Veranstaltungsübersicht -- Sommersemester 2019 (Archiv)

In my talk I will discuss some elements of the proof of the topological recursion for the weighted Hurwitz numbers. The main ingredient is the tau-function - the all genera generating function, which is a solution of the integrable KP or Toda hierarchy. My talk is based on a series of joint papers with G. Chapuy, B. Eynard, and J. Harnad.

In a paper by Harvey, Lee and Murthy, the authers calculated the elliptic genera of ADE singularities as the partition of some gauged linear sigma models using the technique called supersymmetric localization. In this talk, I will give a free field construction of these elliptic genera and talk about the geometric interpretation.

Geometric flows are a natural non-perturbative approach to the construction of special holonomy metrics. Several flows have been proposed in different settings, such as the Kähler--Ricci flow and the Laplacian flow. The spinor flow is a unified approach for all Ricci flat special holonomy manifolds based on the spinorial characterization of such metrics. In this talk I will discuss its definition (due to Ammann, Weiß, Witt), its relationship to other flows and several recent results concerning its behavior.

The dominant energy condition implies an inequality for the induced initial value pair on a spacelike hypersurface of a Lorentzian manifold. In this talk, we want to study the space of all initial value pairs that satisfy this inequality strictly. In order to do so, we introduce a Lorentzian alpha-invariant for initial value pairs, and compare it to its classical counterpart. Recent non-triviality results for the latter will then imply that this space has non-trivial homotopy groups.

It is a fact of life that the Lie algebra of isometries of a riemannian manifold is a filtered Lie algebra whose associated graded Lie algebra is a subalgebra of the euclidean algebra: the Lie algebra of isometries of the flat model of riemannian geometry. It is also a fact of life, more recently understood, that the Lie superalgebras which arise as supersymmetries of supergravity backgrounds too are filtered Lie superalgebras whose associated graded Lie superalgebra is a subalgebra of the Poincaré superalgebra: the supersymmetry algebra of the flat supergravity background. The cohomology theory governing such filtered deformations is generalised Spencer cohomology. In this talk I will review these facts and describe some consequences of the calculations of generalised Spencer cohomology for Poincaré superalgebras in different dimensions: including what could be considered a cohomological derivation of eleven-dimensional supergravity and a determination of the possible lorentzian 4- and 6-dimensional manifolds admitting rigid supersymmetry. This is based on collaborations with Andrea Santi and Paul de Medeiros.

A manifold M is called an E-manifold if it has homology only in even dimensions, ie. H_{2k+1}(M;Z) = 0 for all k. Examples include complex projective spaces and complete intersections. We consider 8-dimensional simply-connected E-manifolds. Those that have Betti numbers b_2 = r and b_4 = 0, and fixed second Stiefel-Whitney class w_2 = w form a group \theta(r;w), which acts on the set of E-manifolds with b_2 = r and w_2 = w. The classification of E-manifolds based on this action consists of 3 steps: computing \theta(r;w), classifying the set of orbits and finding the stabilizers. In this talk I will present results in each of these steps, as well as an application, the proof of a special case of Sullivan's conjecture about complete intersections.

In this talk I will review the parametrix construction in the b-calculus by describing in detail a concrete example, the construction of the resolvent for the Laplacian on functions for an asymptotically Euclidean manifold. I will mostly follow the two papers Resolvent at Low Energy and Riesz Transform for Schrödinger Operators on Asymptotically Conic Manifolds. I & II by C. Guillarmou and A. Hassell.

Ein Schrödinger-Operator auf einer Fläche $S$ ist definiert als Summe aus dem Laplace-Operator mit einem Potential $V \in C_0(S)$. Wir interessieren uns hierbei speziell für die Multiplizität des zweiten Eigenwerts über geschlossenen zusammenhängenden Flächen. Y. Colin de Verdière hat die Vermutung aufgestellt, dass sich deren Supremum explizit über die Färbungszahl der Fläche ausdrücken lässt. Wir wollen dies mit einer Abschätzung über die Eulercharakterisik untermauern, in dem wir uns spezielle zweifache Überlagerungen für die Flächen betrachten und dafür eine verwandte Version des Borsuk Ulam Theorems zeigen.

Moduli spaces of solutions to non-linear elliptic PDE's such as instantons in gauge theory are fundamental for the construction of counting invariants. Using information about the solution space provided by the index theory of an approximating family of linear differential operators, we explain our results on orientations for moduli spaces, including new developments in G2-holonomy.

Spin(7) is one of the exceptional holonomy groups. Spin(7)-manifolds are in particular Ricci flat. The condition for a Spin(7)-structure to be torsion-free gives rise to a complicated system of non-linear differential equations. One of the most fundamental ways to solve differential equations is to use symmetries to reduce the number of variables and complexity. For exceptional holonomy manifolds this can only be used in the non-compact setting. I will explain the construction of Spin(7)-manifolds with cohomogeneity one. Here the non-linear PDE system is reduced to a non-linear ODE system. I will give an overview over previous work and mention recent progress.

After a short introduction to Jacobi related geometries, such as Poisson, symplectic, contact and generalized complex/contact manifolds, and their appearance in mathematical physics,  I want to present some results on their (semi-)local structure around transversal submanifolds, so-called "Normal forms". They can be seen as generalization of the Weinstein splitting theorem for Poisson manifolds and they induce in fact a very explicit local description  of Jacobi-related structures.

The second part of the talk is intended to focus on a special Jacobi related geometry: generalized contact bundles, the odd-dimensional counterparts of generalized complex manifolds. I want to show that their global existence is cohomologically obstructed by means of a spectral sequence. At the end I want to give some classes of examples of generalized contact structures.   

The problem of determining the essential self-adjointness of a differential operator on a smooth manifold, and its powers, is an important and well studied topic. One of the primary motivations for studying the essential self-adjointness of a differential operator $D$, comes from the fact that it allows one to build a functional calculus (of Borel functions) for the closure of that operator. Such a functional calculus is then used to solve partial differential equations on a manifold, defined through the operator. In this talk, I will present joint work with L. Bandara where we consider the question of essential self-adjointness of first order differential operators, and their powers, in the context of non-smooth metrics on noncompact manifolds. Using methods from geometry and operator theory we are able to show that essential self-adjointness, at its heart, is an operator theoretic condition which requires minimal assumptions on the geometry of the manifold. Applications to Dirac type operators on Dirac bundles will be discussed.

Let n>1 be an integer, and B^n be the unit ball in C^n. K\subset B^n is a compact subset or {z_1=0=z_2}. By using developing map and Hartogs' extension theorem, we show that a Kaehler metric on B^n\K with constant holomorphic sectional curvature uniquely extends to the ball. This is a joint work with Si-en Gong and Hongyi Liu.

J. Nitsche showed that an isolated singularity of a hyperbolic metric is either a cone singularity or a cusp one. M. Heins proved on compact Riemann surfaces a classical existence theorem about singular hyperbolic metrics where the Gauss-Bonnet formula is the necessary and sufficient condition. We prove that a developing map of a singular hyperbolic metric on a compact Riemann surface has a Zariski dense monodromy group in PSL(2;R). Moreover, we also provide some evidences to the conjecture that it be also the case on a noncompact Riemann surface which admits no non-trivial negative subharmonic function. This is a joint work with Yu Feng, Yiqian Shi, Jijian Song.

Landau-Ginzburg models are a family of quantum field theories characterized by a polynomial (satisfying some conditions) usually called ‘potential’. Often appearing in mirror-symmetric phenomena, they can be collected in categories with nice properties that allow direct computations. In this context, it is possible to introduce an equivalence relation between two different potentials called `orbifold equivalence’. We will present some recent examples of this equivalence, and discuss the computational challenges posed by the search of new ones. Joint work with Timo Kluck.

We’ll present a refinement of Riemann-Roch-Grothendieck theorem on the level of differential forms for families of curves with hyperbolic cusps. The study of spectral properties of the Kodaira Laplacian on a Riemann surface, and more precisely of its determinant, lies in the heart of our approach.

When our result is applied directly to the moduli space of punctured stable curves, it expresses the extension of the Weil-Petersson form (as a current) to the boundary of the moduli space in terms of the first Chern form of a Hermitian line bundle, which provides a generalisation of a result of Takhtajan-Zograf. 

If time permits, we will explain how our result implies some bounds on the growth of the Weil-Petersson form near the compactifying divisor of the moduli space of punctured stable curves. This would permit us to give a new approach to some well-known results of Wolpert on the Weil-Petersson geometry of the moduli space of curves.

Monopoles are pairs formed of a connection and an endomorphism of the bundle that satisfy the Bogomolny equation. There is ample literature on the study of monopoles on R3 under the constraint that the eigenvalues of the endomorphism on the sphere at infinity are distinct, the so-called maximal symmetry breaking case. In joint work with Ákos Nagy, we are exploring monopoles with arbitrary symmetry breaking on R3, and in particular their Nahm transform.

We will present the main result of an article by Corvino on scalar curvature deformation.