Geometry Seminar

Organizers: Sema Salur, Robert Hladky and Ibrahim Unal.

The seminar will generally be held on Thursdays from 2:00 P. M. to 3:00 P. M. in Hylan 1106A. Any exceptions will be noted below.

Spring 2009 Seminars:

April 9th: Caner Koca, Stony Brook University;"Extremal Kahler, Bach-flat and Einstein Metrics on Ruled Surfaces"
Abstract: In this talk, I will describe a new family of non-compact Einstein 4-manifolds which, under conformal rescaling, glue pairwise to give "nice" Riemannian metrics on compact complex surfaces. These metrics originate from Kahler geometry and their construction is based on Tonnesen-Friedman's work on extremal Kahler metrics on ruled surfaces. I will start with an expository disccussion of Calabi's extremal Kahler metrics, and then go on with the details of the construction.

February 26th: Jason Cantarella, University of Georgia;"Shortest curves of bounded curvature and tight knots"

Abstract: We present a criticality theory for knots tied in tubes of fixed diameter. The theory provides a kind of Euler-Lagrange equation for curves in this class which are critical for minimizing length which balances the first variation of length against self-contacts of the tube and points where the curve has maximum curvature. In regions where the tube has no self-contacts, the problem reduces to finding length-critical curves with curvature bounded above. We call these curves "curvature-critical". There are several interesting conclusions. There are a two-parameter family of curvature-critical curves in R^3. These include a family of non-C^2 curvature-critical curves for the problem with fixed endpoints. However, we show that the only closed C^{1,1} curvature-critical curve is the round circle.

Fall 2008 Seminars:

November 6th: Tom Ivey, College of Charleston; "Some Results on Austere 4-folds"

Abstract: Austere submanifolds in Euclidean space are for which the eigenvalues of the second fundamental form, in any normal direction, are symmetrically arranged around zero. The class of austere submanifolds was introduced by Harvey and Lawson in 1982, as a way of generating special Lagrangian submanifolds via the conormal bundle. Austere 3-folds were classified by Bryant, and austere submanifolds with rank 2 Gauss map were classified by Dajczer and Florit. In this talk, I will discuss joint work with Marianty Ionel on classifying austere submanifolds of dimension 4. These include complex surfaces in C^n, but we show that there are a wealth of new examples with the same type of compex-linear second fundamental form. We classify submanifolds of this type in R^6, in particular finding new examples which can be constructed from holomorphic curves in CP^3. We also obtain finiteness theorems and sharp lower bounds on the dimension of the linear subspace in which the austere submanifold lies, depending on the type of the second fundamental form.

October 24th(Friday): Selman Akbulut, Michigan State University; "Corks, Plugs and exotic smooth structures"

Abstract:'Corks' are codimension zero contractible Stein submanifolds of a smooth 4-manifold which determine all of its exotic structures. 'Plugs' are very different kind of small codimension zero Stein submanifolds which can also be used to change the exotic structures. Corks generalized Mazur manifolds, whereas Plugs generalize 'Gluck construction'. Corks and Plugs can be thought of freely moving particles in a 4-manifold (like``Fermions' and ``Bosons' in physics) functioning like little knobs on a wall which turn on-off the ambient exotic lights. In this talk I will discuss some new Corks and Plugs (discovered this year). I will show that by inflating a Cork (or a Plug) in two different ways we can construct small exotic Stein manifold pairs, and also discuss how you can knot Corks and Plugs inside 4-manifolds (this talk summarizes recent joint work with Yasui, and the previous joint work with Matveyev).

October 23th: Xiaodong Cao, Cornell University;

October 17th (Friday): Bianca Santoro, Duke University; "Complete Kahler metrics on crepant resolutions."

Abstract: In this talk, we plan to explain some existence results for complete Ricci-flat kahler metrics on crepant resolutions of singularities. The method allows us to provide a wider class of examples of complete Ricci-flat kahler metrics with richer topoplogy at infinity.

September 18th: Hyunjoo Cho, U of R; "Contact structures on G_2 manifolds, part 2"

September 25th: Colleen Robles, Texas A & M; "Lie algebra cohomology and the rigidity of homogeneous projective varieties. "
Abstract: It is a classical problem in geometry to determine the differential invariants of a space. Given a homogeneous projective variety G/P, we ask: how many differential invariants are necessary to recognize G/P? That is, if X is an unknown projective variety, how many derivatives do we need to take at a point of X to determine whether or not X = G/P (modulo projective transformation)? This rigidity question is governed by an exterior differential system (EDS, a system of pde). The EDS arises naturally from the geometry G/P. There is a second, closely related, EDS that is determined by the associated representation theory. The rigidity of the second EDS is determined by Lie algebra cohomology (a la Kostant). I will discuss how this helps us answer the original rigidity question. This is joint work with J.M. Landsberg (Texas A&M).

September 18th: Hyunjoo Cho, U of R; "Contact structures on G_2 manifolds, part 1"
Abstract : Geiges shows that every 2-connected closed 7-manifold admits a contact structures. In this talk, we use similar approach to prove the existence of contact structures on non-compact, 2-connected G_2 manifolds.

September 11th: A.J. Todd, U of R; "On Deformations of Asymptotically Cylindrical Special Lagrangian Submanifolds"
Abstract: We will first review the definitions of asymptotically cylindrical Calabi-Yau $3$-folds $(X,\om,J,g,\Om)$ and their asymptotically cylindrical special Lagrangian submanifolds $L$; then we will give a brief outline of the proof that the deformation space of asymptotically cylindrical special Lagrangian submanifolds in $X$ near $L$ is itself a smooth manifold. Finally, since the asymptotically cylindrical linear elliptic partial differential operator $(d+*d^*)^p_{2+l,\ga}$ plays an important role in this proof, specifically when it is Fredholm, we will calculate necessary and sufficient conditions on the decay rate $\ga$ that ensure this operator is Fredholm.