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Series: Geometry Topology Seminar

I will sketch how to detect nontrivial higher homotopy groups of the space of complete nonnegatively curved metrics on an open manifold.

Series: Geometry Topology Seminar

The algebraic K-theory of the sphere spectrum, K(S), encodes significant information in both homotopy theory and differential topology. In order to understand K(S), one can apply the techniques of chromatic homotopy theory in an attempt to approximate K(S) by certain localizations K(L_n S). The L_n S are in turn approximated by the Johnson-Wilson spectra E(n) = BP[v_n^{-1}], and it is not unreasonable to expect to be able to compute K(BP). This would lead inductively to information about K(E(n)) via the conjectural fiber sequence K(BP) --> K(BP) --> K(E(n)). In this talk, I will explain the basics of the K-theory of ring spectra, define the ring spectra of interest, and construct some actual localization sequences in their K-theory. I will then use trace methods to show that it the actual fiber of K(BP) --> K(E(n)) differs from K(BP), meaning that the situation is more complicated than was originally hoped. All this is joint work with Ben Antieau and Tobias Barthel.

Series: Geometry Topology Seminar

We prove that the torsion of any smooth closed curve in Euclidean space which bounds a simply connected locally convex surface vanishes at least 4 times (vanishing of torsion means that the first 3 derivatives of the curve are linearly dependent). This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in 3-space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical 4 vertex theorem for planar curves. The proof follows from an extensive study of the structure of convex caps in a locally convex surface.

Series: Geometry Topology Seminar

These are two half an hour talks.Evgeny's abstract: The most useful approach to a classication of 3-manifolds is the complexity theory foundedby S. Matveev. Unfortunately, exact values of complexity are known for few infinite seriesof 3-manifold only. We present the results on complexity for two infinite series of hyperbolic3-manifolds with boundary.Andrei's abstract: We define coordinates on virtual braid groups. We prove that these coordinates are faithful invariants of virtual braids on two strings, and present evidence that they are also very powerful invariants for general virtual braids.The talk is based on the joint work with V.Bardakov and B.Wiest.

Series: Geometry Topology Seminar

Construction of pseudo-Anosov elements of mapping class groups of surfaces is a non-trivial task. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus. In joint work with Hyunshik Shin, we resolve this conjecture for all surfaces.

Series: Geometry Topology Seminar

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)

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.