Course overview -- Wintersemester 2019/2020 (Archive)

The elliptic genus of K3 is an index for the 1/4-BPS states of its sigma-model. At the torus orbifold point there is an accidental degeneracy of such states. We blow up the orbifold fixed points and show that this fully lifts the accidental degeneracy of the 1/4-BPS states with dimension h=1. Thus, at a generic point near the Kummer surface the elliptic genus measures not just their index, but counts the actual number of these BPS states. Finally, we comment on the implication of this for symmetry surfing and Mathieu moonshine.

Path algebras of quivers with relations naturally appear in algebraic geometry as endomorphism algebras of so-called tilting bundles. In this setting, deformations of such quotients of path algebras can be used to give a concrete description of deformations of the category of (quasi)coherent sheaves as Abelian category, which are known to combine both "classical" deformations of the variety and "noncommutative" algebraic deformation quantizations.

In this talk I will present recent joint work with Zhengfang Wang for describing deformations of path algebras of quivers with relations algebraically / combinatorially. I plan on focussing on examples of geometric origin and will try to explain such deformations from a geometric perspective.

Selberg zeta functions are zeta functions associated with the geodesic flow on a hyperbolic surface and, sometimes, a representation of the fundamental group of the surface. The spectral theory of the Selberg zeta for unitary representations is well-known, but, however, for certain no-unitary representations the Selberg zeta function may even not exist.

In the talk, I would like to suggest a way of the regularization of the Selberg zeta function to such types of non-unitary representations using the transfer operator approach.

Poisson brackets appear in different fields of mathematics and physics, such as the theory of Lie algebras and classical mechanics. This talk is meant to give a short introduction to the subject of Poisson geometry in general and, afterwards, to discuss the connection of Poisson structures and Lie groupoids/algebroids, symplectic geometry and deformation quantization.

In contrast to real and complex line bundles, quaternionic line bundles are not classified by a suitable characteristic class. However, there are results on the classification over four and five dimensional spin manifolds.

This talk is an introduction to quaternionic line bundles and their classification. We will see why the approach to classification in the real and complex cases does not transfer to the quaternionic setting. Based on this, I would like to present different approaches to the classification over spin manifolds of low dimension.

In this talk I will present the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions. We consider potentials in a fixed $L^q$ space with $q\geq 1$. Functional determinants of Sturm-Liouville operators with smooth potentials or with potentials with prescribed singularities have been widely studied, I will present a short review of these results and will explain how to extend the definition of the functional determinant to potentials in $L^q$. The maximization problem turns out to be equivalent to a problem in optimal control. I will explain how we obtain existence and uniqueness of the maximizers. The results presented in the talk are join work with J-B. Caillau (UCDA, CNRS, Inria, LJAD) and P. Freitas (Lisboa).

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Given a hyperbolic 3-manifold M of finite volume, we study a family of twisted Alexander polynomials of M. We show an asymptotic formula for the behavior of those polynomials on the unit circle, and recover the hyperbolic volume as the limit. It extends previous works of Müller (for M closed) and Menal-Ferrer--Porti. This is a joint work with Jerome Dubois, Michael Heusener (Clermont-Ferrand) and Joan Porti (Barcelona).

In this talk I will start by introducing the concept of multipoint Seshadri constants and discuss their relationship with Nagata's conjecture on plane curves. I will then introduce the notion of a Kähler packing and show that there is a direct connection between the multipoint Seshardi constant and the existence of Kähler packings. To end I will provide an example of a Kähler packing of a toric variety which highlights a connection between Kähler packings of certain polytopes and their corresponding varieties.

The Euler characteristic plays a major role as a topological invariant, for example in Euler's polyhedron theorem. It can also be understood as an analytic invariant. The Poincaré-Hopf-Index theorem builds a bridge between these two realms. Furthermore, it is a geometric invariant as in the theorem of Gauß-Bonnet.

In this talk, I will focus on homologies and explain the relation between the Euler characteristic and the so-called Betti numbers, which are the rank of the homology groups of an underlying complex. The Euler characteristic is therefore also an invariant of algebraic objects.

Conformal Field Theory (CFT) is a branch of mathematical physics with many intriguing applications, the most mathematically fruitful of which (to date) has been Moonshine Theory. In physics, CFT plays major roles in the treatment of critical phenomena, String Theory and AdS/CFT correspondence.

In the first part of this talk, I will give a short introduction to unitary two dimensional Euclidean CFTs (in contrast to relativistic QFTs), before truncating them to so called ‘chiral’ CFTs. One crucial element of CFT is the so-called Operator Product Expansion (OPE), which is, in the chiral context, encoded in the structure of a conformal Vertex Operator Algebra (VOA). As an elementary (yet useful) example, the U(1) current of the free boson will be considered and I will show how it fits into a conformal VOA.

The purpose of this talk is to describe the set of generalized complex structures on a maximal flag manifold which are invariant by the adjoint action on the flag, as well as study some of their geometric properties. We will give an explicit expression for the invariant pure spinor line associted with each of these structures. We will characterize all invariant generalized Kähler structures on a maximal flag manifold. Finally, we will describe the quotient spaces determined by the set of all invariant generalized complex (resp. almost Kähler) structures under the action by invariant B-transformations. This is a joint work with Elizabeth Gasparim and Carlos Varea.

The Selberg zeta function is a central object in the study of correlations between spectral and geometric data on hyperbolic orbifolds. Motivated by D. Mayer's seminal investigations of the modular surface, one promising approach relies on a representation of the zeta function as a Fredholm determinant of certain, purpose-built, transfer operators associated with the geodesic flow on the orbifold. In particular, this representation yields a correspondence between the 1-eigenspaces of these operators and zeros of the zeta function, which in turn relate to $L^2$-eigenvalues and resonances of the Laplacian. Based on previous work by A. Pohl and various coauthors, we construct such transfer operator families for a wide class of geometrically finite Fuchsian groups with hyperbolic ends, as well as Banach spaces on which these operators act nuclearly of order 0. This is work in progress, jointly with A. Pohl.

I will explain a construction of symplectic Lefschetz fibrations on adjoint orbits, give examples and an application to mirror symmetry. I will then show that our construction produces a family of examples satisfying the Kontsevich-Katzarkov-Pantev conjecture (this is joint work with Ballico, San Martin, and Rubilar).

Spectral flow, topological twists, chiral rings related to a refinement of the de Rham cohomology and to marginal deformations, spacetime supersymmetry, mirror symmetry. These are some examples of features arising from the N=2 Virasoro chiral algebra of superstrings compactified on Calabi-Yau manifolds. To various degrees of certainty, similar features were also established for compactifications on 7- and 8-dimensional manifolds with exceptional holonomy group G2 and Spin(7) respectively. In this talk, I will explain that these are more than analogies: I will discuss the underlying symmetry connecting exceptional holonomy to Calabi-Yau surfaces (K3) via a limiting process. Based on arXiv:2001.10539.