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/

Poisson transforms for differential forms

In the theory of homogeneous spaces there are two diametral examples, namely Riemannian symmetric spaces and flat parabolic geometries. The first one is naturally endowed with an invariant Riemannian metric and therefore carries a vast variety of invariant differential operators. On the other hand, the geometric structure of parabolic geometries are rather exotic, so invariant differential operators are rare objects. To connect these types of homogeneous spaces we construct Poisson transforms, which are integral operators between vector bundle valued differential forms on homogeneous parabolic geometries and vector valued differential forms on Riemannian symmetric spaces. Via the theory of geometry of homogeneous spaces, we can reduce the question of their existence and their relation with several natural differential operators to computations in a finite dimensional representation of a reductive Lie group.

Supervisor: Andreas Cap

christoph.harrach@univie.ac.at

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

Quantitative Photoacoustic Imaging in the Acoustic Regime

I am working in the field of convex optimisation and related topics, in particular the convergence analysis of algorithms and their applications.

Supervisor: Radu Ioan Boţ

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

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

What kind of financial mathematics should be taught in math classes?

I am a PhD student in the field of mathematics education. My current studies concern creation of fundamental ideas, empirical investigations in math classes and design of a math curriculum including financial literacy.

Supervisors: Hans Humenberger, Mathias Beiglböck

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

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

Some cardinal invariants of the generalized Baire spaces

My research interests deal with set theory, specifically forcing, cardinal invariants of the continuum and its generalizations to the context of uncountable cardinals. These invariants are cardinal numbers describing mostly the combinatorial or topological structure of the real line. We work with the generalization of many of the classical ones, for instance, cardinals in Cichoń's Diagram and try to establish ZFC relationships between them. When there are not such relations, we look for consistency results in some specific forcing extensions.

Supervisor: Sy-David Friedman

Homepage: http://www.logic.univie.ac.at/~montoyd8/

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

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

Geometric Aspects related to Gabor Frames

The study of Gabor frames originates in a 1946 paper of Gabor in which he describes intermediate cases between pure time analysis and pure frequency analysis (Fourier analysis). The idea is to have a two-dimensional representation of a one-dimensional signal, similar to a musical score. This is done by shifting a so-called window function both in the time domain (translation) and Fourier (frequency) domain (modulation). The choice of the window function itself is already a delicate task and also how to perform the time-frequency shifts in order to measure enough information of the signal such that we are able to reconstuct it in a stable way. Due to uncertainty principles and its invariance under the Fourier transform, the Gaussian is often chosen as a window function. For the Gaussian window, we investigate how the shifts in time and frequency should be carried out such that we get a system which allows fast and stable reconstruction from measured samples. The conjecture is that if we assume regular shifts in time and frequency and consider each combined shift as a point in the time-frequency plane (phase-space) we should use a hexagonal pattern as it is used to optimally pack discs in the plane.

Supervisor: Karlheinz Gröchenig

Homepage: homepage.univie.ac.at/markus.faulhuber

**Mathematical studies in nonlinear water wave theory**

My research deals with the mathematical analysis of gravity water waves governed by the incompressible Euler equations. A core part of my work is dedicated to the existence theory and qualitative properties of solutions to highly nonlinear approximative equations modeling the propagation of large amplitude waves. Moreover, I am interested in analyzing models for ocean waves in equatorial regions and wind-induced current fields.

Supervisor: Adrian Constantin