# Online Seminar "Geometric Analysis"

Since the coronavirus pandemic forces us to cancel workshops, conferences and seminars, we need new ways to stay connected. So lets make the best out of it and join forces to build an online seminar able to compete with all our local research seminars. You can find videos of most of the talks either below or on our youtube channel.

## How it (should) work

• We will use zoom to make this seminar work. You can join through a link we will send you through the mailing list you have to subscribe to before.
• Meeting id: 693-964-597. You will get the password through the mailinglist.
• Please be patient, in case the system breaks down

## Mailing list

There is a mailing list which announces the speakers for each week and shares the password to the meeting. To subscribe, send a blank email with the subject "subscribe osga YOUREMAILADRESS YOURNAME" to listserver@sbg.ac.at. You can unsubscribe in a similar way.

## List of Upcoming Talks

12. 10. 2021, 19:00 (CET)
Joan Verdera The regularity of the boundary of a vortex patch and commutators of singular integrals
##### Abstract:
I 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 patches and the problem of regularity of the boundary. I will describe some elements of a short proof I have found recently, which also solves the regularity problem for other transport equations. Commutators of singular integrals play a key role, as it is well-known.
19. 10. 2021, 19:00 (CET)
Andrea Mondino (University of Oxford, UK) Optimal Transport, weak Laplacian bounds and minimal boundaries in non-smooth spaces with Lower Ricci Curvature bounds
26. 10. 2021, 19:00 (CET)
Henrik Schumacher (University of Chemnitz, Germany)
02. 11. 2021, 19:00 (CET)
Asaf Shachar
09. 11. 2021, 19:00 (CET)
Juncheng Wei
16. 11. 2021, 19:00 (CET)
Stephanie Wang Capturing surfaces with differential forms
23. 11. 2021, 19:00 (CET)
Jerome Wettstein Properties of the Half-Harmonic Gradient Flow

## Complete List of Talks

03. 04. 2020, 19:00 (CEST)
Philipp Reiter (University Halle, Germany) A bending‐twist model for elastic rods
07. 04. 2020, 19:00 (CEST)
Remy Rodiac (University Paris-Saclay, France) Inner variations and limiting vorticities for the Ginzburg-Landau equations video
14. 04. 2020, 19:00 (CEST)
Daniel Steenebrügge (RWTH Aachen, Germany) A speed preserving Hilbert gradient flow for generalized integral Menger curvature video
21. 04. 2020, 19:00 (CEST)
Andrew Sageman-Furnas (TU Berlin, Germany) Navigating the space of Chebyshev nets video
21. 04. 2020, 20:00 (CEST)
Siran Li (Rice University, USA) Isometric Immersions of Riemannian Manifolds into Euclidean Spaces, Revisited video
28. 04. 2020, 19:00 (CEST)
Bastian Käfer (RWTH Aachen, Germany) A Möbius invariant energy for sets of arbitrary dimension and codimension video
05. 05. 2020, 19:00 (CEST)
Huy The Nguyen (Queen Mary University London, United Kingdom) High Codimension Mean Curvature Flow and Surgery video
05. 05. 2020, 20:00 (CEST)
Elena Mäder-Baumdicker (TU Darmstadt, Germany) The Morse index of Willmore spheres and its relation to the geometry of minimal surfaces
12. 05. 2020, 19:00 (CEST)
Jesse Ratzkin (University Würzburg) On constant Q-curvature metrics with isolated singularities and a related fourth order conformal invariant video
19. 05. 2020, 19:00 (CEST)
Volker Branding (University Vienna, Austria) Higher order generalizations of harmonic maps video
26. 05. 2020, 19:00 (CEST)
Katharina Brazda (University Vienna, Austria) The Canham-Helfrich model for multiphase biomembranes video
02. 06. 2020, 19:00 (CEST)
Lynn Heller (University Hannover) Area Estimates for High genus Lawson surfaces via DPW video
09. 06. 2020, 19:00 (CEST)
Sven Pistre (RWTH Aachen, Germany) The Radon transform and higher regularity of surfaces minimising a Finsler area video
16. 06. 2020, 19:00 (CEST)
Marc Pegon (University Paris-Diderot, France) Partial regularity for fractional harmonic maps into spheres video
##### Abstract:
Similarly to “classical” harmonic maps, which are critical 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 $s in (0,1)$. In this talk, after a brief reminder on classical harmonic maps, I will present the fractional setting and the partial regularity results we have obtained for maps valued into a sphere. In the case of half harmonic maps ($s= rac{1}{2}$), I will also recall the connection with minimal surfaces with free boundary, which allowed us to improve known regularity results for energy minimizing maps into spheres.
16. 06. 2020, 20:00 (CEST)
Myfanwy Evans (University Potsdam, Germany) Periodic tangling
##### Abstract:
This talk will introduce the use of geometric ideas in the characterisation and analysis of tangled biophysical systems. It will introduce the construction of idealised tangled structures using ideas of both symmetry and homotopy of tangled lines on surfaces. These structures provide an extensive set of tangling motifs for the exploration of the behaviour of tangled microstructures in liquids, and I will show preliminary results working towards this goal, including an example of the geometry-driven swelling of human skin cells.
23. 06. 2020, 19:00 (CEST)
Miles Simon (University Magdeburg, Germany) On the regularity of Ricci flows coming out of metric spaces. video
##### Abstract:
Joint work with Alix Deruelle, Felix Schulze We consider 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 for which the norm of the full curvature tensor at time $t$ is bounded by $c/t$ for some fixed constant $c>1$ for all $t in (0,T)$. From previous works, it is known that if the solution is complete for all times $t>0$, then there is a limit metric space $(M,d_0)$, as time t approaches zero. We show : if there is a open region $V$ on which $(V,d_0)$ is *smooth*, then the solution can be extended smoothly to time zero on $V$.
30. 06. 2020, 19:00 (CEST)
Peter Topping (University Warwick, United Kingdom) Uniqueness of limits in geometric flows video
##### Abstract:
Quite often when considering long-time behaviour of geometric flows, or considering blow-ups of singularities in geometric PDE, we extract limits using soft compactness arguments. For example, a flow might easily be seen to converge to a limit at a *sequence* of times converging to infinity. The more subtle question is then whether the flow converges as time converges to infinity, without having to restrict to a sequence of times. I will outline some of the issues that arise in this subject, focussing on gradient flows for the harmonic map energy, and sketch some recent work with M.Rupflin and J.Kohout.
07. 07. 2020, 19:00 (CEST)
Ruben Jakob (Technion, Israel) Generic full smooth convergence of the elastic energy flow in the 2-sphere video
##### Abstract:
The speaker is going to present his recent investigation of the Moebius invariant Willmore flow'' (MIWF) in the 3-sphere and of some particular version of the elastic energy flow'' (EEF) in the 2-sphere. We will discuss the interaction between these two geometric flows via the Hopf fibration and the resulting possibility to transfer particular insights about the EEF'' to the MIWF'', and vice versa special insights about the MIWF'' back to the EEF''. A big motivation for this parallel investigation is the announced proof (by the speaker) of the ''generic full smooth convergence'' of the EEF'' in the 2-sphere.
14. 07. 2020, 19:00 (CEST)
Christian Bär (University Potsdam, Germany) Counter-intuitive approximations video
##### Abstract:
The Nash-Kuiper embedding theorem is a prototypical example of a counter-intuitive approximation result: any short embedding of a Riemannian manifold into Euclidean space can be approximated by *isometric* ones. As a consequence, any surface can be isometrically $C^1$-embedded into an arbitrarily small ball in $R^3$. For $C^2$-embeddings this is impossible due to curvature restrictions. We will present a general result which will allow for approximations by functions satisfying strongly overdetermined equations on open dense subsets. This will be illustrated by three examples: real functions, embeddings of surfaces, and abstract Riemannian metrics on manifolds.
21. 07. 2020, 19:00 (CEST)
Carla Cederbau (University Tübingen, Germany) On CMC-foliations of asymptotically flat manifolds video
##### Abstract:
In 1996, Huisken and Yau proved existence of foliations by constant mean curvature (CMC) surfaces in the asymptotic end of an asymptotically 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 initial data sets in General Relativity (Metzger). I will present a new foliation by constant spacetime mean curvature surfaces (STCMC), also in the context of asymptotically Euclidean initial data sets in General Relativity (joint work with Sakovich). The STCMC-foliation is well-suited to define a notion of total center of mass in General Relativity.
28. 07. 2020, 19:00 (CEST)
Nadine Große (University Freiburg, Germany) Boundary value problems on singular domains: an approach via bounded geometries video
##### Abstract:
In this talk, we consider boundary value problems on domains with non smooth boundaries. We approach this problem by transferring it to non-compact manifolds with a suffiently nice geometry -- the bounded geometry. This gives a more general framework that allows to handle Dirichlet (or Dirichlet-Neumann mixed) boundary value problems for domains with a larger class of singularities on the boundary and gives a nice geometric interpretation. This is joint work with Bernd Ammann (Regensburg) and Victor Nistor (Metz).
04. 08. 2020, 19:00 (CEST)
Melanie Rupflin (University Oxford, UK) Łojasiewicz inequalities near simple bubble trees for the $H$ surface equation
##### Abstract:
In this talk we discuss a gap phenomenon for critical points ofthe $H$-functional on closed non-spherical surfaces when $H$ is constant, and in this setting furthermore prove that sequences of almost critical points satisfy Łojasiewicz inequalities as they approach the first non-trivial bubble tree. To prove these results we derive sufficient conditions for Łojasiewicz inequalities to hold near a finite-dimensional submanifold of almost-critical points for suitable functionals on a Hilbert space. The presented results are joint work with Andrea Malchiodi and Ben Sharp.
11. 08. 2020, 19:00 (CEST)
Julian Scheuer (University Cardiff, UK) Concavity of solutions to elliptic equations on the sphere video
##### Abstract:
An important question in PDE is when a solution to an elliptic equation is concave. This has been of interest with respect to the spectrum of linear equations as well as in nonlinear problems. An old technique going back to works of Korevaar, Kennington and Kawohl is to study a certain two-point function on a Euclidean domain to prove a so-called concavity maximum principle with the help of a first and second derivative test. To our knowledge, so far this technique has never been transferred to other ambient spaces, as the nonlinearity of a general ambient space introduces geometric terms into the classical calculation, which in general do not carry a sign. In this talk we have a look at this situation on the unit sphere. We prove a concavity maximum principle for a broad class of degenerate elliptic equations via a careful analysis of the spherical Jacobi fields and their derivatives. In turn we obtain concavity of solutions to this class of equations. This is joint work with Mat Langford, University of Tennessee Knoxville.
18. 08. 2020, 19:00 (CEST)
Richard Bamler (University of California Berkeley, USA) Ricci flow in higher dimensions video
##### Abstract:
I will present new results concerning Ricci flows in higher dimensions
25. 08. 2020, 19:00 (CEST)
Max Engelstein (University of Minnesota Winding for Wave Maps video
##### Abstract:
Wave maps are harmonic maps from a Lorentzian domain to a Riemannian target. Like solutions to many energy critical PDE, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming in on these singularities yields a harmonic map (called a soliton or bubble) in the weak limit. One fundamental question is whether this weak limit is unique, that is to say, whether different bubbles may appear as the limit of different sequences of rescalings. We show by example that uniqueness may not hold if the target manifold is not analytic. Our construction is heavily inspired by Peter Topping's analogous example of a winding bubble in harmonic map heat flow. However, the Hamiltonian nature of the wave maps will occasionally necessitate different arguments. This is joint work with Dana Mendelson (U Chicago).
01. 09. 2020, 19:00 (CEST)
Peter McGrath (North Carolina State University) Quantitative Isoperimetric Inequalities on Riemannian Surfaces video
##### Abstract:
In this talk, we introduce a scattering asymmetry which measures the asymmetry of a domain by quantifying its incompatibility with an isometric circle action. We prove a quantitative isoperimetric inequality involving the scattering asymmetry and characterize the domains with vanishing scattering asymmetry by their rotational symmetry. We also give a new proof of the sharp Sobolev inequality for Riemannian surfaces which is independent of the isoperimetric inequality. This is joint work with J. Hoisington.
08. 09. 2020, 19:00 (CEST)
Ryan Alvarado (Amherst College) A characterization of the Sobolev embedding theorem in metric-measure spaces.
##### Abstract:
Historically, 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 underlying domain is intimately linked to the availability of these embeddings. In fact, certain geometrical characterizations of domains which support Sobolev embeddings have been obtained in the Euclidean setting, albeit in the plane. In this talk, we will revisit these embedding theorems in the more general context of metric-measure spaces and discuss some recent work which identifies a measure theoretic condition that is both necessary and sufficient to ensure their veracity. A measure characterization of Sobolev extension domains in the metric setting as well as applications of our methods to spaces supporting $p$-Poincaré inequalities will also be discussed. This talk is based on joint work with Przemysław Górka (Warsaw University of Technology), Piotr Hajłasz (University of Pittsburgh).
22.09.2020, 19:00 (CEST)
Fritz Hiesmayr (University College London, UK) A rigidity theorem for the Allen-Cahn equation in $S^3$
##### Abstract:
We present a recent rigidity theorem for the Allen-Cahn equation in the three-sphere: critical points with Morse index are symmetric 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 solutions: they intersect if they are connected. This in fact holds in all manifolds with positive Ricci curvature. Time permitting we will discuss additional rigidity results in higher-dimensional spheres.
29. 09. 2020, 19:00 (CEST)
Anna Siffert (University Münster, Germany) Constructing explicit p-harmonic functions
##### Abstract:
The study of $p$-harmonic functions on Riemannian manifolds has invoked the interest of mathematicians and physicists for nearly two centuries. Applications within physics can for example be found in continuum mechanics, elasticity theory, as well as two-dimensional hydrodynamics problems involving Stokes flows of incompressible Newtonian fluids. In my talk I will focus on the construction of explicit $p$-harmonic functions on rank-one Lie groups of Iwasawa type. This joint work with Sigmundur Gudmundsson and Marko Sobak.
06. 10. 2020, 19:00 (CEST)
Jonas Hirsch (University Leipzig, Germany) On the regularity of area minimizing currents mod(p) video
##### Abstract:
Joint work with C. De Lellis, A Marches and S. Stuvard In this talk I would like to give a glimpse on the regularity of area minimzing currents mod(p). Motivation: If one considers real soap films one notice that from time to time one can find configurations where different 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. Considering area minimzing currents within 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 $C^{1,lpha}$ regular surface.
After a short introduction 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 dimension of the set. If time permits I will give a short outlook of what we would be the expected result.
13.10.2020, 19:00 (CEST)
Renan Assimos (MPI Leipzig, Germany) On a spherical Bernstein theorem by B. Solomon
##### Abstract:
Joint work with J. Jost: A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 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 neighborhood of an $S^{k−1}$ equator, is totally geodesic in $S^{k+1}$. In this talk, I will present a new proof strategy for Solomon's theorem which allows us to obtain analogous results for higher codimensions. If time permits, we sketch the proof for codimension 2 compact minimal submanifolds of $S^{k+1}$.
20. 10. 2020, 19:00 (CEST)
Daniel Campbell (University of Hradec Kralove, Czeck Republic) Pathological Sobolev homeomorphisms in GFT and NE video
##### Abstract:
Sobolev homeomorphisms are the natural choice for minimization problems in non-linear elasticity. For the regularity of these problems it would be useful to be able to approximate these maps by smooth homeomorphisms in their corresponding Sobolev space (the so-called Ball-Evans problem). We construct a pair of homeomorphisms for which is impossible simultaneously solving the Hajlasz problem. That is we construct a Sobolev homeomorphism equalling identity on the boundary of a cube but with negative Jacobian almost everywhere.
27. 10. 2020, 19:00 (CET)
Simon Brendle (Columbia University, USA) The isoperimetric inequality for minimal surfaces
03. 11. 2020, 19:15 (CET)
Henrik Matthiesen New minimal surfaces from shape optimization video
##### Abstract:
I will discuss the connection between sharp eigenvalue bounds and minimal surfaces in two cases: The first eigenvalue of the Laplacian on a closed surface among unit area metrics, and the first Steklov eigenvalue on a compact surface with non empty boundary among metrics with unit length boundary. In both cases maximizing metrics - if they exist - are induced by certain minimal immersions. More precisely, minimal immersions into round spheres for the closed case and free boundary minimal immersions into Euclidean balls in the bordered case. I 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. This is based on joint work with Anna Siffert in the closed case and Romain Petrides in the bordered case.
10. 11. 2020, 19:00 (CET)
Julia Menzel (University Regensburg, Germany) Boundary Value Problems for Evolutions of Willmore Type
##### Abstract:
The Willmore flow arises as the $L^2$-gradient flow of the Willmore energy which is itself given by the integrated squared mean curvature of the considered surface. After a short introduction and review 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 conditions. We further study the elastic flow of planar networks composed of curves meeting in triple junctions. As a main result we obtain that starting from a suitable initial network the flow exists globally in time if the 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 $2 pi$. This talk is based on my recently submitted PhD thesis and includes joint work with H. Abels, H. Garcke and A. Pluda.
17. 11. 2020, 19:00 (CET)
Behnam Esmayli (University of Pittsburgh) Co-area formula for maps into metric spaces video
##### Abstract:
Co-area formula for maps between Euclidean spaces contains, as its very special cases, both Fubini's theorem and integration in polar coordinates formula. In 2009, L. Reichel proved the coarea formula for maps from Euclidean spaces to general metric spaces. I will discuss a new proof of the latter by the way of an implicit function theorem for such maps. An important tool is an improved version of the coarea inequality (a.k.a Eilenberg inequality) 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 and can thus be viewed as a new (and arguably more geometric) proof in that case as well.
24. 11. 2020, 19:00 (CET)
John Maddocks (EPF Lausanne, Switzerland) Ideal knots: The trefoil, analysis and numerics to experiment video
##### Abstract:
Geometrical knot theory is an area of mathematics that has been growing in activity over the last few decades. It involves the study of specific shapes of knotted curves, rather than their topology, where the specific knot shape is fixed by some criterion, typically minimizing some form of knot energy. In this talk I will introduce some older work of both my collaborators and I, as well as others, on  the specific case of ideal, or tightest, knot shapes. I will start by explaining the analytical difficulties, along with some associated theorems. Then I will describe some numerical results concentrating on the specific case of the ideal trefoil. And finally I will describe some very recent experimental results for the ideal trefoil obtained by the group of Pedro Reis at the EPFL.
01. 12. 2020, 19:00 (CET)
Marius Müller (University Freiburg, Germany) The Willmore Flow of Tori of Revolution video
##### Abstract:
This is a joint work with Anna Dall'Acqua, Adrian Spener and Reiner Schätzle. We study the Willmore flow of tori that have a revolution symmetry - so-called tori of revolution. Luckily, the Willmore flow preserves this symmetry. Because of that we can look at the flow as an evolution of the profile curves - a reduction of the dimension! We will examine the geometry of this curve evolution and understand why it is somewhat natural to look at those curves in hyperbolic geometry. We prove: If the hyperbolic length of the profile curves remains bounded, then the Willmore flow converges. The remaining question: How can the hyperbolic length of those curves be controlled? We use variational methods to $control the hyperbolic length$ by the Willmore energy - but this control is only available below an energy level of $8 pi$. We obtain: If we start the Willmore flow with a torus of revolution of Willmore energy below $8 pi$, then the flow converges. If 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 threshold that was already found by E. Kuwert and R. Schätzle for the Willmore flow of spheres.
08. 12. 2020, 19:00 (CET)
Ursula Ludwig (University Duisburg, Germany) An Extension of a Theorem by Cheeger and Müller to Spaces with Isolated Conical Singularities
##### Abstract:
An important comparison theorem in global analysis is the comparison of analytic and topological torsion for smooth compact manifolds equipped with a unitary flat vector bundle. It has been conjectured by Ray and Singer and has been independently proved by Cheeger and Müller in the 70ies. Bismut and Zhang combined the Witten deformation and local index techniques to generalise the result of Cheeger and Muüller to arbitrary flat vector bundles with arbitrary Hermitian metrics. The aim of this talk is to present an extension of the Cheeger-Müller theorem to spaces with isolated conical singularities by generalising the proof of Bismut and Zhang to the singular setting.
15. 12. 2020, 19:00 (CET)
Herrmann Karcher (University Bonn, Germany) Numerical experiments with closed constant curvature space curves video
12. 01. 2021, 19:00 (CET)
Rupert Frank (TU Munich, Germany) Which magnetic fields support a zero mode? video
##### Abstract:
Motivated by the question 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 present some results supporting them. Those concern, in particular, two novel Sobolev inequalities for spinors and vector fields. The talk is based on joint work with Michael Loss.
19. 01. 2021, 19:00 (CET)
Wilderich Tuschmann (KIT, Germany) Spaces and moduli spaces of Riemannian metrics
##### Abstract:
Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to study is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure 'how many' different metrics and geometries the given manifold actually does exhibit; but one can topologize and view those also in very different manners. In my talk, I will report on some general results and open questions about spaces and moduli 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 metric geometry and analysis to these objects and topics.
26. 01. 2021, 19:00 (CET)
Tobias Weth (University Frankfurt, Germany) Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds
##### Abstract:
The talk is concerned with geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains of a Riemannian manifold. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary value problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case where the first nonzero Neumann eigenvalue is not simple. In the second part of the talk, we focus on product manifolds with euclidean factors, and we classify the subdomains where the associated overdetermined boundary value problem has a solution. If time permits, I will also briefly discuss the first nontrivial Stekloff eigenvalue. This is joint work with Moustapha Fall (AIMS Senegal).
02.02. 2021, 19:00 (CET)
Paul Creutz (University Cologne, Germany) Area minimizing surfaces for singular boundary values video
##### Abstract:
Fix a nonnegative integer g and a finite configuration of disjoint Jordan curves in Euclidean space. Then, by a classical result of Douglas, there is an area minimizer among all surfaces of genus at most g which span the given curve configuration. In the talk I will discuss a generalization of this theorem to singular configurations of possibly non-disjoint or self-intersecting curves. The proof relies on an existence result for minimal surfaces in singular metric spaces and does not seem amenable within classical smooth techniques. This is joint work with M. Fitzi.
09. 02. 2021, 19:00 (CET)
Florian Litzinger (Queen Mary College London, UK) Optimal regularity for Pfaffian systems and the fundamental theorem of surface theory
##### Abstract:
The fundamental theorem of surface theory asserts the existence of a surface immersion with prescribed first and second fundamental forms that satisfy the Gauss–Codazzi–Mainardi equations. Its proof is based on the solution of a Pfaffian system and an application of the Poincaré lemma. Consequently, the regularity of the resulting immersion crucially depends on the regularity of the solution of the corresponding Pfaffian system. This talk shall briefly review both the classical smooth case and the regularity theory and then introduce an extension to the optimal regularity.
16. 02. 2021, 19:00 (CET)
Gerhard Huisken (University Tübingen, MFO Oberwolfach, Germany) Mean curvature flow with surgery
##### Abstract:
The evolution of hypersurfaces in a Riemannian manifold along its mean curvature vector is governed by a quasilinear parabolic system that exhibits smoothing behavior and singularity formation at the same time since the evolution of the geometry is governed by a non-linear reaction diffusion system. The lecture explains how for embedded 2-surfaces of positive mean curvature 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.
23. 02. 2021, 19:00 (CET)
Stephan Wojtowytsch (Princeton University, USA) Optimal transport for non-convex optimization in machine learning video
##### Abstract:
Function approximation is a classical task in both classical numerical analysis and machine learning. Elements of the recently popular class of neural networks depend nonlinearly on a finite set of parameters. This nonlinearity gives the function class immense approximation power, but causes parameter optimization problems to be non-convex. In fact, generically the set of global minimizers is a (curved) manifold of positive dimension. Despite this non-convexity, gradient descent based algorithms empirically find good minimizers in many applications. We discuss this surprising success of simple optimization algorithms from the perspective of Wasserstein gradient flows in the case of shallow neural networks in the infinite parameter limit.
02. 03. 2021, 19:00 (CET)
Ruijun Wu Super Liouville equations on the 2-sphere
##### Abstract:
The 2D super Liouville equations, from the super Liouville field theory, is a conformally invariant system which couples the classical Liouville equation with a Dirac equation. We are interested in the existence of nontrivial solutions. Aside from those known solutions induced from prescribing curvature equations and those from Killing spinors, we introduced an additional (but natural) parameter and obtained new solutions via bifurcation theory. This is a joint work with A. Malchiodi and A. Jevnikar.
09. 03. 2021, 19:00 (CET)
Christian Ketterer (University Toronto, Kanda)
16. 03. 2021, 19:00 (CET)
Verena Bögelein (University Salzburg, Germany) Higher regularity in congested traffic dynamics
##### Abstract:
We consider an elliptic system that is motivated by a congested traffic dynamics problem. It has the form $$\mathrm{div}\bigg((|Du|-1)_+^{p-1}\frac{Du}{|Du|}\bigg)=f,$$ and falls into the context of very degenerate problems. Continuity properties of the gradient have been investigated in the scalar case by Santambrogio & Vespri and Colombo & Figalli. In this talk we establish the optimal regularity of weak solutions in the vectorial case for any $$p>1$$. This is joint work with F. Duzaar, R. Giova and A. Passarelli di Napoli.
23. 03. 2021, 19:00 (CET)
Sonja Hohloch (University Antwerpen, Belgium) On recent advances in semitoric integrable systems video
##### Abstract:
Roughly 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 systems. Semitoric systems have been symplectically classified about a decade ago by Pelayo and Vu Ngoc by means of five invariants. Recently, there has been made considerable progress by various authors concerning the computation of these invariants. In this talk, we will give an introduction to semitoric systems before considering a recent, intuitive family of semitoric systems that allows for explicit observation of bifurcation behaviour such as bifurcations between focus-focus and elliptic-elliptic singularities and other interesting geometric-topological features related to singularities and bifurcations. The latter part is based on a joint work with A. De Meulenaere.
30. 03. 2021, 19:00 (CET)
James Scott (University Pittsburgh,USA) Fractional Korn-Type Inequalities and Applications video
##### Abstract:
We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev-Slobodeckij spaces. The equivalence can be considered a Korn-type characterization of said Sobolev spaces. For vector fields defined on various classes of domains, we obtain a relevant form of the inequality. As an application, we consider variational problems associated to strongly coupled systems of nonlocal equations motivated by a continuum mechanics model known as peridynamics. We use the fractional Korn-type inequalities to characterize vector fields in associated energy spaces and obtain existence and uniqueness of solutions in fractional Sobolev spaces.
06. 04. 2021, 19:00 (CET)
Oded Stein (MIT, USA) The Biharmonic Equation in Geometry Processing
##### Abstract:
The Laplacian has been an extensively used tool of geometry processing and computer graphics for a long time. In this talk we will take a look at a close relative of the Laplacian, the Bilaplacian, as well as its partial differential equation, the biharmonic equation. The Bilaplacian can be used in applications such as smoothing, interpolation, character animation, distance computation, and more. We will examine the biharmonic equation and its use in geometry processing, we will look at ways to discretize it for curved surfaces, and we will discuss different boundary conditions of the biharmonic equation.
13. 04. 2021, 19:00 (CET)
Amy Novick-Cohen Surface diffusion, and surface diffusion coupled with mean curvature motion
##### Abstract:
Surface diffusion as well as mean curvature motion constitute geometric motions relevant to the modelling various phenomena arising in modeling thin poly-crystalline films. We first review some special grooving solutions and traveling wave solutions. Afterwards we focus on certain composite axi-symmetric geometries; here the steady states may be described by piecing together Delaunay surfaces, and related evolutionary questions are pertinent to solid state wetting and dewetting.
20. 04. 2021, 19:00 (CET)
Andrea Malchiodi (SNS Pisa, Italy) On critical points of the Moser-Trudinger functional
##### Abstract:
It is known that in two dimensions Sobolev functions in $$W^{1,2}$$ satisfy critical embedding properties of exponential type. In 1971 Moser obtained a sharp form of the embedding, controlling the integrability of $$F(u) := \int \exp(u^2)$$ in terms of the Sobolev norm of $$u$$. On a closed Riemannian surface, $$F(u)$$ is unbounded above for $$\|u\|_{W^{1,2}} > 4 \pi$$. We are however able to find critical points of $$F$$ constrained to any sphere $$\{ \|u\|_{W^{1,2}} = \beta \}$$, with $$\beta > 0$$ arbitrary. The proof combines min-max theory, a monotonicity argument by Struwe, blow-up analysis and compactness estimates. This is joint work with F. De Marchis, O. Druet, L. Martinazzi and P. D. Thizy.
27. 04. 2021,
Francesco Palmurella (ETH Zürich, Switerland) The parametric approach to the Willmore flow
##### Abstract:
We introduce a parametric framework for the study of Willmore gradient flows which enables to consider a general class of weak, energy-level solutions and opens the possibility to study energy quantization and finite-time singularities. In this first work we restricted to a small-energy regime and proved that, for small-energy weak immersions, the Cauchy problem in this class admits a unique solution.
04. 05. 2021, 19:00 (CET)
Frank Duzaar (University Erlange, Germany) Higher integrability for porous medium type systems
##### Abstract:
In this talk we report on recent developments concerning the higher integrability of the spatial gradient to porous medium type systems of the form $$\partial_ t u- \Delta(|u|^{m-1}u) = \rm{div}\, F.$$
11. 05. 2021,
Hans Knüpfer (University Heidelberg, Germany) Gamma-limit for zigzag walls in thin ferromagnetic films
##### Abstract:
In the continuum theory, the magnetization of a ferromagnetic sample $$\Omega \subset \mathbb R^3$$ is described by a unit vector field $$m \in H^1(\Omega,S^2)$$. The minimization of the underlying micromagnetic energy leads to the formation of extended magnetic domains with uniform magnetization, separated by thin transition layers. One type of such transition layers, observed in thin ferromagnetic films are the so called zigzag walls. We consider the family of energies $$E_\varepsilon[m] \ = \ \frac{\epsilon}{2}\|\nabla m\|_{L^2(\Omega)}^2 + \frac 1{2\varepsilon} \|m \cdot e_2\|_{L^2(\Omega)}^2 + rac{\pi\lambda}{2|\ln \varepsilon|} \|\nabla \cdot (m-M)\|_{\dot H^{-\frac 12}}^2,$$ valid for thin ferromagnetic films. We consider a material in the form a thin strip and enforce a charged domain wall by suitable boundary conditions on $$m$$. Here, $$M$$ is an arbitrary fixed background field to ensure global neutrality of magnetic charges. In the limit $$\varepsilon \to 0$$ and for fixed $$)\lambda > 0$$, corresponding to the macroscopic limit, we show that the energy $$\Gamma$$--converges to a limit energy where jump discontinuities of the magnetization are penalized anisotropically. In particular, in the subcritical regime $$\lambda \leq 1$$ one--dimensional charged domain walls are favorable, in the supercritical regime $$\lambda > 1$$ the limit model allows for zigzaging two--dimensional domain walls.
18. 05. 2021,
Catherine Bandle (Universtiy Basel, Switzerland) Domain variations for boundary value problems.
##### Abstract:
We consider boundary value problems which are Euler-Lagrange equations of certain energy-functionals. Important questions in this context are: How 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? The technique of domain variations studies the changes of functionals under infinitesimal deformations. It is a differential calculus that allows to derive necessary conditions for the geometry of an optimal domain. Its beginnings go back to Hadamard in 1908, who calculated the first variation of Green's functions with Dirichlet boundary conditions. In this talk, the first and second variations of the energy of torsion problem with Robin boundary conditions will be discussed.
25. 05. 2021,
Sun-Yung Alice Chang (Princeton University, USA) On bi-Lipschitz equivalence of a class of non-conformally flat spheres
##### Abstract:
This is a report of some recent joint work with Eden Prywes and Paul Yang. The main result is a bi-Lipschitz equivalence of a class of metrics on 4-shpere under curvature constraints. The proof involves two steps: first a construction of quasiconformal maps between two conformally related metrics in a positive Yamabe class, followed by the step of applying the Ricci flow to establish the bi-Lipschitz equivalence from such a conformal class to the standard conformal class on 4-spheres.
01. 06. 2021,
Shankar Venkataramani On branch points and $$C^{1,1}$$ pseudospherical immersions
##### Abstract:
This is a report of joint work with Toby Shearman. The key result is that one can define a (local) winding number of the Gauss Map for $$C^{1,1}$$ hyperbolic surfaces in $$R^3$$ and this degree is an obstruction for approximation by smooth immersions in $$W^{2,2}_{loc}$$. I will discuss the ideas behind the proof, as well as the motivation for studying this question, which comes from the mechanics of non-Euclidean plates
08. 06. 2021,
Sigurd Angenent Nonuniqueness in mean curvature flow and Ricci flow
##### Abstract:
Reporting 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 dimensions $$5, \dots, 9$$.
15. 06. 2021,
Martin Rumpf (University Bonn, Germany) Riemannian calculus in shape spaces
22. 06. 2021,
Hans-Joachim Hein (University Münster, Germany) Smooth asymptotics for collapsing Calabi-Yau metrics
##### Abstract:
Yau'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 families have revealed many remarkable phenomena over the years. I will review 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 holomorphic fibration of a fixed compact Calabi-Yau manifold. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
29. 06. 2021,
Rob Kusner (University of Massachusetts at Amherst, USA)
06. 07. 2021,
Andreas Bernig (Unversity Frankfurt, Germany) Intrinsic volumes on pseudo-Riemannian manifolds
##### Abstract:
The intrinsic volumes in Euclidean space can be defined via Steiner's tube formula and were characterized by Hadwiger as the unique continuous, translation and rotation invariant valuations. By the Weyl principle, their extension to Riemannian manifolds behaves naturally under isometric embeddings. In a series of papers with Dmitry Faifman and Gil Solanes, we developed a theory of intrinsic volumes in pseudo-Euclidean spaces and on pseudo-Riemannian manifolds. Fundamental results like Hadwiger's theorem, Weyl's principle and Crofton formulas on spheres have their natural analogues in the pseudo-Riemannian setting.
13. 07. 2021,
Hans-Christoph Grunau (University Magdeburg, Germany) Boundary value problems for the Willmore and the Helfrich functional for surfaces of revolution video
20. 07. 2021,
Xavier Cabre (ICREA and UPC (Barcelona, Spain)) Stable solutions to semilinear elliptic equations are smooth up to dimension 9
##### Abstract:
The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with 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.
27. 07. 2021,
Jürgen Jost (MPI Leipzig, Germany) Nonpositive curvature: Geometric and analytic aspects
##### Abstract:
Motivated by questions from data analysis, we develop a new approach to curvature of metric spaces. The approach works also for discrete metric spaces and links curvature to deviations from hyperconvexity.
03. 08. 2021,
Robert Haselhofer (University Toronto) Mean curvature flow through neck-singularities
##### Abstract:
In this talk, I will explain our recent work showing that mean curvature flow through neck-singularities is unique. The key is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms Ilmanen’s mean-convex neighborhood conjecture, and more precisely gives a canonical neighborhood theorem for neck-singularities. Furthermore, assuming the multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed. The two-dimensional case is joint work with Choi and Hershkovits, and the higher-dimensional case is joint with Choi, Hershkovits and White.
10. 08. 2021,
Alex Waldron (University of Wisconsin) Harmonic map flow with low dbar-energy
##### Abstract:
I'll describe some history, recent results, and open problems about harmonic map flow in dimension two. The main result is as follows: let $$\Sigma$$be a compact oriented surface and $$N$$ a compact Kähler manifold with nonnegative holomorphic bisectional curvature (e.g. $$\mathbb{CP}^n$$. For harmonic map flow starting from an almost-holomorphic map $$\Sigma \to N$$ (in the energy sense), the body map'' at each singular time is continuous, and no neck'' appears between the body map and the bubble tree. This is joint work with Chong Song.
24. 08. 2021,
Fabian Rupp (University Ulm, Germany) A dynamic approach to the Canham problem
##### Abstract:
Motivated by the Canham-Helfrich model for lipid bilayers, the minimization of the Willmore energy among surfaces of given topological type subject to the constraint of fixed isoperimetric ratio has been extensively studied throughout the last decade. In this talk, we consider a dynamical approach by introducing a non-local $$L^2$$-gradient flow for the Willmore energy, which preserves the isoperimetric ratio. For topological spheres with initial energy below an explicit threshold, we show global existence and convergence to a Helfrich immersion as $$t\to \infty$$. Our proof relies on a blow-up procedure and a constrained version of the Lojasiewicz--Simon gradient inequality.
31. 08. 2021,
Theodora Bourni (University of Tennessee, USA) Ancient polygonal pancakes
##### Abstract:
Mean curvature flow (MCF) is the gradient flow of the area functional; it moves the surface in the direction of steepest decrease of area. An important 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 curvature regions is of great interest. This is done by studying ancient solutions, solutions that have existed for all times in the past, and which model singularities. In this talk we will discuss their importance and ways of constructing and classifying such solutions. In particular, we will focus on collapsed'' solutions and construct, in all dimensions $$n\ge 2$$, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. Moreover, we will show that collapsed solutions decompose backwards in time'' into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. This is joint work with Mat Langford and Giuseppe Tinaglia.
07. 09. 2021,
Antonio De Rosa (University of Maryland, USA) Regularity of anisotropic minimal surfaces
##### Abstract:
I will present regularity theorems for weak minimal surfaces with respect to anisotropic surface energies, extending the celebrated isotropic counterparts proved by Allard.
14. 09. 2021,
Tobias Lamm (KIT Karlsruhe, Germany) Diffusive stability results for the harmonic map flow and related equations
##### Abstract:
The goal of this talk is to introduce the audience to the theory of diffusive stability in the context of the harmonic map flow. This theory is useful when studying stability results for parabolic equations and we will illustrate its use for geometric equations such as the harmonic map flow. Additionally, we use this theory in order improve various uniqueness results for solutions with rough initial data.
21. 09. 2021,
Felix Schulze (University Warwick, UK) Mean curvature flow with generic initial data
##### Abstract:
Mean curvature flow is the gradient flow of the area functional and constitutes 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 approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a first step in this direction Colding-Minicozzi have shown in fundamental work that spheres and cylinders are the only linearly stable singularity models. As a second step toward Huisken's conjecture we show that mean curvature flow of generic initial closed surfaces in $$\mathbb{R}^3$$ avoids asymptotically conical and non-spherical compact singularities. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact self-similarly shrinking solutions. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.
28. 09. 2021,
Louis Dupaigne (Université Claude Bernard Lyon 1, France) The best constant in Sobolev's inequality
##### Abstract:
Due to its conformal invariance, the sharp Sobolev inequality takes equivalent forms on the three standard model spaces i.e. the Euclidean space, the round sphere and the hyperbolic space. By analogy, we introduce three weighted manifolds named after Caffarelli, Kohn and Nirenberg (CKN) for the following reason: the sharp Caffarelli-Kohn-Nirenberg inequality in the standard Euclidean space can be reformulated as a (sharp) Sobolev inequality written on the CKN Euclidean space. It is equivalent to similar (but new) Sobolev inequalities on the CKN sphere and the CKN hyperbolic space. In addition, the Felli-Schneider condition, that is, the region of parameters for which symmetry breaking occurs in the study of extremals, turns out to have a purely geometric interpretation as an (integrated) curvature-dimension condition. To prove these results, we shall use Bakry's generalization of the notion of scalar curvature, (a weighted version of) Otto's calculus, the reformulation of all the inequalities (and many more) as entropy-entropy production inequalities along appropriate gradient flows in Wasserstein space, and eventually elliptic PDE methods as our best tool for building rigorous and concise proofs.
05. 10. 2021,
Filip Rindler (University of Warwick, UK) Space-time integral currents of bounded variation
##### Abstract:
I will present aspects of a theory of space-time integral currents with bounded variation in time. This is motivated by a recent model for elasto-plastic evolutions that are driven by the flow of dislocations (this model is joint work with T. Hudson). The classical scalar BV-theory can be recovered as the 0-dimensional limit case of this BV space-time theory. However, the emphasis is on evolutions of higher-dimensional objects, most notably 1D loops moving within 3D domains (i.e., the codimension 2 case), which corresponds to dislocation dynamics in a material specimen. Based on this, I will discuss the notion of Lipschitz deformation distance between integral currents, which arises physically as a (simplified) measure of dissipation. In particular, I will explain its relation to the boundaryless Whitney flat metric.
12. 10. 2021,
Joan Verdera The regularity of the boundary of a vortex patch and commutators of singular integrals
##### Abstract:
I 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 patches and the problem of regularity of the boundary. I will describe some elements of a short proof I have found recently, which also solves the regularity problem for other transport equations. Commutators of singular integrals play a key role, as it is well-known.
19. 10. 2021,
Andrea Mondino (University of Oxford, UK) Optimal Transport, weak Laplacian bounds and minimal boundaries in non-smooth spaces with Lower Ricci Curvature bounds
26. 10. 2021,
Henrik Schumacher (University of Chemnitz, Germany)
02. 11. 2021,
Asaf Shachar
09. 11. 2021,
Juncheng Wei
16. 11. 2021,
Stephanie Wang Capturing surfaces with differential forms
23. 11. 2021,
Jerome Wettstein Properties of the Half-Harmonic Gradient Flow

## Organizers

Simon Blatt (University Salzburg)

Philipp Reiter (University Halle)

Armin Schikorra (University Pittsburgh)

Guofang Wang (University Freiburg)