Differential Geometry Serminar

• The seminars are held on Tuesday afternoon in Room 507 of the Harvard Science Center at 4:15 pm.

• The talks are one hour long with some time afterwards for questions and discussion.

• The seminar is organized by Prof. Shing-Tung Yau and Yaiza Canzani.

Fall Schedule

 September 1: Zhengcheng Gu (Perimeter Institute) September 8: Hossein Movasati (IMPA) September 15: Claude LeBrun (Stony Brook University) September 22: The seminar won't run on this date. September 29: Gabor Lippner (Northeastern University) October 6: Song Sun (Stony Brook University) October 13: Adrian Zahariuc (Harvard University) October 20: Michael Anderson (Stony Brook University) October 27: Thomas Walpuski (MIT) November 3: Binglong Chen (Sun Yat-sen University) November 10: Boris Hanin (MIT) November 17: Xin Zhou (MIT) November 24: The seminar won't run on this date. December 1: Bong Lian (Brandeis University)

Abstracts

November 10 : Boris Hanin (MIT)

Nodal Sets of Random Eigenfunctions of the Harmonic Oscillator
Random eigenfunctions at energy E of the isotropic harmonic oscillator in R^d have an U(d) symmetry and are in some ways analogous to random spherical harmonics of fixed degree on S^d, whose nodal sets have been the subject of many recent studies. However, there is a fundamentally new aspect to this ensemble, namely the existence of allowed and forbidden regions. In the allowed region, the Hermite functions behave like spherical harmonics, while in the forbidden region, Hermite functions are exponentially decaying and it is unclear to what extent they oscillate and have zeros. The purpose of this talk is to present several results about the expected volume of the zero set of a random Hermite function in both the allowed and forbidden regions. This is joint work with Steve Zelditch and Peng Zhou.

November 3 : Binglong Chen (Sun Yat-sen University)

On the local regularity of Einstein spacetimes
Given a spacetime manifold satisfying vacuum Einstein field equation, and an observer on the manifold, we will construct a canonical coordinate system around this observer so that the coefficients of the metric tensor can gain the optimal (quantitative) regularity.

October 27: Thomas Walpuski (MIT)

G2–instantons over twisted connected sums
In joint work with H. Sá Earp, we introduced a method to construct G2–instantons over compact G2–manifolds arising as the twisted connected sum of a matching pair of building blocks. I will recall some of the background (including the twisted connected sum construction and a short discussion as to why one should care about G2–instantons), discuss our main result and explain how to interpret it in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface. Finally, I will talk about how to use our construction to produce a rather concrete example of a G2–instanton over twisted connected sum recently discovered by Crowley and Nordström.

October 20: Michael Anderson (Stony Brook University)

Boundary value problems for Einstein metrics and a conjecture of Bartnik
We discuss elliptic boundary value problems for the Einstein equations. We'll then discuss corresponding existence results in dimension 3 and the simplest case in dimension 4, namely static vacuum Einstein metrics. This leads to a partial solution of a conjecture of Bartnik in general relativity (the static extension conjecture).

October 13: Adrian Zahariuc (Harvard University)

Specialization of Quintic Threefolds to the Chordal Variety
We consider a family of quintic threefolds specializing to a certain reducible threefold with two irreducible components. What are the flat limits of the rational curves in this degeneration? I will provide a “first approximation” to the answer by describing the space of genus zero stable morphisms to the central fiber (as defined by J. Li).

October 6: Song Sun (Stony Brook University)

Singularities of Kahler-Einstein metrics and algebraic stability
A natural question in Kahler geometry is to understand degenerations and singularity formations of Kahler manifolds (in particular, of Kahler-Einstein manifolds). Given a sequence of Kahler-Einstein manifolds satisfying certain non-collapsing hypothesis, we proved earlier that a Gromov-Hausdorff limit, which is a priori only a metric space, has a natural structure of a normal projective variety. These limit spaces admit metric tangent cones, which serve as local models for the metric singularities, and it is then interesting to ask for the meaning in terms of algebraic geometry. We will report recent progress towards this question. Our work suggests a notion of stability” for germs of algebraic singularities. This talk is based on joint work with Simon Donaldson.

September 29: Gabor Lipner (Notheastern University)

Sharp convexity estimates for harmonic functions on the lattice, via absolute monotonicity
Propagation of smallness, and convexity-type, estimates play an important role in understanding both local and global behavior of eigenfunctions. Classical methods to prove such results break down in the discrete setting. In this talk I will describe a surprising connection between harmonic functions and absolute monotonicity, and show how this can be used to prove convexity results on the lattice. (Joint work with Dan Mangoubi.)

September 15: Claude LeBrun (Stony Brook University)

Mass in Kähler Geometry
I will describe a simple formula, discovered in joint work with H.-J. Hein, for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), this formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.

September 8: Hossein Movasati (IMPA - Instituto de Matematica Pura e Aplicada)

Why should one compute the periods of algebraic cycles?
Let $X$ be a smooth projective variety of dimension $2k$ over complex numbers and let $Z$ be a subvariety of $X$ of dimension $k$. One says that the infinitesimal Hodge conjecture (IHC) holds if the deformation space of the pair $(X,Z)$ inside the moduli space of $X$ is the same as the deformation space of the Hodge cycle $[Z]$ induced by $Z$. Hodge conjecture does not imply IHC, however, verifications of IHC in many explicit situations imply the Hodge conjecture for deformed Hodge cycles. In this talk I am going to explain how explicit computations of periods of differential forms over cycles $[Z]$ lead to the verifications of IHC. I will also prove that IHC holds for linear projective spaces inside hypersurfaces. This in two dimensional case ($k=1$), where the Hodge conjecture is well-known as Lefschetz $(1,1)$ theorem, is a result of Green and Voisin in 1990. Some partial results concerning complete intersections inside sextic hypersurfaces fourfolds (which are Calabi-Yau) will be given. One of the basic ingredients of the proof is the so called infinitesimal variation of Hodge structures. The talk is partially based on arXiv:1411.1766.

September 1: Zhengcheng Gu (Perimeter Institute)

Topological Quantum Field Theory approach for Bosonic Symmetry-Protected-Topological Phases with Abelian Symmetry in Three Dimensions
Symmetry protected topological(SPT) phase is a generalization of topological insulator(TI). Different from the intrinsic topological phase, e.g., the fractional quantum hall(FQH) phase, SPT phase is only distinguishable from a trivial disordered phase when certain symmetry is preserved. Indeed, SPT phase has a long history in 1D, and it has been shown that the well known Haldane phase of S=1 Heisenberg chain belongs to this class. However, in higher dimensions, most of the previous studies focus on free electron systems. Until very recently, it was realized that SPT phase also exists in interacting boson/spin systems in higher dimensions. In this talk, I will discuss the general mechanism for bosonic SPT phases and propose a corresponding topological quantum field theory(TQFT) descriptions. I will focus on examples in three (spacial) dimensions, including bosonic topological insulators(BTI).