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Seminars winter 2015/16

There are several research seminars in the Potsdam-Berlin area which are closely related to Geometric Analysis:

• Usually biweekly during term, on Thursdays, the joint seminar "Topics in Geometric Analysis" taking place in turns at the University of Potsdam (UP), the Albert Einstein Institute (AEI) and the Free University Berlin (FU). It is organised by Theodora Bourni (FU), Ulrich Menne (AEI), and Jan Metzger (UP). The dates are Oct 29 (FU), Nov 12 (UP), Nov 26 (AEI), Dec 10 (FU), Jan 14 (UP), and Jan 28 (AEI).
• In the weeks where there is no "Topics in Geometric Analysis" seminar there will usually be a reading seminar on "Geometric Measure Theory" at the Albert Einstein Institute (AEI). It will take place on Oct 15, Oct 22, Nov 5, Nov 19, Dec 3, Dec 17, Jan 7, Jan 21, and Feb 4. Details see below.
Timetable of seminars
DatesTypePlaceSpeakerTopic
Oct 15th, 2015, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Oct 15th, 2015, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Oct 22th, 2015, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Oct 22th, 2015, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ananda Lahiri (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Oct 29th, 2015, 16:15-17:15 Topics in Geometric Analysis FU Berlin, Room 140, Arnimallee 7 (Hinterhaus), 14195 Berlin Yoshihiro Tonegawa (Tokyo Institute of Technology)

Existence of Brakke's mean curvature flow starting from general codimension one sets

Suppose we are given a closed set $M$ in the $n$-dimensional Euclidean space, which is of locally finite $n-1$-dimensional Hausdorff measure and countably $n-1$-rectifiable. Assume furthermore that $M$ is realized as a finite union of boundaries of mutually disjoint open sets. Typical $M$ would be, on a plane, arbitrary network of curves with finite number of junctions, and on 3-D, bubble clusters with complicated singularities. Taking such general $M$ as the initial surface, we prove that there exists a non-trivial Brakke's mean curvature flow which exists for all time. This may be seen as a proper weak solution describing motion of grain boundaries driven by excess surface energy. I will explain the results and outline of proof. This is a joint work with Lami Kim.

Oct 29th, 2015, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Room 140, Arnimallee 7 (Hinterhaus), 14195 Berlin Joseph Lauer (FU Berlin)

Length and gradient estimates in equations of curve shortening flow type

In this talk I'll present a length estimate for planar curve shortening flow and apply it to establish results about the evolution of nonsmooth objects. The estimate itself depends on a crude geometric quantity called the r-multiplicity, which is a type of coarse intersection profile. We'll also discuss extending these results to curve shortening flow in an arbitrary surface, the key step in which is an extension of a certain gradient estimate.

Nov 5th, 2015, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ananda Lahiri (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Nov 5th, 2015, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ananda Lahiri (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Nov 12th, 2015, 16:15-17:15 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Alessandro Carlotto (ETH Zurich)

The finiteness problem for minimal surfaces of bounded index in a 3-manifold

Given a closed, Riemannian 3-manifold (N,g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical "generic finiteness" problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. It is this question that we settle: indeed, we prove that when g is a bumpy metric of positive scalar curvature either finiteness holds or N does contain a copy of RP^3 in its prime decomposition, which is a sharp conclusion as we can exhibit specific obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.

Nov 12th, 2015, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Luca Spolaor (Max Planck Institute for Mathematics in the Sciences)

The regularity of 2-dimensional area-minimizing integral currents

Building upon the Almgren's big regularity paper, Chang proved in the eighties that the singularities of area-minimizing integral 2-dimensional currents are isolated. His proof relies on a suitable improvement of Almgren's center manifold and its construction is only sketched. In recent joint works with Camillo De Lellis and Emanuele Spadaro we give a complete proof of the existence of the center manifold needed by Chang and extend his theorem to two classes of currents which are "almost area minimizing", namely spherical cross sections of area-minimizing 3-dimensional cones and semicalibrated currents.

Nov 19th, 2015, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Nov 19th, 2015, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Nov 26th, 2015, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Nicola Fusco (University of Naples Federico II)

Stability and minimality for a nonlocal variational problem

I will discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We will see that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are $L^1$-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy will be also discussed. As a byproduct of the quantitative estimate, I will show new results concerning periodic local minimizers of the area functional. A further application will concern the global and local minimality of certain lamellar configurations.

Nov 26th, 2015, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Paul Bryan (University of Hannover)

Harnack inequalities, Aleksandrov reflection, and ancient solution of curvature flows on the sphere

Since the seminal work of Li and Yau, Harnack inequalities have played an important role in the study of geometric PDE. Many authors have studied such inequalities and used them to obtain convergence results for curvature flows such as the Ricci flow, Mean Curvature Flow and various non-linear hypersurface flows. I will focus on the hypersurface flow situation, which has been studied principally in Euclidean space. There, Harnack inequalities closely related to solitons (self-similar solutions) and ancient solutions. In the sphere, the notion of 'self-similar' is not so clear, yet a Harnack inequality holds and may be used in classifying ancient solutions of curvature flows in the sphere. I will describe the Harnack inequality (and maybe make a remark or two about conformal-solitons) and then discuss a parabolic Aleksandrov reflection argument that may be used to deduce maximal symmetry for convex, ancient solutions of curvature flows in the sphere.

Dec 3rd, 2015, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Sławomir Kolasiński (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Dec 3rd, 2015, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Sławomir Kolasiński (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Dec 10th, 2015, 16:15-17:15 Topics in Geometric Analysis FU Berlin, Room 140, Arnimallee 7 (Hinterhaus), 14195 Berlin Paul Laurain (Paris)

Quantization phenomena for conformally invariant problems

This talk is devoted to a series of papers we have published with T. Rivière. First, we have been interested in giving a unified proof of some "classical" quantization phenomena for problems such as harmonic maps, J-holomorphic curves or prescribed mean curvature. Then, since this proof relies only on the common dominator of theses problems, namely conformal invariance, we have been able to applied this theory to solve open questions in conformal geometry, such as Bi-harmonic maps, Willmore surfaces (with Y Bernard), free-boundary Harmonic maps (with R Petrides).

Dec 10th, 2015, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Room 140, Arnimallee 7 (Hinterhaus), 14195 Berlin James McCoy (Wollongong)

Curvature contraction of convex surfaces by nonsmooth speeds

We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds and nonsmooth convex speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities.

Dec 17th, 2015, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Sławomir Kolasiński (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Dec 17th, 2015, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Sławomir Kolasiński (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Jan 7th, 2016, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Yangqin Fang (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Jan 7th, 2016, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Yangqin Fang (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Jan 14th, 2016, 16:15-17:15 Topics in Geometric Analysis

University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam

Xavier Tolsa (ICREA - Autonomous University of Barcelona)

Quantitative estimates for the Riesz transform and rectifiabilty for general Radon measures

A remarkable theorem of Léger asserts that if $\mu$ is a Radon measure in the Euclidean space and $B$ is a ball such that $\mu(B)=r(B)$ (where $r(B)$ stands for the radius of $B$) satisfying the linear growth condition $\mu(B(x,r))\leq C_0 r$ for all $x,r$, and so that the curvature of $\mu$

$$c^2(\mu)=\iiint \frac1{R(x,y,z)^2}\,d\mu(x)d\mu(y)d\mu(z)$$

is small enough, then a big piece of $\mu$ on $B$ is supported on a Lipschitz graph and is absolutely continuous with respect to arc length measure on the graph.

In my talk I will present a version of this theorem which involves the $L^2$ norm of the codimension $1$ Riesz transform in the Euclidean space, and I will show an application (by Azzam, Mourgoglou and myself) of this result to an old problem on harmonic measure posed by C. Bishop.

Jan 14th, 2016, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Campus Golm, Building 9, Room 2.22, Karl-Liebknecht-Straße 24-25, 14476 Potsdam Panu Kalevi Lahti (University of Oxford)

A notion of quasicontinuity for functions of bounded variation on metric spaces

Sobolev functions are known to be quasicontinuous, meaning that the restriction of a Sobolev function outside a set of small capacity is continuous. The same cannot hold for functions of bounded variation, or BV functions, since they can have jump sets with large 1-capacity. On a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show a weaker notion of quasicontinuity for BV functions. More precisely, we show that the restriction of a BV function outside a set of small 1-capacity is continuous outside the function's jump set and "one-sidedly" continuous in its jump set.

Jan 21th, 2016, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Yangqin Fang (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Jan 21th, 2016, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Yangqin Fang (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Jan 28th, 2016, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ananda Lahiri (AEI)

Local regularity for weak mean curvature flow

In this talk I want to present a new version of Brakke's local regularity theorem. We consider a weak mean curvature flow given for times in [0,T] in some ball. Suppose the flow lies in a narrow slab, has density ratios less than two planes and does not vanish at time T. Then Brakke's local regularity theorem says that in some small neighbourhood the flow is smooth and graphical for times in (C,T-C) for some constant C. Here we will discuss that this actually holds for times in (C,T). One key observation is that a non-vanishing weak mean curvature flow that is initially locally graphical with small Lipschitz constant will stay graphical for some time.

Jan 28th, 2016, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Daniele Valtorta (Swiss Federal Institute of Technology in Lausanne)

Structure of the singular sets of harmonic maps

In this talk we present the new regularity results proved for the singular sets of minimizing and stationary harmonic maps in collaboration with Aaron Naber (see arXiv:1504.02043). We prove that the singular set of a minimizing harmonic map is rectifiable with effective n-2 volume estimates. The results are based on an improved quantitative stratification technique, which consists in a detailed analysis of the symmetries and almost symmetries of the map u and its blow-ups at different scales, and rely on a new W^{1,p} version of Reifenberg's topological disk theorem. The application of this theorem in the situation of harmonic maps hinges on the monotonicity formula for the normalized energy. Similar results are available for minimizing and stationary currents (see arXiv:1505.03428).

Feb 4th, 2016, 13:15-14:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Feb 4th, 2016, 15:15-16:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Feb 10th, 2016, 09:15-10:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ananda Lahiri (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Feb 10th, 2016, 11:15-12:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.
Feb 10th, 2016, 14:15-15:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mario Santilli (AEI) William K. Allard, "On the first variation of a varifolds", 1972.