The geometric analysis seminar is held on Tuesdays at 11am in the Zoom Meeting with Meeting ID 933 0554 5643 and Passcode 324863.

Fall Quarter, 2021

• September 28, Laura Fredrickson (UO)
  ALG Gravitational Instantons and Hitchin Moduli Spaces
  Abstract: Four-dimensional complete hyperkaehler manifolds can be classified into ALE, ALF, ALG, ALG*, ALH, ALH* families. It has been conjectured that every ALG or ALG* hyperkaehler metric can be realized as a 4d Hitchin moduli space. I will describe ongoing work with Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss to prove a special case of the conjecture, and some consequences. The hyperkaehler metrics on Hitchin moduli spaces are of independent interest, as the physicists Gaiotto—Moore—Neitzke give an intricate conjectural description of their asymptotic geometry.
• October 5, Peng Lu (UO)
  Conformal Bach flow
  Abstract: We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi’s type -estimate of derivatives of curvatures are derived.

  This is a joint work with Jiaqi Chen and Jie Qing.

• October 12, Otis Chodosh (Stanford University)
  Special Time : 10am
  Stable minimal hypersurfaces in R^4
  Abstract: I will explain why a stable minimal hypersurface in R^4 is flat. This is joint work with Chao Li.
• October 19, Demetre Kazaras (Duke)
  If Ricci is bounded below, then mass is in control!
  
  Abstract: The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I’ll use this formula to consider the following question: How flat is an asymptotically flat manifold with very little total mass? We make progress on this problem and confirm special cases of conjectures made by Huisken-Ilmanen and Sormani. The main results asserts that in the class of reasonably-behaved asymptotically flat manifolds with non-negative scalar curvature satisfying a uniform lower bound on Ricci curvature, small mass implies Gromov-Hausdorff closeness to flat space.
• October 26, Aleksander Doan (Columbia University)
  The Gopakumar-Vafa finiteness conjecture
  
  Abstract: Gromov-Witten invariants, which encode information about pseudo-holomorphic curves contained in a given symplectic manifold, provide powerful tools for solving problems in symplectic geometry. However, their geometric meaning is not always clear. These invariants are rational numbers and therefore cannot be simply interpreted as a count of pseudo-holomorphic curves. Moreover, a single curve can contribute to infinitely many Gromov-Witten invariants. The Gopakumar-Vafa conjecture predicts that for symplectic manifolds of dimension six, the information encoded in the Gromov-Witten invariants can be repackaged into a collection of numbers, called the BPS invariants, which don’t suffer from these drawbacks: they are integers and only finitely many of them are nonzero in every homology class. The first part of the conjecture was proved in 2018 by Ionel and Parker. I will discuss a proof of the second part, which relies on combining Ionel and Parker’s cluster formalism with results from geometric measure theory. The talk is based on joint work with Eleny Ionel and Thomas Walpuski.

Previous Schedule: 2020, 2019, 2018, 2017, 2016, 2015 Spring, 2015 Winter, 2014 Fall, 2014 Spring, 2014 Winter, 2012 Winter, 2011 Fall, 2010 Spring, 2010 Winter