Time: Monday 4 pm

Location: Krieger 308

Organizer(s): Jacob Bernstein and Yi Wang

Date |
Speaker |
Title |
Abstract |

(TBA) |
TBA | TBA | |

(Irvine) |
A flow of conformally balanced metrics | We discuss the development on geometric and analytic as- pects of the Anomaly flow. While the flow was originally motivated by the study of Hull-Strominger system from string theory, its zero slope case is potentially of considerable interest in non-Kähler geometry, as it is a flow of (n−1, n−1) form with Kähler stationary points and preserves the conformally balanced property of the initial metric. We establish its long time existences and convergence on Kähler manifolds for suitable initial data by reducing it to a parabolic complex Monge-Ampère type equation which has no concave property. This is joint work with D. Phong and S. Picard. | |

(Rutgers) |
The Loewner-Nirenberg problem in domains with conic singularities | We talk about the boundary behavior of solutions to the Loewner-Nirenberg problem in domains with conic singularities. To analyze the boundary behavior of solutions with respect to multiple normal directions, we first derive the lower eigenvalue growth estiamtes for certain singular elliptic operators. | |

(LSU) |
Logarithmically energy-supercritical Nonlinear Wave Equations: axial symmetry and global well-posedness. | In nonlinear dispersive PDE, radial symmetry often plays a key role in allowing for more refined analysis of the nonlinear interactions which could lead to possible blowup. We will describe recent work where we have recently introduced a mechanism for relaxing assumptions of radiality by considering symmetry in a subset of the variables (for instance, assuming that the initial data is axially symmetric). We applied this philosophy to show global well-posedness and scattering in for the nonlinear wave equation in the logarithmically energy-supercritical setting, generalizing a result of Tao which was established for the radial case. The uses Morawetz and Strichartz estimates that have been adapted to the new symmetry assumption. These methods in fact bring a new perspective to sharp estimates for the energy-critical problem, along the lines of the influential work of Ginibre, Soffer, and Velo. This is joint work with Ben Dodson. | |

(Syracuse) |
On the diameter rigidity of Kahler manifolds with positive bisectional curvature | It follows from the comparison theorem that if the Kahler manifold has bisectional curvature at least 1, then the diameter is no greater than the diameter of the standard complex projective space. I will discuss a joint result with G. Liu regarding the rigidity when the diameter reaches the maximum. | |

Special SeminarTuesday 4-5pm in Krieger Laverty Lounge |
(UCL) |
On the existence of translators in slabs | In this talk I will present joint work with Bourni and Langford. We prove that for any A\in (0,\pi/2) there exists a strictly convex translating solution of mean curvature flow in R^n contained in a slab of width \pi/cosA and in no smaller slab. |

(JHU) |
Wave maps on (1+2)-dimensional curved spacetimes | In this talk we will discuss joint work with Casey Jao and Daniel Tataru where we initiate the study of 2+1 dimensional wave maps on a curved space time in the low regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical regularity. We proceed by generalizing the classical optimal wave bilinear L^2 estimates to variable coefficients by means of wave packet decompositions and characteristic energy estimates. This allows us to iterate in a curved X^{s,b} space. | |

Kempf LectureMihalis Dafermos |
No Seminar |
TBA | |

(UFF Brazil) |
The mass of asymptotically hyperbolic manifolds with non-compact boundary. | We define a mass-type geometric invariant for Riemannian manifolds asymptotic to the hyperbolic half-space and discuss a positive mass theorem. This is a joint work with Levi de Lima (UFC-Brazil). | |

(Purdue) |
The geometry of the free boundary near the fixed boundary generated by a fully nonlinear uniformly elliptic operator | This talk focuses on the regularity problem of the free boundary near the fixed boundary for the fully nonlinear obstacle problem in higher dimensions. | |

(UCL) |
Generic uniqueness of expanders with vanishing relative entropy | We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same smooth cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that generically expanders with vanishing relative entropy are unique. This also implies that generically locally entropy minimizing expanders are unique. This is joint work with A. Deruelle. | |

(MIT) |
Uniform level set estimates for the first Dirichlet eigenfunction | In this talk, we will discuss the first Dirichlet eigenfunction and the torsion function on a convex planar domain of high eccentricity. Our aim will be to obtain estimates on the shape of the level sets of these functions, which are uniform in this high eccentricity setting. The first Dirichlet eigenfunction is log-concave in the whole domain, but we will see that in a level set around its maximum it satisfies a stronger quantitative concavity property, consistent with the shape of its intermediate level sets. We will end by establishing an approximation for the torsion function, and then use this to construct examples illustrating contrasting behaviour for the eigenfunction and torsion function near their respective maxima. | |

(Michigan State) |
n exponential convergence result for parabolic optimal transport with boundary | The optimal transport, or Monge-Kantorovich problem is a classical optimization problem, and it is known that solutions to this problem can be obtained by solving an elliptic PDE of Monge-Ampère type. A natural approach is to find a related parabolic flow, whose solution will converge to a steady state which is the solution of the elliptic problem. In this talk, I will discuss an exponential convergence rate result for solutions of such a parabolic problem, in the case when the domains involved have nonempty boundary. The approach is based on a differential Harnack inequality, but there are complications arising from the presence of a boundary which confounds a proof for the full Harnack inequality. Instead, we will use some features specific to optimal transport to bypass these difficulties and obtain exponential convergence without the full Harnack inequality. Additionally, I will discuss an interesting connection with the pseudo-Riemannian framework for optimal transport introduced by Kim and McCann. The content of this talk is based on joint work with Farhan Abedin (MSU). |

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