z UID:OSGA/93 DESCRIPTION:Title: H ölder continuity for a doubly nonlinear equation\nby Leah Schätzler (University Salzburg) as part of Online Seminar "Geometric Analysis"\n\n\n Abstract\nThe prototype of the partial differential equations considered i n this talk is\n$$\n\\partial_t \\big( |u|^{q-1} u \\big) - \\operatorname {div} \\big( |Du|^{p-2} Du \\big) = 0\n\\quad \\text{in } E_T = E \\times (0\,T] \\subset \\mathbb{R}^{N+1}\n$$\nwith parameters $q>0$ and $p>1$.\nW ell-known special cases of this doubly nonlinear equation are the porous m edium equation ($p=2$)\, the parabolic $p$-Laplace equation ($q=1$) and Tr udinger's equation ($q=p-1$).\nI will present H\\"older continuity results based on joint work with Verena B\\"ogelein\, Frank Duzaar and Naian Liao .\n LOCATION:https://researchseminars.org/talk/OSGA/93/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Stanin (University Salzburg) DTSTART:20220125T170000Z DTEND:20220125T180000Z DTSTAMP:20260422T225637Z UID:OSGA/94 DESCRIPTION:Title: Gl obal continuity of variational solutions weakening the one-sided bounded s lope condition\nby Thomas Stanin (University Salzburg) as part of Onli ne Seminar "Geometric Analysis"\n\n\nAbstract\nIn this talk\, we have a lo ok at regularity properties of variational solutions to a class of Cauchy- Dirichlet problems of the form\n\n$$\n\\begin{cases}\n\\partial_t u - \\ma thrm{div}_x (D_\\xi f(Du)) = 0 & \\text{in}\\ \\Omega_T\, \\\\\nu = u_0 & \\text{on}\\ \\partial_\\mathcal{P}\\Omega_T.\n\\end{cases}\n$$\n\nWe do n ot impose any growth conditions from above on $f \\colon \\R^n \\to \\R$ b ut require it to be convex and coercive. The domain $\\Omega \\subset \\R^ n$ is supposed to be bounded and convex and for the time-independent bound ary datum $u_0 \\colon \\overline\\Omega \\to \\R$\, we require continuity . These assumptions on $u_0$ are weaker than a one-sided version of the bo unded slope condition. We present a result showing variational solutions $ u \\colon \\Omega_T \\to \\R$ to these problem class to be globally contin uous.\n LOCATION:https://researchseminars.org/talk/OSGA/94/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandra Pluda (University of Pisa) DTSTART:20220208T170000Z DTEND:20220208T180000Z DTSTAMP:20260422T225637Z UID:OSGA/95 DESCRIPTION:Title: Re solution of singularities of the network flow\nby Alessandra Pluda (Un iversity of Pisa) as part of Online Seminar "Geometric Analysis"\n\n\nAbst ract\nThe curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature (at any point and time) and can be interpreted as the gradient flow of the length. We consider the same flow for networks (finite unions of sufficiently smooth\ncurves whos e end points meet at junctions). Because of the variational nature of the problem\, one expects that for almost all the times the evolving network w ill possess only triple junctions where the unit tangent vectors forms ang les of 120 degrees (regular junctions). However\, even if the initial netw ork has only regular junctions\, this property is not preserved by the flo w and junctions of four or more curves may appear during the evolution.\nT he aim of this talk is first to describe the process of singularity format ion and then\nto explain the resolution of such singularities and how to c ontinue the flow in a classical PDE framework.\n\nThis is a research in co llaboration with Jorge Lira (Universidade Federal do Ceará)\, \nRafe Maz zeo (Stanford University) and  Mariel Saez (P. Universidad Catolica de Ch ile).\n LOCATION:https://researchseminars.org/talk/OSGA/95/ END:VEVENT BEGIN:VEVENT SUMMARY:Vyron Vellis (University of Tennessee\, Knoxville) DTSTART:20220215T170000Z DTEND:20220215T180000Z DTSTAMP:20260422T225637Z UID:OSGA/96 DESCRIPTION:Title: Bi -Lipschitz embeddings\nby Vyron Vellis (University of Tennessee\, Knox ville) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nTo im prove our understanding of a metric space\, it is often\nhelpful to realiz e the space within some Euclidean space. The embedding\nproblem is concern ed with recognizing those spaces which admit an embedding\ninto some Eucli dean space that does not distort its geometry too much. The\nbi-Lipschitz emebedding problem is concerned with identifying those metric\nspaces for which such an embedding exists. The embedding problem has\nrecently genera ted great interest in theoretical computer science and\, more\nspecificall y\, in graphic imaging and storage and access issues for large\ndata sets. In the first part of the talk we will examine the embeddability\nof two w ell-known sub-Riemannian manifolds\, the Grushin plane and the\nHeisenberg group. In the second part we will discuss the embeddability of\nmetric tr ees with good geometry. The talk is based on joint works with\nRomney (201 7)\, Li\, Chousionis\, and Zimmerman (2020)\, David (2020)\, and David\nan d Eriksson-Bique (2021).\n LOCATION:https://researchseminars.org/talk/OSGA/96/ END:VEVENT BEGIN:VEVENT SUMMARY:Azahara DelaTorre (Sapienza Università di Roma) DTSTART:20220222T170000Z DTEND:20220222T180000Z DTSTAMP:20260422T225637Z UID:OSGA/97 DESCRIPTION:Title: Th e fractional Yamabe problem with singularities\nby Azahara DelaTorre ( Sapienza Università di Roma) as part of Online Seminar "Geometric Analysi s"\n\n\nAbstract\nThe so called Yamabe problem in Conformal Geometry consi sts in finding a metric conformal to a given one and which has constant sc alar curvature. From the analytic point of view\, this problem becomes a s emilinear elliptic PDE with critical (for the Sobolev embedding) power non -linearity. If we study the problem in the Euclidean space\, allowing the presence of nonzero-dimensional singularities can be transformed into redu cing the non-linearity to a Sobolev-subcritical power. A quite recent noti on of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic PDE. \n\nIn this talk\, we will focus on metrics which are singular along nonz ero-dimensional singularities. In collaboration with Ao\, Chan\, Fontelos\ , González and Wei\, we covered the construction of solutions which are s ingular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods c oming from conformal geometry and Scattering theory for the study of non-l ocal ODEs. Due to the limitations of the techniques we used\, the particul ar case of ``maximal’’ dimension for the singularity was not covered. In a recent work\, in collaboration with H. Chan\, we cover this specific dimension constructing and studying singular solutions of critical dimensi on.\n LOCATION:https://researchseminars.org/talk/OSGA/97/ END:VEVENT BEGIN:VEVENT SUMMARY:Tomasz Adamowicz (Polish Academy of Sciences in Warsaw) DTSTART:20220301T170000Z DTEND:20220301T180000Z DTSTAMP:20260422T225637Z UID:OSGA/98 DESCRIPTION:Title: Is operimetric inequalities and curvature of level sets for harmonic function s on smooth and singular surfaces\nby Tomasz Adamowicz (Polish Academy of Sciences in Warsaw) as part of Online Seminar "Geometric Analysis"\n\n \nAbstract\nOne of the main themes of the talk are monotonicity formulas f or\nlevel sets of harmonic functions in Euclidean domains and Riemannian\n surfaces\, including the singular Alexandrov surfaces. Such formulas allow \nfor studying the logarithmic convexity of the length of the level curves and\nrelated isoperimetric type inequalities. Related are the studies of the\ngeodesic curvature of the level curves and of the steepest descent.\n LOCATION:https://researchseminars.org/talk/OSGA/98/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohameden Ahmedou (Universitty of Giessen) DTSTART:20220308T170000Z DTEND:20220308T180000Z DTSTAMP:20260422T225637Z UID:OSGA/99 DESCRIPTION:Title: Ne w Blow up phenomena for the Nirenberg's problem on half spheres\nby Mo hameden Ahmedou (Universitty of Giessen) as part of Online Seminar "Geomet ric Analysis"\n\n\nAbstract\nIn this talk I will report on a refined blo w up analysis of finite energy approximated solutions to the Nirenberg's problem on half spheres. The later consists of prescribing under minimal boundary conditions the scalar curvature to be a given function. In parti cular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg's problem on half spheres is far more complicated that in the case of closed spheres. I ndeed besides the combination of interior and boundary blow ups\, there ar e "non simples blow ups" for subcritical solutions having zero or nonzer o weak limit. The formation of such non simple blow ups is governed by a vortex problem\, unveiling an unexpected connection with Euler equation s in fluid dynamic and mean fields type equations in mathematical physics. \nThis is joint work with Mohamed Ben Ayed (Sfax University).\n LOCATION:https://researchseminars.org/talk/OSGA/99/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Young (New York University) DTSTART:20220322T170000Z DTEND:20220322T180000Z DTSTAMP:20260422T225637Z UID:OSGA/101 DESCRIPTION:Title: M etric differentiation and embeddings of the Heisenberg group\nby Rober t Young (New York University) as part of Online Seminar "Geometric Analysi s"\n\n\nAbstract\nThe Heisenberg group is the simplest example of a noncom mutative nilpotent Lie group. In this talk\, we will explore how that nonc ommutativity affects geometry and analysis in the Heisenberg group. We wil l describe why good embeddings of $\\mathbb{H}$ must be bumpy at many scal es\, how to study embeddings into $L_1$ by studying surfaces in $\\mathbb{ H}$\, and how to construct a metric space which embeds into $L_1$ and $L_4 $ but not in $L_2$. This talk is joint work with Assaf Naor.\n LOCATION:https://researchseminars.org/talk/OSGA/101/ END:VEVENT BEGIN:VEVENT SUMMARY:Maria G. Westdickenberg (RWTH Aachen University) DTSTART:20220426T160000Z DTEND:20220426T170000Z DTSTAMP:20260422T225637Z UID:OSGA/103 DESCRIPTION:Title: C onvergence and metastability of (weakly) nonconvex gradient flows\nby Maria G. Westdickenberg (RWTH Aachen University) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nTogether with Felix Otto\, Richard Sc hubert\, and other collaborators\, we have\ndeveloped two different energy -based methods to capture convergence and metastability. We have\nused the se methods to establish optimal\, algebraic convergence for the Mullins-Se kerka (MS)\nproblem in the plane and the Cahn-Hilliard equation on the lin e. After a general introduction of the central ideas\, we comment in parti cular on the role of curvature in the MS problem. Work in progress with Ri chard Schubert and Felix Otto extends the L1-based method to the Mullins-S ekerka evolution in three dimensions.\n LOCATION:https://researchseminars.org/talk/OSGA/103/ END:VEVENT BEGIN:VEVENT SUMMARY:Christian Gerhards (TU Freiberg) DTSTART:20220329T160000Z DTEND:20220329T170000Z DTSTAMP:20260422T225637Z UID:OSGA/104 DESCRIPTION:Title: S ome inverse magnetization problems motivated from geoscience\nby Chris tian Gerhards (TU Freiberg) as part of Online Seminar "Geometric Analysis" \n\n\nAbstract\nWe present an overview on some aspects of geomagnetic inve rse problems related to Hardy spaces on the sphere/Lipschitz surfaces\, He lmholtz Hodge decomposition on Lipschitz domains\, and spatial localizatio n. A particular focus is on uniqueness issues and the influence of discret ization (e.g.\, if the geometry of the discretization influences uniquenes s). The talk will on the on hand try to motivate the geophysical backgroun d of these problems and provide some basic examples\, and on the other han d present a proper analysis of the problems.\n LOCATION:https://researchseminars.org/talk/OSGA/104/ END:VEVENT BEGIN:VEVENT SUMMARY:Michał Miśkiewicz (Institute of Mathematics\, Polish Academy of Sciences) DTSTART:20220405T160000Z DTEND:20220405T170000Z DTSTAMP:20260422T225637Z UID:OSGA/105 DESCRIPTION:Title: S truggles with the regularity of $n$-harmonic maps\nby Michał Miśkiew icz (Institute of Mathematics\, Polish Academy of Sciences) as part of Onl ine Seminar "Geometric Analysis"\n\n\nAbstract\nLet us consider the Dirich let $n$-energy $\\int_{\\mathcal{M}} |\\nabla u|^n$ for maps $u \\colon \\ mathcal{M}^n \\to \\mathcal{N}^m$ between two Riemannian manifolds. Its Eu ler-Lagrange system – known as the $n$-harmonic equation – is an examp le of a conformally invariant system with critical nonlinearity. In genera l such systems can have discontinuous solutions\, but the regularity of $n $-harmonic maps is an open problem for $n > 2$. \n\nPartial results in thi s direction rely on the jacobian structure of the $n$-harmonic equation\, together with the theory of Hardy and BMO spaces. After a brief review of these methods\, I will describe a new application\, which leads to yet ano ther partial result – regularity for $W^{n/2\,2}$-solutions – but also gives some hope for further progress. \n\nThis is joint work with Paweł Strzelecki and Bogdan Petraszczuk.\n LOCATION:https://researchseminars.org/talk/OSGA/105/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Chen (Georg-August-Universität Göttingen) DTSTART:20220412T160000Z DTEND:20220412T170000Z DTSTAMP:20260422T225637Z UID:OSGA/106 DESCRIPTION:Title: G luing constructions of minimal surfaces: Recent progress and future plans< /a>\nby Hao Chen (Georg-August-Universität Göttingen) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nI will review our recent const ructions of minimal surfaces by gluing catenoids\, helicoids\, and saddle towers. In particular\, we recently resolved some technical issues in pre vious similar constructions and revealed surprising connections between mi nimal surfaces and fluid dynamics. Moreover\, I will discuss possibilitie s of further improving the gluing techniques. The talk covers joint works with Martin Traizet and Daniel Freese.\n LOCATION:https://researchseminars.org/talk/OSGA/106/ END:VEVENT BEGIN:VEVENT SUMMARY:Jasmin Hörter (Karlsruhe Institut of Technology) DTSTART:20220802T160000Z DTEND:20220802T170000Z DTSTAMP:20260422T225637Z UID:OSGA/107 DESCRIPTION:Title: R igidity of $\\epsilon$-harmonic maps of low degree\nby Jasmin Hörter (Karlsruhe Institut of Technology) as part of Online Seminar "Geometric An alysis"\n\n\nAbstract\nIn 1981 Sacks and Uhlenbeck introduced their famous alpha-approximation of the Dirichlet energy for maps from surfaces and sh owed that critical points converge (away from finitely many points) to a h armonic map. Now one can ask whether every harmonic map is captured by thi s limiting process. Lamm\, Malchiodi and Micallef answered this for maps f rom the two sphere into the two sphere and showed that the Sacks-Uhlenbeck method produces only constant maps and rotations if the energy lies below a certain threshold. We investigate the same question for the epsilon-app roximation of the Dirichlet energy.\nJoint work with Tobias Lamm and Mario Micallef.\n LOCATION:https://researchseminars.org/talk/OSGA/107/ END:VEVENT BEGIN:VEVENT SUMMARY:Sara Daneri (Gran Sasso Science Institute) DTSTART:20220927T160000Z DTEND:20220927T170000Z DTSTAMP:20260422T225637Z UID:OSGA/108 DESCRIPTION:Title: S ymmetry breaking and pattern formation for functionals with competing inte ractions\nby Sara Daneri (Gran Sasso Science Institute) as part of Onl ine Seminar "Geometric Analysis"\n\n\nAbstract\nIn this talk we will revie w some recent results we obtained on the one-dimensionality of the minimiz ers\nof a family of continuous local/nonlocal interaction functionals in g eneral dimension. Such functionals have a local term\, typically the perim eter or its Modica-Mortola approximation\, which penalizes interfaces\, an d a nonlocal term favouring oscillations which are high in frequency and i n amplitude. The competition between the two terms is expected by experime nts and simulations to give rise to periodic patterns at equilibrium. Func tionals of this type are used  to model pattern formation\, either in mat erial science or in biology. The difficulty in proving the emergence of su ch structures is due to the fact that the functionals are symmetric with r espect to permutation of coordinates\, while in more than one space dimens ions minimizers are one-dimensional\, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnet ic systems\, respectively  colloidal suspensions)\, both in sharp interfa ce and in diffuse interface models\, minimizers are one-dimensional and pe riodic\, in general dimension and also while imposing a nontrivial volume constraint. The results are contained in a series of joint works with Eris Runa and Alicja Kerschbaum.\n LOCATION:https://researchseminars.org/talk/OSGA/108/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Lagemann (RWTH Aachen University) DTSTART:20220503T160000Z DTEND:20220503T170000Z DTSTAMP:20260422T225637Z UID:OSGA/109 DESCRIPTION:Title: T angent-point energies as Gamma-limit of discrete tangent-point energies on biarc curves\nby Anna Lagemann (RWTH Aachen University) as part of On line Seminar "Geometric Analysis"\n\n\nAbstract\nUsing interpolation with biarc curves we prove $\\Gamma$-convergence of discretized tangent-point e nergies to the continuous tangent-point energies in the $C^1$-topology. As a consequence\, discrete almost minimizing biarc curves converge to minim izers of the continuous tangent-point energies. In addition\, taking point -tangent data from a given $C^{1\,1}$-curve $\\gamma$\, we establish conve rgence of the discrete energies evaluated on biarc curves interpolating th ese data\, to the continuous tangent-point energy of $\\gamma$\, together with an explicit convergence rate. This is joint work with Heiko von der M osel.\n LOCATION:https://researchseminars.org/talk/OSGA/109/ END:VEVENT BEGIN:VEVENT SUMMARY:Gael Yomgne Diebou (Uni Bonn) DTSTART:20220510T160000Z DTEND:20220510T170000Z DTSTAMP:20260422T225637Z UID:OSGA/110 DESCRIPTION:Title: W ell-posedness theory for the weakly harmonic maps problem subject to irreg ular data\nby Gael Yomgne Diebou (Uni Bonn) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nWe study the existence\, uniqueness an d regularity of weakly harmonic maps\ninto a closed Riemannian manifold. I n this talk\, I will emphasize on the\nnovel ideas\, based on intrinsic fe atures of the problem and modern\nharmonic analysis tools which allow us t o prescribe Dirichlet data with\ninfinite energy. More precisely\, we prov e that under a mere smallness\nhypothesis on the boundary data measured in the $L^{\\infty}$ or $BMO$\nnorm\, there exists a unique solution which i s locally infinitely smooth.\nWhile this regularity feature fails in absen ce of the smallness assumption\,\nexistence still persists for large data provided the domain is bounded and\nthere exist smooth stable weakly harm onic maps.\nThis is a joint work with Herbert Koch.\n LOCATION:https://researchseminars.org/talk/OSGA/110/ END:VEVENT BEGIN:VEVENT SUMMARY:Ali Hyder (TIFR-CAM\, Bangalore) DTSTART:20220524T160000Z DTEND:20220524T170000Z DTSTAMP:20260422T225637Z UID:OSGA/111 DESCRIPTION:Title: B low-up analysis and partial regularity results for Liouville type equation s\nby Ali Hyder (TIFR-CAM\, Bangalore) as part of Online Seminar "Geom etric Analysis"\n\n\nAbstract\nDue to the presence of the exponential nonl inearity\, the Liouville equation in dimension three and higher is supercr itical. In particular\, it admits several singular solutions. We will talk about asymptotic behavior of a family of stationary solutions\, and how t o use it to obtain partial regularity results.\n LOCATION:https://researchseminars.org/talk/OSGA/111/ END:VEVENT BEGIN:VEVENT SUMMARY:Hannes Matt (RWTH Aachen University) DTSTART:20220531T160000Z DTEND:20220531T170000Z DTSTAMP:20260422T225637Z UID:OSGA/113 DESCRIPTION:Title: B anach gradient flows for various families of knot energies\nby Hannes Matt (RWTH Aachen University) as part of Online Seminar "Geometric Analysi s"\n\n\nAbstract\nThis is joint work with Daniel Steenebrügge and Heiko v on der Mosel. We establish long-time existence of Banach gradient flows fo r generalised integral Menger curvatures and tangent-point energies\, and for O'Hara's self-repulsive potentials $E^{\\alpha\,p}$. In order to do so \, we employ the theory of curves of maximal slope in slightly smaller spa ces compactly embedding into the respective energy spaces associated to th ese functionals\, and add a term involving the logarithmic strain\, which controls the parametrisations of the flowing (knotted) loops. As a prerequ isite\, we prove in addition that O'Hara's knot energies $E^{\\alpha\,p}$ are continuously differentiable.\n LOCATION:https://researchseminars.org/talk/OSGA/113/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Laux (University of Bonn) DTSTART:20220621T160000Z DTEND:20220621T170000Z DTSTAMP:20260422T225637Z UID:OSGA/114 DESCRIPTION:Title: L ocal minimizers of the area functional based on a concept of local paired calibrations\nby Tim Laux (University of Bonn) as part of Online Semin ar "Geometric Analysis"\n\n\nAbstract\nCalibrations are an elegant tool to prove that a given surface (or surface cluster) minimizes the area functi onal. In this talk\, I will present a way to extend the notion of (paired) calibrations to the setting when one can only hope for local minimality. Based on this notion\, one can show that any partition of the plane\, whos e network of interfaces consists of finitely many straight segments with a singular set made up of finitely many triple junctions at which the Herri ng angle condition is satisfied\, is a local minimizer of the interface en ergy with respect to $L^1$ perturbations of the phases. This result not on ly holds for the case of the area functional but for a general class of su rface tension matrices.\n\nThis is based on joint work with Julian Fischer \, Sebastian Hensel\, and Theresa Simon.\n LOCATION:https://researchseminars.org/talk/OSGA/114/ END:VEVENT BEGIN:VEVENT SUMMARY:Simon Bortz (University of Alabama) DTSTART:20220614T160000Z DTEND:20220614T170000Z DTSTAMP:20260422T225637Z UID:OSGA/115 DESCRIPTION:Title: C aloric Measure and Parabolic Uniform Rectifiability\nby Simon Bortz (U niversity of Alabama) as part of Online Seminar "Geometric Analysis"\n\n\n Abstract\nIn the late 70's Dahlberg showed that harmonic measure and surfa ce measure are mutually absolutely continuous in Lipschitz domains in $\\m athbb{R}^d$ (this was a long standing conjecture). In fact\, he showed a s tronger quantitative version of mutual absolute continuity \, $A_\\infty$\ , which is equivalent to certain $L^p$ estimates on solutions. It was conj ectured by Hunt that the same is true in the parabolic setting\, that is\, for parabolic Lipschitz graph domains\; however\, this turned out to be f alse as a counterexample was produced by Kaufman and Wu. On the other hand \, it was later shown by Lewis and Murray that if the graphs had a little more time-regularity then Dahlberg's theorem holds.\n\nTogether with my co -authors\, we have shown the work of Lewis and Murray is sharp. In particu lar\, if a domain is given by the region above a parabolic Lipschitz graph the $A_\\infty$ property of caloric measure is equivalent to this extra t ime regularity. These `regular' parabolic Lipschitz graphs are the prototy pical parabolic uniformly rectifiable (P-UR) sets and this project is part of a larger program to characterize P-UR sets by properties of caloric fu nctions/measure.\n LOCATION:https://researchseminars.org/talk/OSGA/115/ END:VEVENT BEGIN:VEVENT SUMMARY:Augusto Ponce (Université catholique de Louvain) DTSTART:20220628T160000Z DTEND:20220628T170000Z DTSTAMP:20260422T225637Z UID:OSGA/116 DESCRIPTION:Title: A topological toolbox for Sobolev maps\nby Augusto Ponce (Université c atholique de Louvain) as part of Online Seminar "Geometric Analysis"\n\n\n Abstract\nClassical works by F. Bethuel and by F. Hang and F-H. Lin have\n identified the local and global topological obstructions that prevent smoo th\nmaps from being dense in the Sobolev space \\(W^{1\, p}(M^{m}\; N^{n}) \\)\nbetween two Riemannian manifolds when \\(p < m\\). They are related t o the\nextension of continuous maps from subsets of \\(M^{m}\\) to \\(N^{n }\\).\n\nIn this talk I will present some work in progress with P. Bousque t\n(Toulouse) and J. Van Schaftingen (UCLouvain)\, inspired from the notio ns of\nmodulus introduced by B. Fuglede and degree for VMO maps by H. Brez is and L.\nNirenberg.\nI shall explain how one can decide whether a specif ic Sobolev map \\(u :\nM^{m} \\to N^{n}\\) can be approximated or not by s mooth ones\, even in the\npresence of topological obstructions from \\(M^{ m}\\) or \\(N^{n}\\).\n LOCATION:https://researchseminars.org/talk/OSGA/116/ END:VEVENT BEGIN:VEVENT SUMMARY:Xavier Tolsa (ICREA - Universitat Autònoma de Barcelona - CRM) DTSTART:20220705T160000Z DTEND:20220705T170000Z DTSTAMP:20260422T225637Z UID:OSGA/117 DESCRIPTION:Title: T he regularity problem for the Laplace equation and boundary Poincaré ineq ualities in rough domains\nby Xavier Tolsa (ICREA - Universitat Autòn oma de Barcelona - CRM) as part of Online Seminar "Geometric Analysis"\n\n \nAbstract\nGiven a bounded domain $\\Omega \\subset \\mathbb R^n$\, one s ays that the\n$L^p$-regularity problem is solvable for the Laplace equatio n in $\\Omega$\nif\, given any continuous function $f$ defined in $\\parti al \\Omega$ and the\nharmonic extension $u$ of $f$ to $\\Omega$\, the non- tangential maximal\nfunction of the gradient of $u$ can be controlled in $ L^p$ norm by the\ntangential derivative of $f$ in $\\partial\\Omega$. Up t o quite recently this\nwas only known to hold for Lipschitz domains (in so me range of $p$'s). \nIn my talk I will explain a recent result with Mihal is Mourgoglou where we\nshow that the $L^p$-regularity is also solvable in more general domains\,\nsuch as 2-sided chord-arc domains. In the solutio n of this problem\, the\nPoincaré inequality in the boundary of the domai n plays an important role. I\nwill also discuss this issue and a related j oint result with Olli Tapiola\nwhere we show that the boundaries of 2-side d chord-arc domains support\n1-Poincaré inequalities.\n LOCATION:https://researchseminars.org/talk/OSGA/117/ END:VEVENT BEGIN:VEVENT SUMMARY:Guozhen Lu (University of Connecticut) DTSTART:20220920T160000Z DTEND:20220920T170000Z DTSTAMP:20260422T225637Z UID:OSGA/118 DESCRIPTION:Title: H elgason-Fourier analysis techniques in sharp geometric inequalities on hyp erbolic spaces\nby Guozhen Lu (University of Connecticut) as part of O nline Seminar "Geometric Analysis"\n\n\nAbstract\nIn recent years\, we hav e developed a new approach to establish sharp geometric and functional ine qualities using the Helgason-Fourier analysis techniques on symmetric spac es. Such inequalities include sharp higher order Hardy-Sobolev-Maz'ya and Hardy-Adams inequalities on hyperbolic spaces on all Riemannian symmetric spaces of noncompact type of rank one. Precise expressions of Green's func tions of GJMS operators on real hyperbolic spaces in terms of hypergeometr ic functions are established as well.This is based on a series of joint wo rks with Joshua Flynn\, Jungang Li\, and Qiaohua Yang.\n LOCATION:https://researchseminars.org/talk/OSGA/118/ END:VEVENT BEGIN:VEVENT SUMMARY:Max Goering DTSTART:20220719T160000Z DTEND:20220719T170000Z DTSTAMP:20260422T225637Z UID:OSGA/119 DESCRIPTION:Title: F inslerian regularity theory in Euclidean space\nby Max Goering as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nIn the setting of s ets of finite perimeter\, the regularity of\nsurfaces minimizing $\\| \\cd ot \\|_{p}$-surface energies is entirely unknown.\nSince these energies do not satisfy Almgren's ellipticity condition\, the PDE\nthat arises (as th e partial linearization in the small gradient regime of\nthe anisotropic m inimal surface) is very degenerate elliptic. In this\nexample\, the releva nt PDE is the Finsler $\\gamma$-Laplacian. This motivates\na discussion of the state-of-the-art regularity theory for the very\ndegenerate elliptic and non-linear Finsler $\\gamma$-Laplacian. Pending time\,\nsome potential applications to classical questions in geometric measure\ntheory will als o be discussed. This talk discusses joint work.\n LOCATION:https://researchseminars.org/talk/OSGA/119/ END:VEVENT BEGIN:VEVENT SUMMARY:Ulrich Menne (National Taiwan Normal University and National Cente r for Theoretical Sciences) DTSTART:20220726T160000Z DTEND:20220726T170000Z DTSTAMP:20260422T225637Z UID:OSGA/120 DESCRIPTION:Title: I ntegral chains with coefficients in a complete normed commutative group\nby Ulrich Menne (National Taiwan Normal University and National Center for Theoretical Sciences) as part of Online Seminar "Geometric Analysis"\n \n\nAbstract\nAs a service to the community\, in joint work with Christian Scharrer\, we provide—for Euclidean space—a basic treatment of locall y rectifiable chains and of the complex of locally integral chains. In thi s setting\, we may beneficially develop the idea of a complete normed comm utative group bundle over the Grassmann manifold whose fibre is the coeffi cient group of the chains. Our exposition also sheds new light on some alg ebraic aspects of the theory. Finally\, we indicate an extension to a geom etric approach to locally flat chains centring on locally rectifiable chai ns rather than completion procedures.\n\nThe virtual whiteboard https://miro.com/ app/board/uXjVOpucY68=/ will be employed\; its password protection wil l be removed for the duration of the presentation. There will be no recor ding for this talk.\n LOCATION:https://researchseminars.org/talk/OSGA/120/ END:VEVENT BEGIN:VEVENT SUMMARY:Yannick Sire (Johns Hopkins University) DTSTART:20220913T160000Z DTEND:20220913T170000Z DTSTAMP:20260422T225637Z UID:OSGA/121 DESCRIPTION:Title: H armonic maps with free boundary and beyond\nby Yannick Sire (Johns Hop kins University) as part of Online Seminar "Geometric Analysis"\n\n\nAbstr act\nI will introduce a new heat flow for harmonic maps with free boundary . After giving some motivations to study such maps in relation with extrem al metrics in spectral geometry\, I will construct weak solutions for the flow and derive their partial regularity. The introduction of this new flo w is motivated by the so-called half-harmonic maps introduced by Da Lio an d Riviere\, which provide a new approach to the old topic of harmonic maps with free boundary. I will also state some open problems and possible gen eralizations.\n LOCATION:https://researchseminars.org/talk/OSGA/121/ END:VEVENT BEGIN:VEVENT SUMMARY:Julian Fischer (IST Austria) DTSTART:20220906T160000Z DTEND:20220906T170000Z DTSTAMP:20260422T225637Z UID:OSGA/122 DESCRIPTION:Title: M ultiphase Mean Curvature Flow: Uniqueness Properties of Weak Solution Conc epts and Phase-Field Approximations\nby Julian Fischer (IST Austria) a s part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nTopology chan ges occur naturally in geometric evolution equations like mean curvature f low. As classical solution concepts break down at such geometric singulari ties\, the use of weak solution concepts becomes necessary in order to des cribe topological changes. For two-phase mean curvature flow\, the theory of viscosity solutions by Chen-Giga-Goto and Evans-Spruck provides a conce pt of weak solutions with basically optimal existence and uniqueness prope rties. In contrast\, the uniqueness properties of weak solution concepts f or multiphase mean curvature flow had remained mostly unexplored.\n\nBy in troducing a novel concept of "gradient flow calibrations"\, we establish a weak-strong uniqueness principle for multiphase mean curvature flow: Weak (BV) solutions to multiphase mean curvature flow are unique as long as a classical solution exists. In particular\, in planar multiphase mean curva ture flow\, weak (BV) solutions are unique prior to the first topological change. As basic counterexamples show\, the uniqueness of evolutions may f ail past certain topology changes\, demonstrating the optimality of our re sult.\n\nIn the last part of the talk\, we discuss further applications of our new concept\, including the quantitative convergence of diffuse-inter face (Allen-Cahn) approximations for multiphase mean curvature flow.\n LOCATION:https://researchseminars.org/talk/OSGA/122/ END:VEVENT BEGIN:VEVENT SUMMARY:Denis Brazke (University of Heidelberg) DTSTART:20221011T160000Z DTEND:20221011T170000Z DTSTAMP:20260422T225637Z UID:OSGA/126 DESCRIPTION:Title: Γ–limit for a sharp interface model related to pattern formation on bio membranes\nby Denis Brazke (University of Heidelberg) as part of Onlin e Seminar "Geometric Analysis"\n\n\nAbstract\nWe derive a macroscopic limi t for a sharp interface version of a model proposed by Komura\, Shimokawa and Andelman to investigate pattern formation in biomembranes due to compe tition of chemical and mechanical forces. We identify sub- and supercrital parameter regimes and show with the introduction of the autocorrelation f unction that the ground state energy leads to the isoperimetric problem in the subcritical regime\, which is interpreted to not form fine scale patt erns\n\nThis is joint work with Hans Knüpfer and Anna Marciniak--Czochra. \n LOCATION:https://researchseminars.org/talk/OSGA/126/ END:VEVENT BEGIN:VEVENT SUMMARY:Ernst Kuwert (University of Freiburg i. Br.) DTSTART:20221018T160000Z DTEND:20221018T170000Z DTSTAMP:20260422T225637Z UID:OSGA/127 DESCRIPTION:Title: C urvature varifolds with orthogonal boundary\nby Ernst Kuwert (Universi ty of Freiburg i. Br.) as part of Online Seminar "Geometric Analysis"\n\n\ nAbstract\nThe talk is concerned with the existence of upper mass bounds f or $m$-dimensional surfaces in terms of curvature integrals. We focus on t he case of surfaces confined to a set $\\Omega$ in ${\\mathbb R}^n$ meetin g $\\partial \\Omega$ orthogonally along their boundary (joint work with M arius Müller\, Freiburg). In a previous paper with Victor Bangert (Freibu rg) there is a related result for $2$-dimensional surfaces in Riemannian m anifolds.\n LOCATION:https://researchseminars.org/talk/OSGA/127/ END:VEVENT END:VCALENDAR