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# Seminars summer 2014

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), Klaus Ecker (FU), Ulrich Menne (AEI), and Jan Metzger (UP). The dates are April 24 (UP), May 15 (AEI), June 5 (FU) and 19 (UP) and July 3 (AEI) and 17 (FU).
• In the weeks where there is no "Topics in Geometric Analysis" seminar there will be a reading seminar on "Geometric Measure Theory" at the Albert Einstein Institute (AEI) or the Freie Universität Berlin (FU). It will take place on April 17 (AEI), May 8 (FU), May 22 (AEI), June 12 (FU), June 26 (AEI), and July 10 (AEI). Details see below.
Timetable of seminars
DatesTypePlaceSpeakerTopic
April 17th, 2014, 14:15-15:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Alexander Volkmann (AEI) Michael Grüter and Jürgen Jost, "
April 17th, 2014, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Alexander Volkmann (AEI) Michael Grüter and Jürgen Jost, "
April 24th, 2014, 16:15-17:15 Topics in Geometric Analysis University of Potsdam, Raum 1.14, Haus 9, Am Neues Palais 10, 14469 Potsdam Tobias Lamm (KIT Karlsruhe)
Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

In this talk we describe recent joint work with R. Schätzle in which we extend a rigidity result of De Lellis and Müller to arbitrary codimensions. More precisely, we show that every immersion of a two-dimensional surface into $R^n$, whose tracefree second fundamental form is small in $L^2$ has to be close to a round sphere in the $W^{2,2}$-norm.

April 24th, 2014, 17:45-18:45 Topics in Geometric Analysis University of Potsdam, Raum 1.14, Haus 9, Am Neues Palais 10, 14469 Potsdam Christopher Nerz (Universität Tübingen)
Foliations of asymptotically flat manifolds and their time evolution

For the study of asymptotically flat manifolds in mathematical general relativity, surfaces of constant mean curvature (CMC) haven proved to be a useful tool. In 1996, Huisken-Yau showed that any asymptotically flat Riemannian manifold can be uniquely foliated by closed CMC surfaces. Furthermore, they interpreted this foliation as a definition of the center of mass. We prove that this definition is compatible with the definition of linear momentum by Arnowitt-Deser-Misner: The evolution of this foliation (asymptotically) corresponds to a translation with direction given by the quotient of (ADM) linear momentum and mass - equivalent to the center of mass in Newtonian systems.

May 8th, 2014, 14:15-15:45 GMT reading seminar FU Berlin, Raum 9, Arnimallee 6, 14195 Berlin Alexander Volkmann (AEI) Michael Grüter and Jürgen Jost, "
May 8th, 2014, 16:15-17:45 GMT reading seminar FU Berlin, Raum 9, Arnimallee 6, 14195 Berlin Alexander Volkmann (AEI) Michael Grüter and Jürgen Jost, "
May 15th, 2014, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mattias Dahl (Kungliga Tekniska Högskolan)
An initial data version of topological censorship in higher dimensions

The principle of topological censorship states that the region outside all black holes in a spacetime should be topologically simple. Classical results show for example that this region is simply connected in 3+1 dimensions.

Recently, an initial data version of topological censorship has been found by Eichmair, Galloway, and Pollack. This states that for a 3-dimensional asymptotically flat initial data set for general relativity the domain outside the outermost marginally trapped surface is diffeomorphic to Euclidean space minus a number of balls.

In this talk I will describe another approach to the initial data version of topological censorship which gives constraints on the topology also in higher dimensions. This is joint work with L. Andersson, G. Galloway, and D. Pollack.

May 15th, 2014, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Kirk Lancaster (Wichita State University) Boundary Behavior of PMC Surfaces: The Concus-Finn Conjecture

Capillary surfaces are interesting geometric objects which turn out to be important in microgravity environments (e.g. in space) and in tiny devices (e.g. electronic, "lab on a chip"). This talk will focus on the mathematical theory of capillary surfaces in vertical cylinders and sketch the proof of the "Concus-Finn conjecture." Related open questions will be mentioned.

May 22th, 2014, 14:15-15:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Alexander Volkmann (AEI) Michael Grüter and Jürgen Jost, "
May 22th, 2014, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI) Michael Grüter and Jürgen Jost, "
June 5th, 2014, 16:15-17:15 Topics in Geometric Analysis FU Berlin, Raum 31, Arnimallee 6, 14195 Berlin Knut Smoczyk (Universität Hannover)
On the topology of translating solitons of the mean curvature flow

This is joint work with Francisco Martin (Granada) and Andreas Savas-Halilaj (Hannover). We obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.

June 5th, 2014, 17:45-18:45 Topics in Geometric Analysis FU Berlin, Raum 31, Arnimallee 6, 14195 Berlin Andrea Malchiodi (SISSA)
A variational approach to Liouville Equations

We consider Liouville equations arising from curvature prescription problems and from models in Electroweak and Chern-Simons theory. We show how improved versions of the Moser-Trudinger inequality, combined with min-max theory, may reduce these PDEs to the study of finite-dimensional objects consisting of measures supported at finitely-many points. These are joint works with D. Bartolucci, A. Carlotto, F. De Marchis and D. Ruiz.

June 12th, 2014, 14:15-15:45 GMT reading seminar FU Berlin, Raum 137, Takustraße 9, 14195 Berlin Ulrich Menne (AEI) Michael Grüter and Jürgen Jost, "
June 12th, 2014, 16:15-17:45 GMT reading seminar FU Berlin, Raum 137, Takustraße 9, 14195 Berlin Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
June 19th, 2014, 16:15-17:15
Topics in Geometric Analysis University of Potsdam, Raum 1.14, Haus 9, Am Neues Palais 10, 14469 Potsdam Frank Duzaar (Universität Erlangen-Nürnberg) A variational approach to the total variation flow

Abstract (PDF file)

June 19th, 2014, 17:45-18:45
Topics in Geometric Analysis University of Potsdam, Raum 1.14, Haus 9, Am Neues Palais 10, 14469 Potsdam Stefano Pigola (Università dell'Insubria) Some geometric aspects of parabolicity, stochastic completeness and Feller property

Using the viewpoint of partial differential equations, we take a rapid tour into the main stochastic properties of a Riemannian manifold and show how they can be used to obtain some information on the geometry of submanifolds with prescribed mean curvature.

June 26th, 2014, 14:15-15:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
June 26th, 2014, 16:15-17:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Theodora Bourni (FU Berlin) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
July 3rd, 2014, 16:15-17:15 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Mariana Smit Vega Garcia (Universität Düsseldorf)
New developments in the lower dimensional obstacle problem

We will describe the lower-dimensional obstacle problem for a uniformly elliptic, divergence form operator with Lipschitz continuous coefficients. We will give an overview of what is known about this problem, new developments and the role of a new monotonicity formula for an appropriate generalization of Almgren's frequency functional in the optimal regularity of the solution.

July 3rd, 2014, 17:45-18:45 Topics in Geometric Analysis Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Guy David (Université Paris-Sud)
Sets that minimize the $d$-dimensional Hausdorff measure under a sliding boundary condition

The main point of the lecture should concern regularity properties of minimal or almost minimal sets that satisfy a sliding boundary condition of Plateau type, and in particular near the boundary. We should in particular discuss some monotonicity property of density, and simple applications in low dimensions.

July 10th, 2014, 14:15-15:45 GMT reading seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Ulrich Menne (AEI) Theodora Bourni: "Allard type boundary regularity theorem for $C^{1,\alpha}$ boundary", 2010.
July 10th, 2014, 16:15-17:45 GMT seminar Albert Einstein Institute, Room 0.01, Am Mühlenberg 1, 14476 Potsdam Harrison Pugh (Stony Brook University)
Plateau's Problem and Cohomological Spanning

Plateau's problem is to find a surface with minimal area spanning a given boundary. I will discuss a new theory in which the usual homological definition of span is replaced with a cohomological one. If M is a connected, oriented compact manifold of dimension n-2 in R^n, we say a compact set X "spans" M if X intersects every Jordan curve whose linking number with M is one. This definition generalizes to more general boundaries, and to higher codimension. Let S be the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of "size," we prove:

There exists an X_0 in S with smallest size. Any such X_0 contains a "core" Y_0 with the following properties: It is a subset of the convex hull of M and is a.e. a real analytic (n-1)-dimensional minimal submanifold. Furthermore, Y_0 supports a de Rham current S_0 whose boundary has support M. If n=3, then Y_0 has the local structure of a soap film.

A key new idea in this approach is that of a "film chain." There are interesting parallels between physical properties of actual soap films and mathematical properties of their film chain models. This work is joint with J. Harrison.

July 17th, 2014, 16:15-17:15
Topics in Geometric Analysis FU Berlin, Raum 31, Arnimallee 6, 14195 Berlin Claus Gerhardt (Universität Heidelberg)
A unified quantum theory: gravity interacting with Yang-Mills and spinor fields

We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Using canonical quantization we obtain solutions of the Wheeler-DeWitt equation in a vector bundle and the method of second quantization leads to a symplectic vector space $(V,\omega)$ and a corresponding CCR representation for the bosonic components and a CAR relation for the fermionic part. The solution space of the Wheeler-DeWitt equation is invariant under gauge transformations and under isometries in the spacelike base space $\mathcal S_0$ of a given Riemannian metric $\rho_{i,j}$. We also define a net of local subalgebras which satisfy four of the Haag-Kastler axioms.

July 17th, 2014, 17:45-18:45
Topics in Geometric Analysis FU Berlin, Raum 31, Arnimallee 6, 14195 Berlin
Apostolos Damialis (Athens & Berlin) Plateau's laws for diffused interfaces

We present some old and new results on the problem of deriving Plateau's laws at junctions of diffused interfaces via the vector-valued Allen--Cahn equation. We begin with the simplest case of a triple junction on the plane and present in detail a rigorous derivation in the case of triple and quadruple junctions in three-dimensional space. As a conclusion, we discuss some aspects of the related problem of deriving Plateau's laws from static balance of forces relations.