# Speakers

This is the complete list of speakers in alphabetical order containing title and abstract of their talks.

• Bellettini, Costante (Princeton)

Title: Tangent cones to pseudo-holomorphic cycles

Abstract: After some motivational background, we will sketch the proof of the uniqueness of tangent cones to positive-(p,p) integral cycles in an arbitrary almost complex manifold. If time allows it, we will compare positive-(p,p) integral cycles to positive-(p,p) normal cycles: the proof discussed above clearly shows why the uniqueness of tangent cones might fail in the latter case.

• Breiner, Christine (MIT)

Title: A Variational Characterization of the Catenoid

Abstract: We show that the catenoid is the unique surface of least area within a geometrically natural class of minimal surfaces. The proof relies on a result of Osserman and Schiffer that provides a convexity condition on the length of certain level sets of the surface that is sharp on catenoids. An alternate approach that avoids the Weierstrass representation may also be discussed. As an application of the techniques, we give a sharp condition on the lengths of a pair of connected, simple closed curves lying in parallel planes that precludes the existence of a connected minimal surface with boundary the two curves. This is joint work with J. Bernstein.

• David, Guy (Université Paris-Sud)

Title: Should we solve Plateau’s problem again?

Abstract: Plateau’s problem, in its simplest version, consists in considering a simple curve in 3-space, and looking for a surface spanned by this curve and with minimal area. Many different definitions of this have been given, often leading to beautiful solutions of Plateau problems. I will try to mention some of those, and maybe spend more time on a variant with “sliding boundary conditions”. This will be an excuse for describing a first attempt in the direction of boundary regularity (as in J. Taylor’s theorem).

• De Pauw, Thierry (Université Paris Diderot)

Title: Homology of normal chains and cohomology of charges

Abstract: I will review the notion of a charge as it pertains to solving the equation div v = F and insisting that a solution v be continuous. Charges make sense in higher codimensions as linear functionals defined on normal currents and continuous with respect to an appropriate notion of variational convergence. I will then describe the complex cochain of charges in a compact metric space X. This is a complex of Banach spaces, carrying topological and metric information about X. The corresponding cohomology spaces are well-defined iff X carries a linear isoperimetric inequality, in which case the cohomology of charges is naturally equivalent to the dual of the homology of normal chains. Various parts of this work are in collaboration with combinations of co-authors R. Hardt, L. Moonens and W.F. Pfeffer.

• Duzaar, Frank (University Erlangen-Nürnberg)

Title: A sharp quantitative isoperimetric inequality in higher codimension

Abstract: We establish a quantitative isoperimetric inequality in higher codimension. In particular, we prove that for any closed (n − 1)-dimensional manifold Γ in Rn+k the following inequality

D(Γ) ≥ Cd2(Γ)

holds true. Here, D(Γ) stands for the isoperimetric gap of Γ, i.e. the deviation in measure of Γ from being a round sphere and d(Γ) denotes a natural generalization of the Fraenkel asymmetry index of Γ to higher codimensions. This is joint work with Verena Bögelein and Nicola Fusco.

• Eichmair, Michael (ETH Zürich)

Title: Isoperimetric structure of initial data sets

Abstract: In recent joint work with Jan Metzger we have determined the exact large scale isoperimetric structure of complete $n$-dimensional Riemannian manifolds $(M, g)$ that are asymptotically flat and such that

g_{ij} (x) = (1 + \frac{m}{2r^{n-2}})^{\frac{4}{n-2}} \delta_{ij} + O(r^{1-n}) \text{ as } r = |x| \to \infty

in asymptotically flat coordinates. I will provide an overview of the proof and discuss applications of our results in mathematical relativity, in particular in the context of G. Huisken’s isoperimetric mass.

• Hardt, Robert (Rice University)

Title: Some Classes of Chains and Cochains for Metric Spaces

Abstract: Around 1960 (in work of DeGiorgi,Federer-Fleming, and Reifenberg) questions concerning the existence, in R^n (or in a smooth Riemannian manifold) of mass minimizing geometric objects of dimensions greater than 2 led to consideration of various weak limits of submanifolds and stimulated the growth of geometric measure theory. The Federer-Fleming normal and integral currents corresponded to topological chains with real or integer coefficients. Ziemer (1962) and Fleming (1966) also introduced finite mass chains with coefficients in a finite group which gave a useful notion of boundary for minimal surfaces which are nonorientable or have triple junctures. These results were extended significantly by B. White in 1999 to general normed abelian groups and in 2000 by L. Ambrosio and B. Kirchheim to currents in a metric space. Our work with T. DePauw in 2008 used and extended both of these and led to consideration of various classes of flat chains whose homologies measured different geometric aspects of spaces. For example, a space with vanishing 0 dimensional normal chains homology is Lipschitz path-connected. These are considered in recent work with De Pauw and W. Pfeffer which also treat real cochains called “charges” that are topologized variationally and that give a geometric cohomology.

• Krummel, Brian (Cambridge University)

Title: Structure of the branch set of harmonic functions and minimal submanifolds

Abstract: I will discuss some recent results (from ongoing work) on the structure of the branch set of multiple valued solutions to the Laplace’s equation and the minimal surface system. In particular, I will describe a method for establishing asymptotics near branch points, which is based on a modification of the frequency function monotonicity formula due to F. J. Almgren and an adaptation to higher-multiplicity of a “blow up” method due to L. Simon that was originally applied to “multiplicity one” classes of minimal submanifolds satisfying an integrability hypothesis. This is joint work with Neshan Wickramasekera.

• Malý, Jan (Charles University Prague)

Title: Nonabsolutely convergent integrals in metric spaces

Abstract: We introduce integrals of functions with respect to distribution-like functionals on metric spaces. The integral is nonabsolutely convergent, similarly to the Denjoy-Perron (or Henstock-Kurzweil) integral. The process of integration is new even for integration with respect to the Lebesgue measure on the real line.

With the aid of this integral, we can study structures resembling currents or varifolds and establish far reaching generalizations of the Stokes theorem.

This is a joint work with Kristýna Kuncová.

• Marques, Fernando (Instituto de Matemática Pura e Aplicada)

Title: Min-max theory and the Willmore conjecture

Abstract: In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of any torus immersed in Euclidean three-space is at least 2\pi^2. In this talk I will describe a solution to the Willmore conjecture which uses the Almgren-Pitts min-max theory of minimal surfaces. This is joint work with Andre Neves (Imperial College, UK).

• Rivière, Tristan (ETH Zürich)

Title: Yang-Mills Fields in super-critical dimensions, a variational approach

Abstract: The framework of Sobolev connections on smooth bundles is perfectly adapted to the study of variations of Yang-Mills lagrangian in critical dimensions. It has been successfully developed about 3 decades ago in particular for studying spaces of instantons or solutions to various vortex equations in relation with geometric invariants.

There are also strong geometric motivations for studying Yang-Mills Fields in super-critical dimension (Moduli spaces of Hermitte-Einstein connections, SU(4)-instantons on Calabi-Yau manifolds…etc). We will explain how the setting of Sobolev connections on smooth bundles is incomplete in supercritical dimension for applying fundamental principles of the calculus of Variations to Yang-Mills Lagrangian.

We will then present a ”measure theoretic notion” of bundles on which some space of weak connections, that we will define, is weakly closed for the L^p norms and provides a satisfying framework for studying the variations of Yang-Mills in super-critical dimension.

We will give in particular a resolution of the Yang-Mills-Plateau Problem in super critical dimension for abelian bundles.

Finally we will draw the lines of what are the future developments that we forsee for this geometric measure theory notion of bundles.

This project is part of an ongoing collaboration with Mircea Petrache.

• Schulze, Felix (FU Berlin)

Title: The half-space property and entire positive minimal graphs in M x R

Abstract: We show that a properly immersed minimal hypersurface in M × R^+ equals some M × {c} when M is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. If on the other hand, M has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over M. This is joint work H. Rosenberg and J. Spruck.

• Simon, Leon (Stanford University)

Title: Some open questions on the singular set of minimal hypersurfaces

Abstract: We examine some of the basic open questions about the singular set of minimal hypersurfaces in the special context of singular solutions of the symmetric minimal surface equation.

Title: Regularity of area-minimizing integral currents

Abstract: In this talk I present the main points of a new revisited proof of Almgren’s partial regularity result for area-minimizing integral currents up to codimension two. This is joint work with C. De Lellis (University of Zurich).

• Tonegawa, Yoshihiro (Hokkaido University)

Title: A local regularity theorem for weak mean curvature flow

Abstract: Brakke defined his version of MCF for varifold in his seminal book published in 1978 where he studied its existence and regularity issues. I discuss a new proof of local regularity theorem which, among other things, proves almost everywhere smoothness for Brakke’s MCF. The result contains a natural parabolic generalization of Allard regularity theorem.

• White, Brian (Stanford University)

Title: Curvature Blowup in Sequences of Embedded Minimal Surfaces

Abstract: Any sequence of smooth embedded minimal surfaces in a 3-manifold converges subsequentially to a minimal lamination away from the set where the curvature blows up. We discuss what kinds of laminations and blowup sets can occur.