# Fridays 3:00 - 4:00pm, SH 6617

### 5/30, Julie Marie Rowlett, UCSB The Fundamental Gap Conjecture for Triangles"

Abstract: The Fundamental Gap Conjecture due to S. T. Yau and also M. van de Berg states that for a convex planar domain of diameter d, the first two positive eigenvalues of the Laplacian satisfy $$\lambda_2 - \lambda_1 \geq \frac{3 \pi^2}{d^2}.$$ The gap between the first two eigenvalues has been studied by several authors and while it is always interesting to understand the interaction between the eigenvalues of a differential operator and the geometry of the domain, beyond purely mathematical implications, the gap also has physical implications. For example, the gap controls the rate of collapse of any initial state toward a state dominated by the first eigenvalue and is of central interest in statistical mechanics and quantum field theory. The gap is important to numerical methods because it can be used to control the rate of convergence of numerical computation methods. I will discuss joint work with Zhiqin Lu and present a "compactness theorem" which shows that the fundamental gap for triangular domains becomes unbounded as the domain collapses to a segment. This result is a key step to proving the conjecture for triangular domains and for our larger goal of proving the conjecture for all convex domains by approximating polygonal domains.

### 6/6, Zhigang Han, University of Massachusetts A nonextension result on the spectral metric"

Abstract: The spectral metric is a bi-invariant metric on the Hamiltonian diffeomorphism group defined by Schwarz and Oh using Floer-theoretical method. In this talk, I will first explain the definition of the spectral metric for symplectically aspherical manifolds. Then I will explain why for certain symplectic manifolds, this metric does not extend to bi-invariant metric on the full group of symplectomorphisms.

# Fridays 3:00 - 4:00pm, SH 6617

### 1/18, Seungsu Hwang, Chung-Ang University and visiting UCSB Critical points of the total scalar curvature functional"

Abstract: On a compact n-dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation. In 1987 Besse proposed a conjecture that a solution of the critical point equation is Einstein. Since then, a number of mathematicians have given partial proof of the conjecture and obtained many geometric consequences. However, none has given a complete proof. The purpose of the talk is to show that a compact 3-dimensional manifold M is isometric to the round 3-sphere $S^3$ if ker $s_g'* \not= 0$ and its second homology vanishes. Note that this Theorem implies that M is Einstein and hence that Conjecture holds on 3--dimensional compact manifold under certain topological conditions.

### 1/25 Jeffery Case, UCSB The Bakry-Emery tensor in Lorentzian geometry"

Abstract: In this talk, we will consider extensions of the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem to the Bakry-Emery-Ricci tensor. We will point out similarities and differences to the setting in Riemannian geometry, and spend the most time on ideas less familiar in Riemannian geometry.

### 2/1 Colin Hinde, UCLA Synthetic Ricci Curvature and Optimal Transportation"

Abstract: We will review the solution of the optimal transportation problem on Riemannian manifolds. The investigation leads to a synthetic curvature condition equivalent to a lower bound on Ricci curvature yet requiring only the structure of a locally compact length space with a Borel measure. Yet, as in the case of Alexandrov Space, the synthetic curvature condition secures additional structure for free. We will discuss some results in this direction as well as localization of the condition, as time permits.

# Fridays 3:00 - 4:00pm, SH 6617

### 10/19 Yujen Shu, UCSB Constant scalar curvature Kahler metrics on ruled surface"

Abstract: We will introduce the notion of constant scalar curvature Kahler (cscK) metrics, and investigate the existence of such metrics on ruled surfaces.

### 10/26 Zhenghan Wang, Microsoft/UCSB On Exotic (2+1)-TQFTs"

Abstract: G. Moore and N. Seiberg conjectured in 1990s that every (2+1)-TQFT could be realized as a Chern-Simons-Witten theory based on a pair (G,\lamda), where G is a compact (not necessarily connected) Lie group, and \lambda a cohomology class in H^4(BG;Z). I will discuss examples of TQFTs constructed from Jones' subfactor theory which might not be realized in this way. This is a joint work with Seung-moon Hong of Indiana Univ and Eric Rowell of Texas A&M.

### 11/16 Peter Petersen, UCLA Rigidity of shrinking Ricci solitons"

Abstract: I will survey several recent results on Ricci solitons. The goal is to find suitable conditions that make it possible to classify these objects. Several different conditions on curvature and symmetry allow us to classify these objetcs. Best of all in dimension 3 we can classify all shrinking solitions without further assumptions.

### 12/7 Andreas Malmendier, UCSB Rational elliptic surfaces and the u-plane integral"

Abstract: I will give a brief review of how the Seiberg-Witten families of elliptic curves with N_f=0,1,2,3,4 massive hypermultiplets define certain rational elliptic surfaces, p: Z -> CP^1, with non-stable singularities. I will then explain how the masses of the hypermultiplets are closely related to the homology of this surface: the root lattice T corresponding to the singularity type is embedded into H_2(Z). Oguiso and Shioda classified all possible structures for T and the Mordell-Weil group of p. Roughly, the evaluation of the holomorphic two-form on classes perpendicular to T then gives the masses of the hypermultiplets. If time permits, I will explain how the Donaldson invariants for CP^2 can be computed by evaluating a certain ratio of functional determinants of a family of elliptic operators parametrized by the rational elliptic surface for N_f =0. Return to Conference and Seminars Page