Mean curvature flow is the negative gradient flow of the area
functional, and it has attracted a lot of interest in the past few years. In
this talk, we will discuss a PDE-based, gauge theoretic, construction of
codimension-two mean curvature flows based on the Yang-Mills-Higgs
functionals, a natural family of energies associated to sections and metric
connections of Hermitian line bundles. The underlying idea is to approximate
the flow by the solution of a parabolic system of equations and study the
corresponding singular limit of these solutions as the scaling parameter
goes to zero. This is based on joint work with A. Pigati and D. Stern.

In 1983 Hartle and Hawking put forth that signature type-change may be conceptually interesting, leading to the so-called no-boundary proposal for the initial conditions for the universe, which has no beginning because there is no singularity or boundary to the spacetime. But there is an origin of time. In mathematical terms, we are dealing with signature type-changing manifolds where a positive definite Riemannian region is smoothly joined to a Lorentzian region at the surface of transition where time begins.

We utilize a transformation prescription to transform an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Then we prove the transformation theorem saying that locally the metric \tilde{g} associated with a signature-type changing manifold (M, \tilde{g}) is equivalent to the metric obtained from a Lorentzian metric g via the aforementioned transformation prescription. By augmenting the assumption by certain constraints, mutatis mutandis, the global version of the transformation theorem can be proven as well.

The transformation theorem provides a useful tool to quickly determine whether a singular signature type-changing manifold under consideration belongs to the class of transverse type changing semi-Riemannian manifolds.

Gravitational instantons are defined as non-compact hyperKahler
4-manifolds with L^2 curvature decay. They are all bubbling limits of K3
surfaces and thus serve as stepping stones for understanding the K3 metrics.
In this talk, we will focus on a special kind of them called
ALH*-gravitational instantons. We will explain the Torelli theorem, describe
their moduli spaces and some partial compactifications of the moduli spaces.
This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi,
X. Zhu and S. Soundararajan.

Special date/time and joint with Geometry and Topology Seminar.

In this talk, we will start by introducing quiver representations
and some of their applications.  Then we will review noncommutative crepant
resolutions of singularities of Van den Bergh.  We will find that the notion
of quiver stacks will be useful in unifying geometric and quiver
resolutions.  Finally, we will explain our motivation and construction of
these quiver stacks from a symplectic mirror point of view.

Symplectic geometry is a relatively new branch of geometry.
However, a string theory-inspired duality known as “mirror symmetry” reveals
more about symplectic geometry from its mirror counterparts in complex
geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry
called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS
results were then proved for symplectic mirrors to Calabi-Yau and Fano
manifolds. Those mirror to general type manifolds have been studied in more
recent years, including my research. In this talk, we will introduce HMS
through the example of the 2-torus T^2. We will then outline how it relates
to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam,
Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to
hypersurfaces of higher dimensional tori, otherwise known as “theta
divisors.” This is also joint with Azam, Lee, and Liu.

Joint with Geometry and Topology Seminar.