Polynomiality phenomena related to fully packed loops and similar objects

My area of research lies in algebraic and enumerative combinatorics and is centred arround fully packed loops (FPLs). There exists a variety of other combinatorial objects such as alternating sign triangles (ASTs) or alternating sign matrices (ASMs) which are equinumerous to FPLs. My aim is to gain a better understanding of these objects and their relations between each other by studying polynomiality phenomena of refined enumerations.

Supervisor: Ilse Fischer

Eisenstein Cohomology of Arithmetic Groups - The Exceptional Group G_2

Supervisor: Joachim Schwermer

sascha.biberhofer@univie.ac.at

Strange Attractors and Inverse Limit Spaces

From 1960's when Edward Lorenz introduced a mathematical model known as Lorenz system with a so called strange attractor (an attracting set with zero measure and fractal dimension) the study of strange attractors is a topic of ongoing interest in dynamical systems. However, the topological structure of some very simple strange attractors is still poorly understood.

Inverse limit spaces were introduced in a branch of Topology called Continuum Theory (a continuum is a compact connected metric space) as a useful tool to describe interesting continua. In 1990's a connection between strange attractors and inverse limit spaces was established by Marcy Barge et al..

In my PhD project the toplogical sturcture of inverse limits is studied with an emphasis on the inverse limit spaces that appear as strange attractors of some planar diffeomorphisms such as Henon-like attractors.

Supervisor: Henk Bruin

Homepage: www.mat.univie.ac.at/~cinc/

Tiling problems and electrostatics

I am interested in problems that lie at the intersection between combinatorics and statistical physics. More specifically, I study enumerative and asymptotic aspects of rhombus tilings of hexagonal regions that contain holes or defects within their interior. The methods used to approach such problems are often highly combinatorial, while the 'effect' that the holes have on the arrangements of the tiles appears to be governed by well-known physical laws that arise in electrostatics.

Supervisor: Christian Krattenthaler

Homepage: http://homepage.univie.ac.at/tomack.gilmore/

Combinatorics of Affine Weyl Groups

My main research interest is in algebraic and enumerative combinatorics.

In particular, I am interested in the role that combinatorial objects such as integer partitions, Dyck paths and parking functions play in the representation theory of Weyl groups - certain reflection groups with connections to geometry, group theory as well as combinatorics. The simplest and best understood family of Weyl groups are the symmetric groups. My goal is to generalise known results for permutation groups to further families.

Supervisor: Christian Krattenthaler

Innovative Signal Representation in Machine Learning (ISiRIM) - Methods for Music

A huge amount of audio data is available every day which needs to be structured and understood. In machine learning, music information retrieval is an important branch with audio classification as one central application. The advent of convolutional neural networks has led to significantly improved results for typical audio classification tasks. In order to establish a meaningful theory and mathematical understanding of the underlying principles, I will investigate three pillars which decisively influence the success of the classification, namely signal representation, transform domain modeling and dimensionality reduction. Signal representations provide the first step in gaining information from raw audio data, considering the scattering transform of Mallat. Further processing steps will be performed on the obtained coefficients in the transform domain. Therefore, sparsity methods will be used to exploit prior knowledge to gain features to be used as input to the machine learning algorithms. Since a sizable number of training samples is required to achieve satisfying learning results, methods derived from Laplacian and Schrödinger eigenmaps will be used to perform dimensionality reduction. In order to find an appropriate feature representation, my thesis will answer questions considering a possible acceleration of the computational steps in learning algorithms and whether the amount of necessary training data can be reduced by using a special representation.

Supervisor: Monika Dörfler

Quantitative Photoacoustic Imaging in the Acoustic Regime

In Photoacoustic Imaging, the interior of a specimen is analyzed by illuminating it with a short laser pulse and observing the acoustic wave which is induced via the photoacoustic effect. It is a coupled physics imaging modality that combines high resolution of ultrasound and high contrast of electromagnetic absorption. My thesis project concerns the quantitative recovery of acoustic material parameters and the source in photoacoustic tomography.

Supervisor: Otmar Scherzer

Spectral theory of the dbar-Neumann problem

On complex manifolds, there is an analog of the de Rham complex which is useful in studying the complex structure of the manifold, as well as its holomorphic functions. Choosing a Hermitian metric, one may study this complex using techniques from Hilbert space theory. We are interested in the spectral properties of (a certain closed extension of) the Laplacian associated to this complex.

Supervisor: Friedrich Haslinger

Supervisor: Hermann Schichl, Radu Ioan Boţ

Dimension reduction with orthogonal projectors and numerical applications in biomedical image analysis

Dimension reduction is an active field in high-dimensional data analysis to reduce computational time and costs. In my PhD project I am dealing with dimension reduction based on orthogonal projectors with applications in high-dimensional biomedical image data. Deriving new mathematical results that can be applied for clinical studies or even clinical routine in collaboration with the prestigious Vienna Reading Center (at the eye clinic of the Medical University Vienna) are of main interest.

The Grassmannian as the space of orthogonal projectors of rank k offers an useful framework for our mathematical work. On the other hand the Vienna Reading Center provides us with clinical data of the human retina yielding difficult automated image analysis tasks (such as classification, quantification,...) that require dimension reduction. Due to manually annotated training data we are able to work with supervised learning methods, such as (deep) neural networks, for automatization.

Nonorthogonal dimension reducing procedures together with learning techniques have already been successfully applied by us for automated quantification of retinal fluid. In further work we aim to use particular orthogonal projectors for dimension reduction based on our mathematical results.

Supervisor: Martin Ehler

The Geometry of the Space of Arcs

My main research interest is to do algebraic geometry and commutative algebra in infinite-dimensional situations. One example is the study of arc spaces of algebraic varieties in the context of singularity theory. Arc spaces were first introduced by John F. Nash, who famously conjectured a relationship between certain families of arcs and exceptional divisors on resolutions of the original variety. In my dissertation project, I take a slightly different approach by studying the singularities of the arc space itself - motivated, among others, by the Grinberg-Kazhdan-Drinfeld theorem. On the algebraic side, this approach leads to the study of formal series rings in a countable number of indeterminates and their properties.

Supervisor: Herwig Hauser

Post-Lie algebra structures

The aim of my research is to study post-Lie algebra structures. A post-Lie algebra structure is a pair of two Lie algebras (over the same field and with the same dimension) admitting a bilinear product satisfying certain identities.

Post-Lie algebra structures arise naturally as a generalization of pre-Lie algebra structures and LR-structures; however, they also arise in other contexts, e. g. in connection with nil-affine actions on Lie groups, in the theory of Koszul operads and in the study of dynamical systems.

Some of the main questions occurring in this area are: Under which conditions does a pair of Lie algebras admit a post-Lie algebra structure? Under which conditions can one classify all post-Lie algebra structures on a given pair of Lie algebras?

Supervisor: Dietrich Burde

Singularity theorems in Lorentzian Geometry

In my dissertation project we look at the classical singularity theorems of General Relativity and try to generalize them to lower regularity metrics. We also hope to gain a better understanding of the singularity theorems in general by extending results from Riemannian comparison geometry to Lorentzian manifolds.

Supervisor: Michael Kunzinger

Homepage: grafmelanie.wordpress.com

Chaotic behaviour of falling balls

In classical mechanics, a large group of systems considers the behaviour of colliding particles. It is especially interesting to know if such a system exhibits some sort of chaotic or unpredictable behaviour. An example of this is Wojtkowski's system of falling balls; this is a dynamical system in which particles, identified as balls, move up and down a vertical line, colliding with each other elastically and with a fixed rigid floor at the origin. The system is chaotic; it has non-zero Lyapunov exponents (i.e. one of many ways to measure chaotic behaviour of a dynamical system) and in particular sensitive dependence on initial conditions. Natural singularities, such as the simultaneous collisions of three or more particles, make the system even harder to analyse. The question of how to prove statistical properties of this system, despite the chaos and singularities, is a challenging task. A major step to achieve this is ergodicity (proven for two bouncing balls, but not for more). One can think of ergodicity as predictability on average. Whether this system is ergodic or not, is the main question I am working on.

Supervisor: Henk Bruin

Dispersive estimates for radial Schrödinger operators

In my research I currently deal with one-dimensional Schrödinger operators on the positive

halfline with a singularity at zero. The aim is to investigate decay properties of the solutions to the corresponding Schrödinger equation for large times. These kinds of operators arise naturally when considering certain higher dimensional models and therefore another future goal will be, to find out, if results for higher dimensions can be improved by the estimates obtained in dimension one.

Supervisors: Aleksey Kostenko, Gerald Teschl

Costs and the base of the Borel reducibility hierarchy

Over the last decades, countable Borel equivalence relations became objects of great interest in descriptive set theory. Several notions of reducibility, most prominently Borel reducibility and measure reducibility, have been the focus of research. The presence of a measure often allows one to gain much more insight than one is able to get in a purely Borel context. A well-known example of this comes from the notion of the $\mu$-cost of an $E$-invariant Borel probability measure $\mu$, which provides a plethora of theorems, including a positive answer to a weak version of a dynamic analogue of the von Neumann conjecture.

It has been known for some time that there are continuum-many pairwise incomparable countable Borel equivalence relations under Borel reducibility. Nevertheless, both under Borel and measure reducibility it remains unknown whether there exist successors of $E_0$. My research efforts have been focused on generalizing the notion of cost to quasi-invariant probability measures. The other goal of this PhD project is to systematically scrutinize possible candidates that arise from as actions of algebraic groups, using methods from ergodic theory and descriptive set theory.

Supervisor: Benjamin Miller

Supervisor: Sy Friedman

Supervisor: Herwig Hauser

Joint Spectral Radius and Subdivison Schemes

Supervisor: Maria Charina

Homepage: www.tommsch.com

Global variational methods for evolution equations

We are interested in the many aspects of the analysis of nonlinear evolution equations, mostly of parabolic type. We have recently turned attention also to the possibility of formulating suitable global variational principle for evolution.

Supervisor: Ulisse Stefanelli

Orbifold equivalence and knot homologies

The aim of my thesis is to relate certain knot homologies. The latter are homology theories assigned to embeddings of a circle into three-dimensional space. Their construction is such that isotopic embeddings are mapped to isomorphic homologies. In other words, knot homologies are knot invariants. The first such theories arose around the year 2000 and have already entailed significant progress in low dimensional topology and various related fields. We aim to apply higher categorical constructions inspired by physics to broaden the class of knot homologies ideally yielding computationally accessible new knot invariants.

Supervisor: Nils Carqueville

ODE-closedness of function spaces

In my dissertation project I am concerned with the study of ODE-closed spaces. Roughly speaking, those are function spaces with no loss of regularity when solving certain ordinary differential equations.

Since this concept has not yet been studied in an abstract way, even fixing a satisfactory definition is a delicate task. Therefore I plan on exploring various classical function spaces from the viewpoint of ODE-closedness and try to extract a common abstract background yielding hints towards a general theory.

Supervisor: Armin Rainer

Trefftz approximation of time-harmonic wave problems

I am working in the field of numerical mathematics, more specifically on the design and the theoretical analysis of numerical methods for the approximation of time-harmonic wave propagation problems. My focus lies on finite element methods based on Trefftz-type approximating spaces, namely spaces containing functions whose restriction to each element belongs to the kernel of the considered differential operator. Being able to locally reproduce the behaviour of the analytical solutions, these methodologies allow to obtain a given accuracy with less degrees of freedom, as compared to standard polynomial-based finite element methods

Supervisor: Ilaria Perugia

Local adaptation and divergence with gene flow: the role of epistasis

Supervisor: Reinhard Bürger

**Refined Enumeration of Alternating Sign Matrices and Domino Tilings of Aztec Rectangles**

I am interested in finding refined enumeration of alternating sign matrices, a class of combinatorial objects which are fascinating to study because of its inherent simplicity, beauty and connections with other subjects. The methods used to study these objects come not only from combinatorics but also from statistical physics and other areas of mathematics. I am also interested in enumerating tilings of combinatorial objects like domino tilings of Aztec rectangles and lozenge tilings of hexagons.

Supervisor: Ilse Fischer

Homepage: http://www.manjilsaikia.in

Topological Quantum Field Theories and the Tricategory of Bimodule Categories

I am interested in Topological Quantum Field Theories (TQFTs) and higher category theory. TQFTs originated as an object of study in theoretical physics, however in 1988 Atiyah gave an axiomatic definition in terms of categories. More concretely: An n-dimensional TQFT is a symmetric monoidal functor from a category of bordisms representing n-dimensional space-time to a category containing algebraic data, such as the the category of vector spaces.

If the bordisms are decorated with 'defect data' one can associate to an n-dimensional TQFT an n-category (in principle) and study the TQFT in terms of this construction.

My research deals with TQFTs in 3 dimensions. In particular I investigate the tricategory consisting of fusion categories, their module-categories, as well as module- functors and transformations. This categorifies the more classical 2-category of rings, bimodules and intertwiners. A first goal is to identify so called 'orbifolding data' which can be used to build new TQFTs out of old ones. An example of such data should be given by group representations in this tricategory.

Supervisor: Nils Carqueville

Supervisor: Michael Schlosser

Reproducing pairs and flexible time-frequency representations

Real-life signals, like a piece of music, show particular time-frequency characteristics in many cases. Moreover, a nonstationary time-frequency resolution is desirable in many applications for example when mimicing the human auditory system. This calls for adaptive and adaptable representations. From a mathematical point of view, frame theory provides the appropriate toolbox to deal with this. A frame is a family of vectors in a Hilbert space that gives rise to a bounded and invertible analysis/synthesis process. There are however systems which are total but do not satisfy the conditions defining a frame. One possible solution for such a situation is given by the concept of reproducing pairs, i.e., two vector families instead of a single one to construct a bounded and invertible analysis/synthesis process. This PhD project is concerned with studying the properties of reproducing pairs and their application to different time-frequency representations.

Supervisor: Peter Balazs

Homepage: www.kfs.oeaw.ac.at/index.php

Supervisor: Christian Krattenthaler