Colloquium Lecture by Simon Donaldson: Kähler-Einstein metrics, extremal metrics and stabilityAbstract: In the first part of the talk we will give a general outline of the two topics in Kä hler geometry in the title, both growing out of work of Calabi. We will also discuss the parallels with affine differential geometry which arise when one studies toric manifolds. We will explain the standard conjectures in the field, relating the existence of these metrics to algebro-geometric notions of “stability”. In the last part of the talk we will say something about recent work with Chen and Sun which establishes this conjecture in the case of Kä hler-Einstein metrics on Fano manifolds (Yau’s conjecture). |

Lecture by Brian White: Gap theorems for minimal submanifolds of spheresAbstract: The totally geodesic k-sphere is the minimal hypersurface in the (k+1)-sphere of smallest k-dimensional area. What is the next smallest area? This is closely related to the question: what is the smallest density that a minimal variety can have at a singular point? I will discuss these questions and some sharp partial results. |

Lecture by Bo Berndtsson: Variations of Bergman kernels and symmetrization of plurisubharmonic functionsAbstract: I will discuss some applications of a theorem on variation of Bergman kernels. I will concentrate on a problem concerning symmetrization of plurisubharmonic functions and generalizations of the Polya-Szegö theorem. These problems turn out to have some relations to Kähler geometry and the openness conjecture for plurisubharmonic functions. (This is mostly joint work with Robert Berman.) |

Lecture by Hans-Joachim Hein: Singularities of Kähler-Einstein metrics and complete Calabi-Yau manifoldsAbstract: When Einstein metrics form singularities one expects to see a bubble tree structure in which the bubbles are complete Ricci-flat spaces. We still lack a detailed understanding of this process in general. I will discuss several examples - old, new, and speculative - of singularity formation in the Kähler-Einstein case. Topics will include gravitational instantons in real dimension 4, and isolated Einstein singularities whose tangent cones have nonisolated singularities. Partly joint with Ronan Conlon and Aaron Naber. |

Lecture by Andrea Malchiodi: Uniformization of surfaces with conical singularitiesAbstract: We consider a class of singular Liouville equations which arise from the problem of prescribing the Gaussian curvature of a surface imposing a given conical structure at a finite number of points (as well as from models in Chern-Simons theory). The problem is variational, and differently from the "regular" case the Euler-Lagrange functional might be unbounded from below. We will look for critical points of saddle type using a combination of improved geometric inequalities and topological methods. This is joint work with D. Bartolucci, A. Carlotto, F. De Marchis and D. Ruiz. |

Special Lecture by Eugenio Calabi |

Lecture by Yuval Peres: The geometry of fair allocation to random pointsAbstract: Given a random scatter of points (obtained as a limit of uniform picks from a large cube, or as the zeros of a random analytic function) , our goal is to allocate to each point of the process a unit of volume, in a deterministic translation-invariant way, so that the diameter of the region allocated to each point is stochastically as small as possible. One approach to this problem, studied in joint work with C. Hoffman and A. Holroyd, uses the stable marriage algorithm of Gale and Shapley. In dimensions 3 and higher, gravity without inertia yields a satisfying solution. The fairness of the allocation is a consequence of the divergence theorem; The diameters of the allocated regions are analyzed using methods from percolation theory. Finally, I will relate the properties of the allocation to rigidity properties of the underlying point process.References:Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval A stable marriage of Poisson and Lebesgue.Ann. Probab. 34 (2006), no. 4, 1241–1272. Nazarov, Fedor; Sodin, Mikhail; Volberg, Alexander Transportation to random zeroes by the gradient flow. Geom. Funct. Anal. 17 (2007), no. 3, 887–935. Chatterjee, Sourav; Peled, Ron; Peres, Yuval; Romik, Dan Gravitational allocation to Poisson points. Ann. of Math. (2) 172 (2010), no. 1, 617–671. Subhro Ghosh and Yuval Peres, Rigidity and Tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues. Preprint (2013), arXiv:1211.2381 |

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Lecture by Aaron Naber: Characterizations of bounded Ricci curvature and applicationsAbstract: The purpose of this talk is two-fold. First we give new ways of characterizing bounded Ricci curvature on a smooth metric measure space (M^{n},g,e^{-f}dv_{g}). In essence, we show that bounded Ricci curvature controls the infinite dimensional analysis on the path space P(M) of a manifold in a manner analogous to how lower Ricci curvature controls the analysis on M. In particular, we show that the Ornstein-Uhlenbeck operator L on the based path space P_{x,T}(M)= {γ:[0,T] → M; γ(0)=x}, which is a form of infinite dimensional Laplacian, has a spectral gap of (e^{kT}+1)/2 if and only if the Bakry-Emery-Ricci curvature Rc+∇^{2}f is bounded by k. Similarly the Ornstein-Uhlenbeck operator has a log-Sobolev constant of e^{kT}+1 iff the Ricci curvature is bounded by k. We have many other characterizations as well, including notions which relate the Wasserstein geometry of the space of probability measures on M to the metric-measure geometry of path space, and including notions analagous to dimensional Ricci curvature bounds which control |Rc+∇^{2}f-(1/(N-n))∇ f⊗ ∇ f|. In the second part of the talk we build the necessary tools in order to use these new characterizations to define bounded Ricci curvature on an essentially arbitrary metric measure space (X,d,v). The primary technical difficulty is to describe the right notion of parallel translation invariant vector fields along continuous curves in such a setting. Even on a smooth manifold this requires deep ideas from stochastic analysis, but we provide a new approach even in this context which generalizes to arbitrary metric spaces. We spend some time discussing the structure of metric measure spaces with generalized bounded Ricci curvature. In particular we show its possible to define the Ornstein-Uhlenbeck operator on their path spaces, and that these operators also have the desired analytic control proven on smooth spaces. We also show spaces with generalized Ricci curvature bounded by k in this new sense have lower Ricci curvature bounded from below by -k in the sense of Lott-Villani-Sturm. |

Lecture by Peter Kronheimer: Instanton homology for knots and websAbstract: Andreas Floer introduced instanton homology for 3-manifolds in the mid 1980's. He also described a variant he called knot homology, for knots in 3-manifolds. There has now been a lot of progress in understanding the structure of instanton knot homology and its relationship with other, more recent invariants of knots, such as Khovanov homology. Applications have included a proof that Khovanov homology detects the unknot. Replacing the Lie group SU(2) in Floer's construction with SU(N) for larger N leads to an invariant of "webs" (labeled trivalent graphs), with a connection to Khovanov-Rozansky homology. This talk will review some of these developments. |