# Abstracts

### Yamabe Symposium - October 8-10, 2010

#### Nodal Set of Eigenfunctions

*Toby Colding (Massachusetts Institute of Technology)* Sat., Oct. 9, 4 p.m.

*Abstract:* We discuss lower bounds for the Hausdorff measure of nodal sets of eigenfunctions. This is joint work with Bill Minicozzi.

#### Mirror symmetry between Toric A model and Landau-Gizburg B model

*Kenji Fukaya (Kyoto University)* Fri., Oct. 8, 5 p.m.

*Abstract:* In a series of papers (with Oh-Ohta-Ono) we studied Lagrangian Floer theory in the case of Toric manifolds and its Lagrangian fiber of the moment map. I would like to explain how it implies a kind of Mirror symmetry. I also explain how we can use it to obtain information about the Lagrangian submanifold L of Toric manifolds in the case L is not necessarily a Lagrangian fiber. (This part is a joint work with Abousaid, Oh, Ohta, Ono.) The relation of it to cyclic and Hochshild homology and to K. Saito theory over Novikov rings is also discussed.

#### Volumes of Hyperbolic 3-Manifolds

*David Gabai (Princeton University)* Sat., Oct. 9, 9:30 a.m.

*Abstract:* As part of his revolutionary work on hyperbolic geometry in the 1970's, Thurston generalizing work of Jorgensen and Gromov, showed that that the set of volumes of complete finite volume hyperbolic 3-manifolds is closed and well ordered. Recently, Robert Meyerhoff and Peter Milley and the speaker showed that the Weeks manifold is the unique lowest volume closed orientable one, culminating a 30+ year effort by many mathematicians using a wide variety of techniques. In particular, we make use of work of Agol - Dunfield which relies on Perelman's work on Ricci flow. This lecture will survey these developments and discuss various open problems.

#### Smooth Group Actions on 4-Manifolds and Equivariant Gauge Theory

*Ian Hambleton (McMaster University)* Sun., Oct. 10, 9:30 a.m.

*Abstract:* Algebraic automorphisms of algebraic surfaces provide natural examples of smooth finite group actions on 4-manifolds. This talk will describe an equivariant version of the Yang-Mills moduli spaces, and some applications to "rigidity" results for smooth finite group actions on 4-manifolds.

#### On Four-Dimensional Einstein Manifolds

*Claude LeBrun (State University of New York at Stony Brook)* Thurs., Sep. 30, 3:30 p.m.

*Abstract:* A Riemannian metric is said to be Einstein if it has constant Ricci curvature. A central problem in differential geometry is to determine which smooth compact manifolds admit an Einstein metric, and to completely understand the moduli space of all such metrics when they exist . The 4-dimensional case of this problem appears to be highly atypical. This lecture will survey some recent results regarding the special case of 4-manifolds which admit either a complex structure or symplectic structure.

#### Einstein Manifolds and Extremal Kahler Metrics

*Claude LeBrun (State University of New York at Stony Brook)* Fri., Oct. 8, 3:30 p.m.

*Abstract:* This talk will present some new existence results for extremal Kahler metrics on toric surfaces, with applications to the construction of Einstein metrics. In particular, I will explain how an Einstein metric on CP2#2(-CP2) can be constructed by a bubbling-off procedure. The same results also shed new light on the corresponding uniqueness problem.

#### The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties

*Melissa Liu (Columbia University)* Sat., Oct. 9, 2:30 p.m.

*Abstract:* I will discuss (i) SYZ transformation relating equivariant coherent sheaves on a toric variety to Lagrangians in the cotangent of R^n, (ii) microlocalization functor relating the Fukaya category of the cotangent to constructible sheaves on the base (due to Nadler-Zaslow, Nadler), and (iii) a categorification of Morelli's theorem relating equivariant coherent sheaves on a toric variety to constructible sheaves on R^n. This talk is based on joint work with Bohan Fang, David Treumann and Eric Zaslow.

#### Recent Results on Cosmetic Surgeries

*Yi Ni (California Institute of Technology)* Sat., Oct. 9, 11:10 a.m.

*Abstract:* Dehn surgery is a fundamental construction in 3-dimensional topology. Two Dehn surgeries on the same knot are cosmetic if they yield homeomorphic manifolds. Such surgerie s are very rare. I will talk about some recent results on the classification of cosmetic surgeries. The proofs use Heegaard Floer homology and sutured manifold theory. Some of this talk is based on joint work with Zhongtao Wu.

#### Group Actions on Surfaces of General Type

*Ron Stern (University of California at Irvine)* Sun., Oct. 10, 11:00 a.m.

*Abstract:* The automorphism group of an algebraic surfaces of general type is known to be finite. Since a diffeomorphism of such a surface, when viewed as a smooth 4-manifold, must preserve the Seiberg-Witten and Donaldson basic classes, one would suspect that the diffeomorphism group of a surface of general type demonstrate some finiteness properties. For example: Are there only finitely many smoothly distinct but topologically equivalent smooth actions of a fixed cyclic group? We will report on joint work with Ron Fintushel and Nathan Sunukjian that many surfaces of general type have, in fact, infinitely many distinct smooth but topologically equivalent actions of a fixed cyclic group.