Oberseminar Geometrie, Topologie und Analysis

Oberseminar Geometrie, Topologie und Analysis

S. Friedl, H. Geiges, G. Marinescu, G. Thorbergsson

Sommersemester 2011

Freitag, 10:30-11:30, Seminarraum 1

13.5.11 Thomas Püttmann (Bochum)
On the commutator of unit quaternions

Abstract: The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this talk is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S³-actions on S⁷×S³ whose quotients are exotic 7-spheres and give a geometric explanation for the order 24 of the stable homotopy groups π_n+3(Sⁿ).

20.5.11 Hiraku Nozawa (Santiago de Compostela)
Cohomology and the Reeb flows of K-contact manifolds

Abstract: A compact manifold M with a contact form whose Reeb flow preserves a metric on M admits a geometric structure called a K-contact manifold. In this talk, we will discuss cohomological properties of K-contact manifolds and its relation to the topology of the Reeb flows. These are closely related to cohomological properties of Hamiltonian torus actions. The equivariant basic cohomology of the Reeb flow will be introduced as our main technical tool. This talk is based on a joint work with Oliver Goertsches and Dirk Töben, which is available on the arXiv.

3.6.11 Dan Coman (Syracuse University)
Tian's theorem for big line bundles

Abstract: A line bundle is called big if the spaces of sections in its high tensor powers have maximal dimension growth (e.g. ample line bundles are big). We show that the zeros of sections in high powers of a big line bundle are asymptotically uniformly distributed with respect to a certain singular Kähler form.

Mittwoch
8.6.11
17:45
S3 Andrew Lobb (SUNY Stony Brook)
Exact sequences in Khovanov-Rozansky knot homologies

Abstract: Long exact sequences are crucial to computation in any homology theory. We describe some new long exact sequences and show how they can be used to give concrete and interesting structure results.

10.6.11 Laszlo Lempert (Purdue University)
The cohomological method in complex analysis

Abstract: One of the great discoveries of twentieth century mathematics was that there is a large group of mathematical problems, of very different origins, that one can formulate in the same language, that of cohomologies. While the cohomological formulation in itself rarely solves those problems, it is still important, because it prepares the ground to the application of general methods and to the discovery of analogies.
In the talk I will survey a line of research in complex analysis, of cohomological nature, starting with Cousin's investigations in the late nineteenth century up to recent developments in infinite dimensions.

24.6.11 Yuya Koda (Tohoku University)
Automorphisms of the 3-sphere that preserve spatial graphs and handlebody knots

Abstract: A graph and a handlebody embedded in the 3-sphere is called, respectively, a spatial graph and a handlebody knot. In this talk, we consider the groups of isotopy classes of automorphisms of the 3-sphere that preserve spatial graphs or handlebody knots. After a brief review of some relevant history, we discuss the finiteness properties of the groups.

1.7.11 Dan Silver (University of South Alabama)
Lehmer's Question and Reidemeister's Knot

Abstract: The 1361st cyclic resultant of the Alexander polynomial of the figure-eight knot is the square of a 285-digit prime. In 1933 this fact would not have surprised the number theorist D.H. Lehmer, but it would have confounded the topologist Kurt Reidemeister. We discuss Lehmer's related question about integral polynomials and its relationship to knot theory.

Dienstag
5.7.11
16:00 Benjamin Himpel (Aarhus)
The asymptotic expansion of the Witten-Reshetikhin-Turaev invariants

Abstract: Witten's influential invariants for links in 3-manifolds given in terms of a non-rigorous Feynman path integral have been rigorously defined first by Reshetikhin and Turaev. Their combinatorial definition based on the axioms of topological quantum field theory is expected to have an asymptotic expansion in view of the perturbation theory of Witten's path integral with leading order term (the semiclassical approximation) given by formally applying the method of stationary phase. Furthermore, the terms in this asymptotic expansion are expected to be well-known classical invariants like the Chern-Simons invariant, spectral flow, the ρ-invariant and Reidemeister torsion. For mapping tori, the Witten-Reshetikhin-Turaev invariants can also be defined as the characters of representations of central extensions of the mapping class group, constructed using the machinery of geometric Kähler quantisation applied to the moduli space of flat connections on a surface. I will present new results on the expansion for finite order mapping tori, whose leading order terms we identified with classical topological invariants. Joint with Jørgen E. Andersen.

H. Geiges, 5.4.02


13.5.11	Thomas Püttmann (Bochum) On the commutator of unit quaternions Abstract: The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this talk is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S³-actions on S⁷×S³ whose quotients are exotic 7-spheres and give a geometric explanation for the order 24 of the stable homotopy groups π_n+3(Sⁿ).
20.5.11	Hiraku Nozawa (Santiago de Compostela) Cohomology and the Reeb flows of K-contact manifolds Abstract: A compact manifold M with a contact form whose Reeb flow preserves a metric on M admits a geometric structure called a K-contact manifold. In this talk, we will discuss cohomological properties of K-contact manifolds and its relation to the topology of the Reeb flows. These are closely related to cohomological properties of Hamiltonian torus actions. The equivariant basic cohomology of the Reeb flow will be introduced as our main technical tool. This talk is based on a joint work with Oliver Goertsches and Dirk Töben, which is available on the arXiv.
3.6.11	Dan Coman (Syracuse University) Tian's theorem for big line bundles Abstract: A line bundle is called big if the spaces of sections in its high tensor powers have maximal dimension growth (e.g. ample line bundles are big). We show that the zeros of sections in high powers of a big line bundle are asymptotically uniformly distributed with respect to a certain singular Kähler form.
Mittwoch 8.6.11 17:45 S3	Andrew Lobb (SUNY Stony Brook) Exact sequences in Khovanov-Rozansky knot homologies Abstract: Long exact sequences are crucial to computation in any homology theory. We describe some new long exact sequences and show how they can be used to give concrete and interesting structure results.
10.6.11	Laszlo Lempert (Purdue University) The cohomological method in complex analysis Abstract: One of the great discoveries of twentieth century mathematics was that there is a large group of mathematical problems, of very different origins, that one can formulate in the same language, that of cohomologies. While the cohomological formulation in itself rarely solves those problems, it is still important, because it prepares the ground to the application of general methods and to the discovery of analogies. In the talk I will survey a line of research in complex analysis, of cohomological nature, starting with Cousin's investigations in the late nineteenth century up to recent developments in infinite dimensions.
24.6.11	Yuya Koda (Tohoku University) Automorphisms of the 3-sphere that preserve spatial graphs and handlebody knots Abstract: A graph and a handlebody embedded in the 3-sphere is called, respectively, a spatial graph and a handlebody knot. In this talk, we consider the groups of isotopy classes of automorphisms of the 3-sphere that preserve spatial graphs or handlebody knots. After a brief review of some relevant history, we discuss the finiteness properties of the groups.
1.7.11	Dan Silver (University of South Alabama) Lehmer's Question and Reidemeister's Knot Abstract: The 1361st cyclic resultant of the Alexander polynomial of the figure-eight knot is the square of a 285-digit prime. In 1933 this fact would not have surprised the number theorist D.H. Lehmer, but it would have confounded the topologist Kurt Reidemeister. We discuss Lehmer's related question about integral polynomials and its relationship to knot theory.
Dienstag 5.7.11 16:00	Benjamin Himpel (Aarhus) The asymptotic expansion of the Witten-Reshetikhin-Turaev invariants Abstract: Witten's influential invariants for links in 3-manifolds given in terms of a non-rigorous Feynman path integral have been rigorously defined first by Reshetikhin and Turaev. Their combinatorial definition based on the axioms of topological quantum field theory is expected to have an asymptotic expansion in view of the perturbation theory of Witten's path integral with leading order term (the semiclassical approximation) given by formally applying the method of stationary phase. Furthermore, the terms in this asymptotic expansion are expected to be well-known classical invariants like the Chern-Simons invariant, spectral flow, the ρ-invariant and Reidemeister torsion. For mapping tori, the Witten-Reshetikhin-Turaev invariants can also be defined as the characters of representations of central extensions of the mapping class group, constructed using the machinery of geometric Kähler quantisation applied to the moduli space of flat connections on a surface. I will present new results on the expansion for finite order mapping tori, whose leading order terms we identified with classical topological invariants. Joint with Jørgen E. Andersen.