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Thu Sep 20

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Monotonicity formulas and Type I singularities
Huaidong Cao, Ordway Visitor, Lehigh University

In this talk we shall introduce Huisken's monotonicity formula for the mean curvature flow and Perelman's monotonicity formulas for the Ricci flow. We shall discuss their applications, including the role they play in studying Type-I singularities of the flows.

Thu Sep 27

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Second variation of Perelman's entropy and stability of Ricci solitons
Huaidong Cao, Ordway Visitor, Lehigh University

Einstein metrics are critical points of the well-known classical Hilbert action
(namely the total scalar curvature functional) under volume normalization. Likewise,
Ricci solitons are critical points of Perelman's entropy.
In this talk, we shall discuss the second variation of Perelman's entropy and stability
of compact Ricci solitons. It turns out the stability for positive Einstein manifolds is
related to two eigenvalue estimates: the first eigenvalue of the Laplacian on functions,
and that of the Lichnerowicz Laplacian on symmetric 2-tensors.

Thu Oct 04

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Stability of Ricci solitons
Huaidong Cao - Ordway Visitor, Lehigh University

In this talk we continue our discussion of the previous week on stability of
Ricci solitons, especially in four dimensions.

Fri Oct 19

Differential Geometry and Symplectic Topology Seminar

2:30pm - Vincent Hall 6
The Smooth 4-dimensional Poincare Conjecture and Dehn surgery on links
Alex Zupan, University of Nebraska

The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

Thu Dec 06

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Enumerative geometry: old and new
Felix Janda, University of Michigan

For as long as people have studied geometry, they have counted
geometric objects. For example, Euclid's Elements starts with the
postulate that there is exactly one line passing through two distinct
points in the plane. Since then, the kinds of counting problems we are
able to pose and to answer has grown. Today enumerative geometry is a
rich subject with connections to many fields, including combinatorics,
physics, representation theory, number theory and integrable systems.

In this talk, I will show how to solve several classical counting
questions. Then I will describe a more modern problem with roots in
string theory which has been the subject of intense study for the last
two decades, namely the study of the Gromov-Witten invariants of the
quintic threefold, a Calabi-Yau manifold. I will explain a recent
break-through in understanding the higher genus invariants that stems
from a seemingly unrelated problem related to the study of holomorphic
differentials on Riemann surfaces.

Thu Dec 20

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions
Steve Trettel, UC Santa Barbara

The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations. Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories. Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual. This connects the aforementioned deformation theories and provides geometric motivation for a representation-theoretic observation of Cooper, Long, and Thistlethwaite

Tue Apr 23

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Self-homeomorphisms of reducible 3-manifolds and applications in topology, geometry and dynamics.
Christoforos Neofytidis, University of Geneva

We recall the self-homeomorphisms of a closed oriented reducible 3-manifold. Using this description, we discuss various problems in low-dimensional topology and dynamics, such as the existence of Anosov tori in 3-manifolds (joint work with Shicheng Wang), the simplicial volume of mapping tori of 3-manifolds (joint work Michelle Bucher) and the virtual Betti numbers of mapping tori of 3-manifolds.

Thu Apr 25

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Harmonic surfaces and simple loops
Vlad Markovic - Ordway Visitor, Caltech