Department of Mathematics National Cheng Kung University
Topic: On the $F$-rationality and cohomological properties of matrix Schubert varieties
Abstract:
Matrix Schubert varieties (MSVs) are introduced by W. Fulton in his theory of degeneracy loci of maps of flagged vector bundles. They are (up to product of an affine space) the opposite big cells of the corresponding Schubert varieties.
The goal of this talk is to sketch the following two results:
1. A characterization of complete intersection MSVs
2. A new proof of $F$-rationality of MSVs
These results build on the Gr\''obner basis theory of the MSVs developed by A. Knutson and E. Miller.
I will also explain how 2. provides a new proof of the following celebrated fact:
Schubert varieties in flag varieties are normal and have at most rational singularities. The original proofs utilize the Bott-Samelson resolution, which is not needed in our proofs.
Department of Mathematics National Taiwan University
Topic: Parabolic constructions of asymptotically flat 3-metrics of prescribed scalar curvature
Abstract:
The scalar curvature problem arises naturally in general relativity as space-like hypersurface in the underlying space-time. In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. This quasi-spherical ansatz has a background foliation with round metrics and converts the problem into a semi-linear parabolic equation. It is also known by work of R. Hamilton and B. Chow that the evolution under the Ricci flow of an arbitrary initial metric $g_0$ on $S^2$, suitably normalized, exists for all time and converges to the round metric.
In this talk, we describe a construction of metrics of prescribed scalar curvature using solutions to the Ricci flow. Considering background foliations given by Ricci flow solutions, we obtain a parabolic equation similar to Bartnik’s. We discuss conditions of scalar curvature that guarantee the solvability of the parabolic equation, and thus the existence of asymptotically flat 3-metrics with a prescribed inner boundary. In particular, many examples of asymptotically flat 3-metrics with outermost minimal surfaces are obtained.
Department of Mathematics National Central University
Topic: Surgery and invariants of Lagrangian surfaces
Abstract:
In this talk I will report on some recent progress on the Lagrangian Knot Problem which was proposed by Y. Eliashberg and L. Polterovich about two decades ago.
More specifically, I will introduce a new symplectic invariant that can be used to distinguish oriented Lagrangian surfaces immersed in parallelizable symplectic 4-manifolds up to their symplectic isotopy classes. I will also introduce an elementary surgery on Lagrangian surfaces, which preserves the smooth isotopy class of Lagrangian surfaces but changes their symplectic isotopy classes. New examples of smoothly but not symplectically isotopic embedded Lagrangian tori are also constructed.
