Seminars in geometry/topology

September 5, 13:15 - 14:15, room 734, Sentralbygg 2

Håvard Bakke Bjerkevik (NTNU): Stability of Persistence Modules

Abstract: The algebraic stability theorem for pointwise finite dimensional (p.f.d.) \(\mathbb{R}\)-persistence modules is a central result in the theory of stability for persistence modules. We present a stability theorem for \(n\)-dimensional rectangle decomposable p.f.d. persistence modules up to a constant \((2n-1)\) that is a generalization of the algebraic stability theorem. We give an example to show that the constant cannot be improved for \(n=2\). The same technique is then applied to what we call triangle decomposable modules, where we obtain smaller constants. The result for triangle decomposable modules combined with work by Botnan and Lesnick proves a version of the algebraic stability theorem for zigzag modules and the persistent homology of Reeb graphs.

September 12, 13:15 - 14:15, room 734, Sentralbygg 2

Håvard Bakke Bjerkevik (NTNU): Stability of Persistence Modules, part 2

Abstract: The algebraic stability theorem for pointwise finite dimensional (p.f.d.) \(\mathbb{R}\)-persistence modules is a central result in the theory of stability for persistence modules. We present a stability theorem for \(n\)-dimensional rectangle decomposable p.f.d. persistence modules up to a constant \((2n - 1)\) that is a generalization of the algebraic stability theorem. We give an example to show that the constant cannot be improved for \(n = 2\). The same technique is then applied to what we call triangle decomposable modules, where we obtain smaller constants. The result for triangle decomposable modules combined with work by Botnan and Lesnick proves a version of the algebraic stability theorem for zigzag modules and the persistent homology of Reeb graphs.

In the second segment, we look at the proof of the main result. Then we show how the proof can be applied to triangle modules to show stability for zigzag modules.

September 19, 13:15 - 14:15, room 734, Sentralbygg 2

Erik Rybakken (NTNU): Using persistent homology to recover neural coding in head direction neurons

Abstract: Neuroscientists have discovered neurons in mice that encode information about head direction. Depending on which direction the mouse is headed, some neurons will fire more rapidly than usual. The head direction neurons were discovered by comparing neural activity with observed head direction. In this talk I will demonstrate that it is possible to discover the circle of head directions just by analyzing the neural recording, without ever using behavioural data. I will also explain the challenges when there are other types of neurons present in the data, and discuss some potential solutions.

October 3, 13:15 - 14:15, room 734, Sentralbygg 2

Truls Bakkejord Ræder (NTNU): Rational Tambara functors

Abstract: The aim of this talk is to give a new result describing the rational homotopy of the sphere spectrum as a Tambara functor. To do this, I will first describe the additional structure on homotopy groups of orthogonal spectra equipped with both a multiplication and an action of a finite group, when these two types of structure interact closely. Along the way, I will give explicit examples in the more manageable rational setting.

October 24, 13:15 - 14:15, room 734, Sentralbygg 2

Tomasz Rybicki (AGH University of Science and Technology): On Hofer type metrics beyond the symplectic category

Abstract: Given any symplectic manifold the Hofer metric \(\rho_H\) is a bi-invariant metric on the group of compactly supported Hamiltonian symplecto- morphisms of it. Hofer geometry constitutes a basic tool in symplectic topology. The aim of this talk is to define a Hofer metric in a more general context, especially for Poisson manifolds. Concerning contact manifolds, Banyaga and Donato introduced a bi-invariant metric of Hofer type on the strict contactomorphism group of a special kind of contact manifolds. Recently, Müller and Spaeth extended this result to all contact manifolds.

We show that an analogue of the Hofer metric \(\rho_H\) on the Hamiltonian group of a Poisson manifold can be defined in two important classes of Poisson manifolds. First we observe that \(\rho_H\) is a genuine metric when the union of all proper leaves of the corresponding symplectic foliation is dense. Next we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable if it can be realized as the space of units of a symplectic groupoid. Our main result states that \(\rho_H\) is a Hofer type metric for every Poisson manifold which admits a Hausdorff integration.

October 31, 13:15 - 14:15, room 734, Sentralbygg 2

Nathan Perlmutter (Stanford University): Cobordism categories and the moduli spaces of odd dimensional manifolds

Abstract: In this talk I will present recent joint work of mine with Fabian Hebestreit where we prove that certain stable moduli spaces of odd dimensional manifolds are homology equivalent to infinite loopspaces. The main novel ingredient in our work is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces. Our main result can be viewed as the first step toward obtaining an analogue of the Madsen-Weiss theorem for odd dimensional manifolds.

November 7, 13:15 - 14:15, room 734, Sentralbygg 2

Eirik Eik Svanes (ILP Jussieu): Infinitesimal Moduli of Instanton Bundles over G2-Manifolds

Abstract: I will begin by reviewing the infinitesimal combined deformations of a holomorphic bundle over a complex manifold. As shown by Atiyah, the complex structure deformations of the base are obstructed unless the moduli are in the kernel of the Atiyah map. I will then show that there is a very similar story emerging for seven dimensional manifolds with G2 holonomy. I will comment on implications for physics, particularly in the context of compactifications of the heterotic string.

November 21, 13:15 - 14:15, room 734, Sentralbygg 2

Alexander Schmeding (NTNU): Shape spaces of absolutely continuous curves

Abstract: In many applications one seeks to compare shapes, i.e. curves with values in an euclidean space, a Lie group or a Riemannian manifold. For example in computer animation, motions of characters are conveniently described by motions of their underlying skeleton. These motions then correspond to curves on \(\mathrm{SO}(3)\).

Shape comparison is usually done by choosing the geodesic distance of a Riemannian metric on the infinite-dimensional manifold of all shapes. To define such a structure, one can use the so called Square Root Velocity Transform (SRVT). The SRVT framework is convenient for numerical computations, but these computations often "leave" the shape manifold. If the target manifold is a vector space, it has been argued that these computations are more natural in the geodesic and metric completion, which is a space of absolutely continuous curves.

Using some classical constructions from Riemannian geometry, we will show in the talk that this idea generalises to the manifold valued case. As a consequence the numerical computations are thus natively carried out in a manifold of absolutely continuous curves.