Course overview -- Wintersemester 2018/2019 (Archive)

I will report on the progress of my phd thesis, presenting a result on the behaviour of the eta invariant of the Dirac and Hodge operator under conic degeneration: under favorable assumptions, realized when e.g. the link of a cone is a space form, the eta invariant tends to the invariant of the degenerated space plus a constant term. I´ll discuss part of the proof and, if time allows, ongoing work for simple edge spaces.

I will review the definition of the Casson-Walker invariant of rational homology spheres and its connection to Feynman graphs. Then I will discuss some recent computations involving the cutting and gluing of these diagrams, and some conjectures that result from these computations.

In my talk I will discuss how one can, in the spirit of some classical results due to Laptev, Safarov and Vassiliev, write the propagator of a class of hyperbolic operators on manifolds as one single oscillatory integral with complex-valued phase function, global both in space and in time. In particular, a refined, geometric version of the method will be presented, in the Riemannian setting: the adoption of a distinguished complex-valued phase function, naturally dictated by the geometric framework, will allow us to visualise the process of circumventing topological obstructions. The microlocal method is explicit and constructive; the calculation of the subprincipal symbol of the propagator enables us to recover asymptotic spectral properties of the operators at hand. I will discuss explicit formulae and recent results for the wave operator. Time permitting, the extension of the method to Lorentzian spacetimes will be briefly analysed. This is joint work with D. Vassiliev (UCL) and M. Levitin (Reading).

The Gaussian distribution is a central object in mathematics and it can be characterised as the unique probability on the real numbers that maximises entropy, for fixed mean and variance. It turns out that the same property can be used to define a discrete Gaussian distribution on the integers. Moreover, the discrete Gaussian is parametrised naturally by the Riemann theta function, and, as such, it has a natural connection to the geometric theory of complex tori, or, more precisely, abelian varieties. The aim of the talk is to present this connection and to show how question in probability give rise to natural problems in geometry and viceversa. This is joint work with Carlos Amendola (TU Munich)

Almost by definition, every Hyperkähler manifold M comes with a 2-sphere of complex structures. These all combine to a family of Kähler manifolds over the complex projective line CP1. Its total space, the twistor space Z of M, has many interesting properties. For example, it carries a real involution covering the antipodal map of CP1. A section of the projection from Z to CP1 is called a real section if it is invariant under the real involution of Z. In case M is the moduli space of Hitchin’s self-duality equations, every solution of these equations determines a real section. Simpson called these preferred sections and posed the question if every real section is a preferred section. A positive answer would imply a complex-analytic way to determine solutions of the self-duality equations. This would be surprising because these are non-linear PDEs. However, a negative answer has recently been given by Heller-Heller by constructing so-called higher sections. In this talk, we show that a symmetric subspace of higher sections is a Hyperkähler manifold which is unexpected because they are a priori unrelated to points of the initial Hyperkähler manifold M. Moreover, we explain the implications to a complex-analytic approach to solutions of the self-duality equations as envisioned by Simpson.

We study $G_2$-orbifolds whose singularities are modelled on $\mathbb R^3\times\mathbb R^4/G$, where $G$ is a finite subgroup of $SU(2)$. Orbifolds of this kind have applications in M-theory and they may define boundary components of the  moduli space of parallel $G_2$-structures. We show how the existing construction methods for $G_2$-manifolds can be  modified such that they produce $G_2$-orbifolds. A recent result of D. Joyce and S. Karigiannis allows us to resolve  the singularities under certain circumstances. Therefore, we are able to construct smooth $G_2$-manifolds with new  values of the second and third Betti number.   

We consider Poisson problems on manifolds with boundary and bounded geometry and assume that they have finite width (that is, that the distance from any point to the boundary is bounded uniformly). We include Robin boundary conditions. As an application, we establish the connection to the Poisson problem on certain domains in the plane and higher dimensional stratified spaces. In particular we get the well-posedness of strongly elliptic equations on domains with oscillating conical singularities, a class of domains that generalizes the class of bounded domains with conical points.
This is joint work with Bernd Ammann (Regensburg) and Victor Nistor (Metz).

I will discuss invariant geometric structures on certain homogeneous spaces, known as $G_2$ flag manifolds. Flag manifolds are known to carry interesting invariant structures, such as a complex structure with compatible Kähler-Einstein metric, as well as other (possibly non-integrable) almost complex structures. These are typically studied from a Lie-theoretic point of view, and are well-understood in that context. However, such algebraic methods shed little light on their geometric origin. In this talk, we will take a complementary, differential-topological approach to studying invariant geometric structures on these manifolds. Besides recovering results typically obtained using Lie theory, we will see that this more geometric approach reveals connections to interesting topics in complex geometry, such as rigidity theorems for Kählerian complex structures, and twistor theory for quaternionic Kähler manifolds.

Factorization algebras are a powerful tool to encode observables in classical and quantum field theory. As suggested by Costello and Gwilliam, to the formal moduli problem describing deformations of flat G-bundles with connections on a manifold M, one can associate a factorization algebra F on M which describes the perturbative aspects of classical Chern-Simons theory on M with structure group G. In the talk I will concentrate on the case of G an abelian group, and show that the factorization algebra F comes naturally equipped with a (homotopy) action of the gauge group Maps(M,G), which can be regarded as a genuine nonperturbative aspect of Chern-Simons theory. Joint work with Corina Keller.

The Bär-Ballmann framework is a comprehensive framework to consider elliptic boundary value problems (and also their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that the induced operator on the boundary is symmetric. Many operators satisfy this requirement including the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general operators with the quintessential example being the Rarita-Schwinger Dirac operator, which is an operator that fails to satisfy this hypothesis.

In this talk, I will present recent work with Bär where we dispense the symmetry assumption and consider general elliptic operators. The ellipticity of the operator still allows us to understand the spectral theory of the induced operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to recover many of the results of the Bär-Ballman framework.

If time permits, I will also touch on the non-compact boundary case, and potential extensions of this to the L^p setting and Lipschitz boundary.

Early hints of mirror symmetry for Calabi-Yau manifolds arose from studying certain vertex operator algebras, intimately connected to string dynamics on these geometries. In my recent article 1809.06376, which the talk will be about, I perform a similar analysis, replacing Calabi-Yau manifolds by 7-dimensional $G_2$ holonomy spaces constructed via the so-called "twisted connected sum" method of Corti, Haskins, Nordstrom, and Pacini. Besides connecting nicely with recent results on the conjectured "mirror symmetry" for $G_2$, this work is a necessary step for applications of twisted connected sums in string theory.

Singular Riemannian foliations are generalizations of smooth isometric group actions. In the setting of compact group actions, torus actions by isometries on a fixed Riemannian  manifold have been studied to understand the topology of the manifold or properties the Riemannian metric might have.

We extend this study to  the setting  of singular Riemannian folaitons by tori. We show that some techniques developed for the study of torus actions can be carried to the foliated setting. In  particular we focus on the case where the foliation has codimenision 2, in order to fix notions.

In this particular case we obtain the following result:

If (M,F) is an singular Riemannian foliation of codimension 2 by tori, on a compact, simply-connected Riemannian manifold, then the foliation is induced by a smooth torus action.

In 1956, John Milnor exhibited the first examples of manifolds homeomorphic, but not diffeomorphic to spheres, since then called exotic spheres. Interesting results on exotic manifolds were obtained through explicit geometric constructions. Here we present a new construction that relates exotic manifolds, such as exotic spheres, flag manifolds and connected sums of them, to their standard counterparts. Such relation is established through a Morita equivalence of action groupoids and is used to produce/reproduce metrics with positive Ricci and almost non-negative curvature. Joint work with L. Cavenaghi.

Already in 1995, physicists Harvey and Moore suggested a multiplication on the space of so-called Bogomol'nyi-Prasad-Sommerfield (BPS) states in certain string theories and claimed to obtain Borcherds-Kac-Moody Lie algebras. These were introduced as a generalization of finite-dimensional semisimple Lie algebras, and I will present a theorem showing that they are indeed the widest generalization retaining existence and some well-behavedness of a grading, (Killing) bilinear form and (Cartan) involution. One direction of this theorem was proven and used by Borcherds in his work on monstrous moonshine. The other direction is considered known as well, but its proof has seemingly never been written down completely due to analogy to the existing literature on Kac-Moody algebras. Its verification, however, turned out not to be as easy as expected. I will sketch the main points of what I felt should be added compared to the literature and give some background on the relation to string theory.

In this talk, we will discuss the spectral properties of the perturbation of the generalized anharmonic oscillator. We consider a piecewise Hölder continuous perturbation, and investigate how the Hölder constant might affect on the eigenvalues. More precisely, we derive the first several terms in the asymptotic expansion for the eigenvalues.

Persistent homology is a tool for topological data analysis, that can help to analyse deformed geometric shapes like connected components, circles, voids and higher dimensional homology. The computation of persistent homology is based on the construction of a filtered cell complex and scales roughly cubic in the number of cells. Discrete Morse theory reduces the number of cells in a complex without changing its homology. In 2013 Vidit Nanda and Konstantin Mischaikow used filtration-wise Morse reductions to proof a speed up for certain persistent homology computations and implemented the software Perseus.

In practice, many filtered cell complexes grow by one simplex per filtration value and cannot be reduced by Nanda and Mischaikow's approach, e.g. Cech complexes. This talk will show some ideas how to trade off an approximated result for a faster computation. This effect can be explained by allowing small deviations from the elder rule. The new construction of an induced filtered acyclic matching helps for an informed choice of the approximation parameter. Also, the theoretical construct of pairings on a graded multiset of real numbers unifies persistent homology and filtered acyclic matchings. As an aside this allows the purely combinatorial proof of a filtered Euler formula for all such pairings.