Time: Monday 4 pm

Location: Shaffer 303

Organizer(s): Jacob Bernstein and Yi Wang

Date |
Speaker |
Title |
Abstract |

(IAS) |
0,01% Improvement of the Liouville property for discrete harmonic functions on Z^2 | We prove that if u is a discrete harmonic function on a lattice Z^2, and |u| < 1 on 99,99% of Z^2, then u is a constant function. Based on a joint work with L.Buhovsky, Eu.Malinnikova and M.Sodin. | |

(JHU) |
Yamabe problem on compact manifolds with boundary | I will be talking about generalizing Yamabe problem to compact manifolds with boundary under different situations. We will prove the existence of conformal metrics having constant scalar curvature and constant boundary mean curvature under some conditions.Variation is the main tool. If time permits, we will discuss some flow aspects of these problems. | |

(MIT) |
Essential and nonessential singularities for positive scalar curvature | As a stepping stone toward a theory of positive scalar curvature for C^0 metrics, Gromov has proposed a framework of polyhedral comparisons that hinges on understanding which smooth obstructions to positive scalar curvature carry through to settings where specific singularities are allowed. In this talk we discuss known and conjectured obstructions to positive scalar curvature in the presence of such singular sets with codimensions 1, 2, and 3 or higher. This is joint work with Chao Li. | |

(Howard) |
Methods of Algebraic Topology for the Non-local Yamabe problem | In this talk, we will present a solution of the fractional Yamabe problem for locally flat conformal infinities of Poincare-Einstein manifolds. This case is the counterpart of the locally conformally flat case of the classical Yamabe problem, however no non-local version of the Schoen-Yau Positive Mass Theorem is known. We will show how one can bypass such an issue and in a natural way, by using the Algebraic Topological argument of Bahri-Coron. | |

(Paris 7) |
Fractional harmonic maps and local/nonlocal minimal surfaces | In this talk, I will report on some recent regularity results for fractional harmonic maps into a manifold. The equation for this kind of maps is the fractional analogue of the classical harmonic map system where the Laplacian is replaced by a fractional Laplacian as defined in Fourier space. I will explain the relations between fractional harmonic maps and local or nonlocal minimal surfaces depending on the fractional exponent, and how this relations lead to an improved regularity for sphere targets. | |

(MIT) |
2-bubble dynamics for the wave maps equation. | This talk is about joint work with Jacek Jendrej where we considered energy-critical wave maps taking values in the 2-sphere. It is known that initial data of topological degree zero and energy less than twice that of the ground state harmonic map lead to a global solution that scatters in both time directions. Here we’ll consider data at the threshold energy. For any k-equivariant data with exactly twice the energy of the degree k equivariant harmonic map we prove that the solution is defined globally in time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines classical concentration-compactness techniques with a precise modulation theoretic analysis of interactions between two harmonic maps in the absence of excess radiation. | |

(U Penn) |
Rotational Symmetry of asymptotically conical mean curvature flow expanding solitons | Beginning with work of Ecker-Huisken on long term existence for mean curvature flow of entire graphs, several authors have considered expanding solitons which are asymptotic to rotationally symmetric cones. Chopp-Angenent-Ilmanen showed that each such cone with sufficiently large cone angle possesses at least two rotationally symmetric, self-expanding MCF evolutions, and Helmensdorfer subsequently discovered a third. In joint work with Frederick Fong, we prove that any mean-convex, asymptotically conical self-expanding soliton is actually rotationally symmetric. | |

(NYU) |
Rectifiability theorems for anisotropic energies and Plateau problem. | We present our recent extension of Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a necessary and sufficient condition on the integrand to obtain the rectifiability of every d-dimensional varifold with locally bounded first variation and positive d-dimensional density. We can apply this result to the minimization of anisotropic energies among families of d-rectifiable closed subsets of $\R^n$. Easy corollaries of this compactness result are the solutions to three formulations of the Plateau problem: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by Guy David. Moreover, we apply the rectifiability theorem to prove a compactness result of integral varifolds in the anisotropic setting. | |

(U Chicago) |
Sharp estimates for oscillatory integral operators via polynomial partitioning | I will discuss recent joint work with Larry Guth and Marina Iliopoulou concerning sharp L^p bounds for Hormander-type oscillatory integral operators. The results provide an answer to a question of Hormander from the mid 70s and have applications to the study of the convergence of Fourier series. The talk will be accessible and assume no prior knowledge of this area. | |

(TBA) |
TBA | TBA | |

(Princeton) |
Deformations of Q-curvature | Stability (local surjectivity) and rigidity of the scalar curvature have been studied in an early work of Fischer-Marsden on ``vacuum static spaces''. Inspired by this line of research, we seek similar properties for Q-curvature by studying ``Q-singular spaces'', which were introduced by Chang-Gursky-Yang. In this talk, we investigate deformation problems of $Q$-curvature on closed Riemannian manifolds with dimensions $n\geq 3$. In particular, we prove local surjectivity for non-$Q$-singular spaces and local rigidity of flat manifolds. For global results, we show that any smooth functions can be realized as a $Q$-curvature on generic $Q$-flat manifolds. However, a locally conformally flat metric on $n$-tori with nonnegative $Q$-curvature has to be flat. This is joint work with Wei Yuan. | |

(TBA) |
TBA | TBA | |

(Cornell) |
The helicoidal method | Not long ago we discovered a new method of proving vector valued inequalities in Harmonic Analysis. With the help of it, we have been able to give complete positive answers to a number of questions that have been circulating for some time. The plan of the talk is to describe (some of) these, and to also explain how this method implies sparse domination results for various multi-linear operators and their (multiple) vector valued extensions. Joint work with Cristina BENEA. |

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