Organizers: S.-T. Yau, T. Collins

The seminar meets on Tuesdays from 4:15 to 5:15 in SC 507, except as noted below.

**September 13, 2016**

Aruna Kesavan (CMSA) -Asymptotic structure of space-time with a positive cosmological constant

[abstract] Cosmological observations suggest that our universe is best described on large scales by Einstein's equations with a positive cosmological constant. In this talk, we discuss the implications of this fact for the asymptotic structure of space-time, symmetries and conservation laws. This is joint work with Abhay Ashtekar and Beatrice Bonga. **September 20, 2016**

Gao Chen (Stony Brook) -Classification of graivtational instantons

[abstract] A gravitational instanton is a complete non-compact Calabi-Yau surface, i.e. hyperk\"ahler 4-manifold with an appropriate curvature decay condition on the end. In this talk, I will discuss the classification of gravitational instantons with faster than quadratic curvature decay condition. It's a joint work with Xiuxiong Chen. **September 27, 2016**

Tian-Jun Li (Minnesota) -Symplectic divisor compactifications in dimension 4

[abstract] We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type bound- ary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an ω-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. We also observe that, up to Stein cobrodisms, every contact 3-manifold admits a divisor cap. Joint work with Cheuk Yu Mak. **October 4, 2016**

Alexander Logunov (St. Petersburg & Tel Aviv) -Zero set of a non-constant harmonic function in R^3 has infinite surface area

[abstract] Nadirashvili conjectured that for any non-constant harmonic function in R^3 its zero set has infinite surface area. This question was motivated by the Yau conjecture on zero sets of Laplace eigenfunctions. We will give a sketch of the proof of Nadirashvili's conjecture. **October 11, 2016**

Mark Stern (Duke) -Instantons on ALF spaces

[abstract] I will discuss progress on establishing Cherkis's Nahm transform for multicenter Taub NUT spaces. This is joint work with Sergey Cherkis and Andres Larrain. **October 18, 2016, ** 3-4pm in Math 507 ****

Jian Xiao (Northwestern) -Positivity in the convergence of the inverse sigma-k flow

[abstract] By relating the existence of special K\"ahler metrics with with algebro-geometric stability conditions, such as the Yau-Tian-Donaldson conjecture on the existence of constant scalar curvature, Lejmi and Sz ekelyhidi proposed a conjectural numerical criterion for the solvability of the inverse $\sigma_k$ equation, or equivalently, for the convergence of the inverse $\sigma_k$ flow. Their conjecture looks like a non-linear version of Demailly-Paun's numerical characterization of the K\" ahler cone. In this talk, we discuss the positivity aspects, partially verifying their conjecture for the case k=n-1. **October 25, ** 3-4pm, room G10, CMSA (20 Garden Street) ****

Jonathan Zhu (Harvard) -Entropy and self-shrinkers of the mean curvature flow

[abstract] The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set. **November 1, 2016, ** 3-4pm in Math 507 ****

Robert Haslhofer (Toronto) -The moduli space of 2-convex embedded spheres

[abstract] We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture M_2 is contractible, but the topology of M_n for n\geq 3 is highly nontrivial. In this talk, I will explain how geometric analysis can be used to study the topology of M_n respectively some of its variants. I will start by explaining the topological background and sketching a geometric analytic proof of Smale’s theorem that M_1 is contractible. In the second half of my talk, I’ll describe recent work on space of smoothly embedded spheres in the 2-convex case, i.e. when the sum of the two smallest principal curvatures is positive. Our main theorem (joint with Buzano and Hershkovits) proves that this space is path-connected, for every n. The proof uses mean curvature flow with surgery. **November 8, 2016**

Nicolaos Kapouleas (Brown) -Gluing constructions for minimal surfaes and other geometric objects **November 15, 2016**

Sebastien Picard (Columbia) -Geometric flows and Strominger systems

[abstract] The anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. I will discuss criteria for long time existence and convergence of the flow on toric fibrations with the Fu-Yau ansatz. This is joint work with D.H. Phong and X.W. Zhang. **November 22, 2016**

No Speaker -**November 29, 2016**

CMSA Homological Mirror Symmetry Workshop -**December 6, 2016**

Mattias Jonsson (Michigan) -A variational approach to the Yau-Tian-Donaldson conjecture

[abstract] I will present joint work with Robert Berman and Sebastien Boucksom, on a new proof of a uniform version of the Yau-Tian-Donaldson conjecture for Fano manifolds with finite automorphism group. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory. Instead, the proof is variational and uses pluripotential theory and certain non-Archimedean considerations.