Zeit und Ort

Montag, 20.7.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Prime geodesic theorems describe the asymptotic behavior of the distribution of prime closed geodesics. In particular, for a hyperbolic surface, the prime geodesic counting function has the same leading asymptotic as the prime number counting function. The optimal bound on the remainder term remains, however, an open conjecture. In this talk, we will discuss the so-called Gallagherian PGT; that means, we allow the asymptotics to fail on a set of finite logarithmic measure. We improve and extend recent results of this type established by Gušić for compact locally symmetric spaces to new settings. This is a joint work with Jørgen Olsen Lye.

Zeit und Ort

Montag, 13.7.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Given a Morse function on a closed smooth manifold and a Morse-Smale pseudo-gradient vector field adapted to it, one can construct a topological category called the flow category associated with this data. Its objects are the critical points of the function and its morphism spaces are the spaces of possibly broken gradient trajectories connecting critical points. This topological category models an infinity category, which can be thought of as providing a homotopy coherent notion of composition of unbroken trajectories connecting critical points. On the other hand, such trajectories determine paths on the manifold, that have the property to be exit paths with respect to the stratification by the stable manifolds of the gradient. A construction of Lurie associates an infinity category to this stratification, which can be thought of as providing a notion of homotopy coherent composition of exit paths. I will define these objects more precisely and then present a result I obtained asserting that these two infinity categories are equivalent.

Zeit und Ort

Montag, 6.7.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

For a function \(f\) on a topological space \(X\), Cohen-Steiner--Edelsbrunner--Harer introduced the notion of extended barcode of \(f\). It is a combinatorial gadget that compactly stores information about, in particular, homology of any interlevel set \(f^{-1}([a,b])\) (over a fixed field \(\mathbb F\)). When \(X\) is a manifold (possibly with boundary) and \(f\) is Morse without multiple critical values, we propose an equivalent, geometrical definition of extended barcode. It relies on a Bruhat decomposition, which describes the relative position of two complete flags in a vector space. This approach allows us to answer the following question. Given two manifolds each equipped with a Morse function, choose a diffeomorphism identifiying (parts of) their boundaries. How to obtain the barcode for the function on the glued manifold in terms of two initial barcodes and a given diffeomorphism?

To make the talk (more) accessible to non-topologists, a fraction of the time will be devoted to Bruhat decomposition, which only requires linear algebra.

Zeit und Ort

Montag, 29.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

An irreducible \(G_2\)-manifold is a Riemannian 7-manifold \(M\) with holonomy group equal to the exceptional Lie group \(G_2\). When \(M\) is closed, the Teichmüller space \(T(M)\) of \(G_2\) metrics on \(M\) divided by diffeomorphisms isotopic to the identity is a smooth, finite-dimensional manifold by a result of Joyce. Yet its topology, and that of its quotient by the smooth mapping class group, remains elusive. Using ideas of Crowley, Goette, and Hertl, we exhibit the first known example of a \(G_2\)-manifold \(M\) together with infinitely many diffeomorphisms that both act freely on \(T(M)\) and preserve a connected component. The diffeomorphisms are 7-dimensional analogs of diffeomorphisms of K3 surfaces constructed recently by Farb and Looijenga, and much like the Farb-Looijenga examples, these diffeomorphisms minimize topological entropy among their isotopy class.

Zeit und Ort

Montag, 22.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

A waveguide is a spatial region \(\Omega \subset \mathbb{R}^{n}\) of tubular form, obtained as a local perturbation of a straight tube \(\Omega_{0} = \mathbb{R} \times B_{1}(0)\). Classically, a particle moving inside such a waveguide and undergoing regular reflections at the boundary will, for almost all initial conditions, eventually leave every bounded region in finite time. In contrast, the quantum-mechanical behaviour can be drastically different. One of the central and perhaps surprising phenomena in the theory of quantum waveguides is that the existence of bound states is closely related to the geometry of the underlying tube.

In the non-relativistic setting, the system is typically described by the Dirichlet Laplacian, and the relation between geometry and discrete spectrum has been studied extensively since the 1980s. However, the study of the relativistic case has just began in recent years.

In this talk, we consider non-uniform relativistic quantum waveguides, modelled by a Dirac operator subject to infinite mass boundary conditions. Under certain assumptions on a localised geometric deformation of the waveguide, we prove the existence of at least one discrete eigenvalue in the spectral gap of the straight waveguide. This eigenvalue corresponds to a geometrically induced bound state, giving a relativistic analogue of a well-known phenomenon from the non-relativistic theory.

Zeit und Ort

Montag, 15.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Morse theory serves as a way of studying the topology of manifolds using critical points of smooth functions. For example, Morse homology - which is defined by counting gradient flow lines between critical points of index difference 1 - recovers the singular homology of the manifold. However, it is known that the collection of moduli spaces of gradient flow lines of arbitrary index difference captures even the homotopy type of the manifold. It is therefore a natural question how a given invariant of the manifold can be understood through Morse theoretical constructions. In this talk we consider transport functions as a way of studying the isomorphism classes of principal bundles over a manifold. We will further see that a transport function with values in a topological group G and a right G-space F gives rise to a chain complex. The homology of this chain complex is isomorphic to the singular homology of an associated bundle. If G is a Lie group then a transport function can be obtained from actual parallel transport and the induced chain complex is isomorphic to one defined in the style of Barraud-Damian-Humilière and Oancea.

Zeit und Ort

Montag, 18.5.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Llarull proved that the round sphere is extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized this result and proved that scalar-rigid maps are Riemannian submersions. In this talk, I present a recent generalization showing that a scalar-rigid f: M → N is not only a Riemannian submersion, but that M is essentially a Riemannian products of the base manifold with a Ricci-flat fiber. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator. This is joint work with Oskar Riedler.

Zeit und Ort

Montag, 11.5.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

The existence of closed geodesics on compact Riemannian manifolds has been studied for a long time. In this talk, I present the ideas of Asselle and Mazzucchelli to generalize results for the compact case to the larger class of manifolds without close conjugate points at infinity, i.e. manifolds such that every sufficiently short geodesic outside a given compact set does not contain any conjugate points. Furthermore, I compare their approach with a similar one by Benci and Giannoni involving the sectional curvature of the manifold.