# Student Steering Committee-Members

# Sascha Biberhofer

# Sascha Biberhofer

Eisenstein Cohomology of Arithmetic Groups - The Exceptional Group G_2

Supervisor: Joachim Schwermer

sascha.biberhofer@univie.ac.at

# Axel Böhm (Student Speaker)

# Axel Böhm (Student Speaker)

Supervisor: Hermann Schichl, Radu Ioan Boţ

# Christopher Chiu

# Christopher Chiu

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

# Laura Kanzler (Vice Student Speaker)

# Laura Kanzler (Vice Student Speaker)

Large-time and macroscopic asymptotics in kinetic transport

Supervisor: Christian Schmeiser

# Manjil Saikia

# Manjil Saikia

**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

# Daniel Scherl

# Daniel Scherl

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

# Student Members

# Peter Allmer

# Peter Allmer

Relativistic wave equations

My PhD thesis deals with modeling, (asymptotic) analysis and numerics of time dependent PDEs in quantum mechanics and cosmology. At the core is the Klein-Gordon equation (KG), a wave equation used e.g. in relativistic quantum theory, where speed of propagation is bounded.

We shall work with linear and non-linear versions of the KG equation, including the KG-Maxwell system, a "self-consistent" model for the electromagnetic field. Also, we consider a relativistic quantum equation for fermionic four-spinors, the Dirac equation, which is mathematically closely linked to the KG equation. We will put emphasis also on the "non-relativistic" limit of the KG equation (c → ∞) which yields non-linear Schrödinger (NLS) type equations, e.g. the Gross Pitaevskii equation (GP), a cubic NLS with confinement potential, as the simplest "mean field" model of a Bose-Einstein Condensate (BEC). There is a striking analogy of the fast expansion of a BEC when the confinement is released, with "cosmological inflation" and we shall contribute to the deep modeling question if the universe, including baryonic matter, dark matter and dark energy, can indeed be modelled as a (relativistic) BEC using the KG equation and GP type non-linear PDEs.

Supervisor: Norbert J. Mauser

# Roswitha Bammer-Zimmer

# Roswitha Bammer-Zimmer

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

# Alexander Beigl

# Alexander Beigl

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

# Julius Berner

# Julius Berner

Mathematical analysis of deep learning based methods for solving high-dimensional problems

Supervisor: Philipp Grohs

# Katharina Brazda

# Katharina Brazda

Variational models for biological membranes

Supervisor: Ulisse Stefanelli

Co-Supervisor: Christian Schmeiser

Homepage: https://www.mat.univie.ac.at/~brazda/

# Anna Breger

# Anna Breger

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

# Giancarlo Castellano

# Giancarlo Castellano

Whittaker periods of isobaric sums over totally real fields and special L-values

Supervisor: Harald Grobner

# Mark Jason Celiz

# Mark Jason Celiz

Space of Functions of Variable Bandwidth Parametrized by Piecewise Constant Functions

In my dissertation project I am working on the sampling and reconstruction of signals on the space of functions of variable bandwidth with a piecewise constant bandwidth-parametrizing function. Generally, variable bandwidth spaces are defined via spectral subspaces of an elliptic differential operator (defined by a chosen parametrizing function) corresponding to a given spectral set. For piecewise constant parametrizing functions, signals belonging to these spaces may be viewed as having different local bandwidths on different signal segments, hence the name variable bandwidth.

Supervisor: Karlheinz Gröchenig, Andreas Klotz

# Julia Delacour

# Julia Delacour

Cargo Clustering and Membrane Bending in Cellular Autophagy

Supervisor: Christian Schmeiser

# Dennis Elbrächter

# Dennis Elbrächter

Approximation theory of deep neural networks

Supervisor: Philipp Grohs

# Gianluca Favre

# Gianluca Favre

Supervisor: Christian Schmeiser

# Markus Gabl

# Markus Gabl

Copositive Aproach to Adjustable Robust Optimization with Uncertain Recourse

The research interests of Markus Gabl comprise applications of copositve programming approaches such as quadratic and robust programming. Early work was on applications of semidefinite programming in finance.

Supervisor: Immanuel Bomze

Homepage: vgsco.univie.ac.at/people/phd-students/markus-gabl/

# Caroline Geiersbach

# Caroline Geiersbach

Stochastic Models in PDE Constrained Optimization

I am focusing on stochastic approximation techniques for solving optimization problems constrained by partial differential equations subject to uncertainty.

Supervisor: Georg Ch. Pflug

Homepage: vgsco.univie.ac.at/people/phd-students/caroline-geiersbach/

# Susanna V. Haziot

# Susanna V. Haziot

Mathematical aspects of water waves and physical oceanography

Supervisors: Adrian Constantin

# Hans Höngesberg

# Hans Höngesberg

The monotone triangle perspective on alternating sign arrays: operator formulae and constant term expressions

When Mills, Robbins and Rumsey introduced alternating sign matrices in the early 1980s, they conjectured a simple closed enumeration formula. It took more than ten years to prove it and yet no bijective proof has been found so far. The search of bijective proofs and refined enumeration formulae and the study of related and generalised objects have become of great interest to combinatorialists.

One promising attempt to deal with these objects are so-called monotone triangles and operator formulae. My aim is to explore these methods to provide new insights into refined enumerations and to gain a better combinatorial understanding of alternating sign arrays.

Supervisors: Ilse Fischer, Christian Krattenthaler

# Michael Hofbauer-Tsiflakos

# Michael Hofbauer-Tsiflakos

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

# Manuel Inselmann

# Manuel Inselmann

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

# Michael Kahr

# Michael Kahr

Optimization in Social Networks: Problems on Influence Propagation and Community Detection

During the last decades a multitude of new application areas for rigorous quantitative data-driven network analysis has appeared. Thereby, a notable example is the field of social network analysis. Since the proliferation of online social networks has provided vast data sets about the structure of such networks, as well as about the attributes of its users, a strong increase of research dedicated to that area can be observed, recently. Fundamental problems arising in the context of social network analysis include (i) the identification of key players that may be targeted to support influence propagation throughout the network, and (ii) the identification of communities. In my PhD-project, I tackle the aforementioned problems by methods of Mathematical Optimization including (Mixed-) Integer Linear Programming, Nonconvex Quadratic Optimization, and Conic Optimization. Another important issue (in numerous optimization problems) is that the underlying input data may be affected by uncertainty, e.g., the strength of social ties can only be estimated. I address this uncertainty with the framework of Robust Optimization, and identify optimal solutions to the aforementioned problems, which are robust against all data realizations within in the estimated bounds (i.e., uncertainty sets). Moreover, I investigate the implications for computational complexity as well as for the solution structures, of considering different uncertainty sets.

Supervisors: Immanuel M. Bomze, Markus Leitner

# Marlene Koelbing

# Marlene Koelbing

Gaps in the generalised Baire space

Supervisor: Sy Friedman

# Sarah Koppensteiner

# Sarah Koppensteiner

Obstructions in the Approximation Theory of Shallow Neural Networks

Supervisor: Philipp Grohs

# Josef Küstner

# Josef Küstner

Elliptic Combinatorics

Supervisor: Michael Schlosser

# Paola Lopez

# Paola Lopez

Supervisor: Harald Grobner

# Hana Melánová

# Hana Melánová

Supervisor: Herwig Hauser

# Dennis Meier

# Dennis Meier

Supervisor: Radu Ioan Boţ

# David Melching

# David Melching

Variational structures in thermomechanics of solids

I am interested in partial differential equations and the way they model

various physical phenomena like elasticity, plasticity, damage or thermodynamics in solids. In particular, I investigate and exploit variational structures in

these models to analytically show existence (and sometimes uniqueness)

of solutions for the underlying evolution equations.

Supervisor: Ulisse Stefanelli

Homepage: http://www.mat.univie.ac.at/~melching/

# Melanie Melching

# Melanie Melching

Regularization with integral representations of Sobolev norms of manifold-valued functions

One of the most famous image denoising tools is the Total Variation regularization. Unwanted noise is removed whilst important details such as edges are preserved. We are interested in generalizing this principle to manifold-valued functions respecting the given geometry. Hereby we use an alternative approach to Sobolev spaces.

Supervisor: Otmar Scherzer

# Jakob Möller

# Jakob Möller

Self-consistent modeling in relativistic and non-relativistic quantum mechanics

My research interests lie among PDEs arising from relativistic and non-relativistic Quantum Mechanics, e.g. the linear Pauli equation and the self-consistent Pauli equation, also known as Pauli-Poiswell system and my aim is to understand these equations analytically and numerically. Furthermore I’m interested the behavior of these equations in the non-relativistic and semi-classical limit.

Supervisor: Norbert J. Mauser

# David Nenning

# David Nenning

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

# Dang-Khoa Nguyen

# Dang-Khoa Nguyen

Numerical Algorithms for solving Nonconvex Optimization Problems

Supervisor: Radu Ioan Boţ

# Noema Nicolussi

# Noema Nicolussi

Quantum Graphs

The main objective of my research are so-called quantum graphs, i.e. Schrödinger-type differential operators on networks. More precisely, my work concerns the case of infinite networks, spectral theory for the corresponding differential operators and and relations to difference Laplacians on graphs.

Supervisor: Aleksey Kostenko, Co-supervisor: Gerald Teschl

# Alexander Pichler

# Alexander Pichler

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

# Giulia Pilli

# Giulia Pilli

PDE models for transportation networks

Supervisor: Peter Markowich

# Mateusz Piorkowski

# Mateusz Piorkowski

Riemann-Hilbert approach in asymptotic analysis

My reasearch interest lies on the boundary between nonlinear integrable systems and complex analysis. To be precise, I use the Riemann-Hilbert approach for inverse scattering to extract the asymptotical behaviour of nonlinear PDEs like the KdV equation. Other applications where the Riemann-Hilbert method has been useful include the asymptotics of orthogonal polynomials and eigenvalue distributions of random matrices.

Supervisor: Gerald Teschl

# Martin Pontz

# Martin Pontz

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

Supervisor: Reinhard Bürger

# Ziping Rao

# Ziping Rao

Nonlinear Wave Equations

Supervisor: Roland Donninger

# Jonathan Schilhan

# Jonathan Schilhan

Forcing, infinitary combinatorics and definability

Supervisor: Vera Fischer

# Benedict Schinnerl

# Benedict Schinnerl

Non-Smooth spacetime geometry: Focusing energy conditions and singularity theorems

I study Lorentzian geometry for spacetime metrics of low Regularity (less than twice continuously differentiable) and aim to extend recent results of singularity theorems for this class to weaker energy assumtions. Such assumptions are in particular interesting to general relativity from a quantum point of view. Further I also work on proving a Gannon-Lee type singularity theorem in a low regularity setting.

Supervisor: Roland Steinbauer

# Denise Schmutz

# Denise Schmutz

Computational Algorithms for Imaging of Trapped Particles

In my research I consider various models for imaging of a trapped cell with optical tomography. In this setup the cell is moved out of the standard position by rotating and shifting the sample with optical tweezers. This causes uncertainties in the directional information in the data. In my thesis I will use and develop new mathematical and computational methods for the compensation of these uncertainties.

Supervisor: Otmar Scherzer

# Michael Sedlmayer

# Michael Sedlmayer

My research will treat design, convergence analysis and implementation of numerical methods for non-smooth convex or non-convex optimisation problems that are relevant for machine learning with applications in the area of Digital Humanities.

Supervisor: Radu Ioan Boţ, Tara Andrews

# Diksha Tiwari

# Diksha Tiwari

Supervisor: Paolo Giordano, Michael Kunzinger

# Jordy Timo van Velthoven

# Jordy Timo van Velthoven

Frames generated by representations of locally compact groups

Supervisor: José Luis Romero

# Leopold Veselka

# Leopold Veselka

Quantitative Coupled Physics Imaging

Supervisor: Peter Elbau

# Chen Wang

# Chen Wang

Some enumerative problems in group theory

Supervisor: Christian Krattenthaler

# Claudia Wytrzens

# Claudia Wytrzens

Kinetic theory applied to the study of the interaction between capillary networks and cell clustering in adipose tissue

Supervisor: Sara Merino Aceituno