Seminars and Colloquia by Series

Progress on homogeneous Einstein manifolds

Series
Geometry Topology Seminar
Time
Monday, October 13, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andreas Arvanitoyeorgos – University of Patras –
A Riemannian manifold (M, g) is called Einstein if the Ricci tensor satisfies Ric(g)=\lambda g. For a Riemannian homogeneous space (M=G/H,g), where G is a Lie group and H a closed subgroup of G, the problem is to classify all G-invariant Einstein metrics. In the present talk I will discuss progress on this problem on two important classes of homogeneous spaces, namely generalized flag manifolds and Stiefel manifolds. A generalized flag manifold is a compact homogeneous space M=G/H=G/C(S), where G is a compact semisimple Lie group and C(S) is the centralizer of a torus in G. Equivalently, it is the orbit of the adjoint representation of G. A (real) Stiefel manifold is the set of orthonormal k-frames in R^n and is diffeomorphic to the homogeneous space SO(n)/SO(n-k).One main difference between these spaces is that in the first case the isotropy representationdecomposes into a sum of irreducible and {\it non equivalent} subrepresentations, whereas in thesecond case the isotropy representation contains equivalent summands. In both cases, when the number of isotropy summands increases, various difficulties appear, such as description of Ricci tensor, G-invariant metrics, as well as solving the Einstein equation, which reduces to an algebraic system of equations. In many cases such systems involve parameters and we use Grobner bases techniques to prove existence of positive solutions.Based on joint works with I. Chrysikos (Brno), Y. Sakane (Osaka) and M. Statha (Patras)

Some contact embeddings to the standard 5-sphere

Series
Geometry Topology Seminar
Time
Monday, October 6, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ryo Furukawa – University of Tokyo
In this talk we consider the contact embeddings of contact 3-manifolds to S^5 with the standard contact structure.Every closed 3-manifold can be embedded to S^5 smoothly by Wall's theorem. The only known necessary condition to a contact embedding to the standard S^5 is the triviality of the Euler class of the contact structure. On the other hand there are not so much examples of contact embeddings.I will explain the systematic construction of contact embeddings of some contact structures (containing non Stein fillable ones) on torus bundles and Lens spaces.If time permits I will explain relation between above construction and some polynomials on \mathbb C^3.

Trisections of 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, September 22, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Gay – University of Georgia
This is joint work with Rob Kirby. Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds; a Heegaard splitting splits a 3-manifolds into 2 pieces each of which looks like a regular neighborhood of a bouquet of circles in R^3 (a handlebody), while a trisection splits a 4-manifold into 3 pieces of each of which looks like a regular neighborhood of a bouquet of circles in R^4. All closed, oriented 4-manifolds (resp. 3-manifolds) have trisections (resp. Heegaard splittings), and for a fixed manifold these are unique up to a natural stabilization operation. The striking parallels between the two dimensions suggest a plethora of interesting open questions, and I hope to present as many of these as I can.

4-manifolds can be surface bundles in many ways

Series
Geometry Topology Seminar
Time
Monday, September 15, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nick Salter – University of Chicago
An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the story is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on a quantitative version of the “how many?” question.

Transverse Surgery on Knots in Contact 3-Manifolds

Series
Geometry Topology Seminar
Time
Wednesday, August 27, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
James Conway – Georgia Tech
We will define transverse surgery, and study its effects on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

Dynnikov’s Coordinates

Series
Geometry Topology Seminar
Time
Monday, August 25, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 00-TBA
Speaker
Oyku Yurttas – Georgia Tech
In this talk I will explain the Dynnikov’s coordinate system, which puts global coordinates on the boundary of Teichmuller space of the finitely punctured disk, and the update rules which describe the action of the Artin braid generators in terms of Dynnikov’s coordinates. If time permits, I will list some applications of this coordinate system. These applications include computing the geometric intersection number of two curves, computing the dilatation and moreover studying the dynamics of a given pseudo-Anosov braid on the finitely punctured disk.

Tightness and Legendrian surgery

Series
Geometry Topology Seminar
Time
Thursday, July 10, 2014 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andy Wand – University of Nantes
A well known result of Giroux tells us that isotopy classes ofcontact structures on a closed three manifold are in one to onecorrespondence with stabilization classes of open book decompositions ofthe manifold. We will introduce a characterization of tightness of acontact structure in terms of corresponding open book decompositions, andshow how this can be used to resolve the question of whether tightness ispreserved under Legendrian surgery.

An ODE associated to the Ricci flow

Series
Geometry Topology Seminar
Time
Monday, June 16, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Atreyee Bhattacharya – Indian Institute Of Science
In this talk we will discuss an ODE associated to the evolution of curvature along the Ricci flow. We talk about the stability of certain fixed points of this ODE (up to a suitable normalization). These fixed points include curvature of a large class of symmetric spaces.

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