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On Unitarity for sl(m/n)-Supermodules: Dirac Cohomology, Indices, Superdimension
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This thesis investigates unitarizable supermodules over special linear Lie superalgebras
sl(m|n) and their basic classical counterparts A(m|n), denoted g, with a focus on their
structure, classification, and applications in both mathematics and theoretical physics. It
is structured in four main parts, each exploring a distinct but interrelated aspect of the
theory.
The first part develops a general framework for understanding unitarity and provides a concise classification of unitarizable simple g-supermodules, derived using the Dirac inequality
and decomposition under the even Lie subalgebra. The Dirac operator and its associated Dirac cohomology serve as central tools in this study, capturing essential aspects of
unitarity. We demonstrate that Dirac cohomology can uniquely determine unitarizable
supermodules, and compute it explicitly of unitarizable simple supermodules. This
leads to a refined characterization of unitarity, forming the basis for our novel classifica-
tion of unitarizable simple supermodules. Furthermore, we establish a connection between
Dirac cohomology and Kostant’s cohomology of Lie superalgebras, derive a decomposition
of formal characters, and introduce a Dirac index.
In the second part, we construct a formal superdimension for infinite-dimensional unitarizable supermodules, inspired by the theory of relative discrete series representations. We
show that this superdimension vanishes for most simple supermodules but is non-trivial
precisely when the infinitesimal character has maximal degree of atypicality. In particular,
our result aligns with the Kac–Wakimoto conjecture for finite-dimensional supermodules.
The third part investigates applications to theoretical physics, focusing in particular on
the so-called “superconformal index” – a character-valued invariant assigned by physicists
to unitarizable supermodules of Lie superalgebras, such as su(2, 2|n), which appear in the
context of certain quantum field theories. The index is computed as a supertrace over
a Hilbert space and remains constant across families of representations that arise from
varying physical parameters. This invariance is due to the fact that only “short” simple
supermodules contribute to the index, making it stable under recombination phenomena
occurring at the boundary of the unitarity region. We develop these notions for unitarizable
supermodules over g. Along the way, we provide a precise dictionary between various
notions from theoretical physics and mathematical terminology. Our final result is a kind of
“index theorem” that relates the counting of atypical constituents in a general unitarizable
g-supermodule to the character-valued Q-Witten index, expressed as a supertrace over the
full supermodule. The formal superdimension of part 2 can also be formulated in this
framework.
The final part is an addendum that extends the Dirac operator and cohomology to their
cubic counterparts. We develop a theory of cubic Dirac operators associated to parabolic
subalgebras and prove a super-analog of the Casselman–Osborne theorem. We show that
Dirac cohomology is trivial unless for highest weight supermodules, and demonstrate, under suitable conditions, an embedding of Dirac cohomology into Kostant’s (co)homology. This
embedding becomes an isomorphism in the unitarizable case. We also provide complete
computations of Dirac cohomology for finite-dimensional simple supermodules with typical
highest weight and for supermodules in the parabolic BGG category
Pattern Formation and Supersolid Sound Modes in a Driven Superfluid
Full text linkSystems driven far from equilibrium can show radically different properties from the same system at equilibrium. In some cases, new steady-states can emerge, enabling the application of theoretical frameworks typically developed in equilibrium. In this thesis, we discuss the emergence of self-stabilized, square lattice patterns in a Bose-Einstein condensate (BEC) with periodically modulated interactions. We show that despite the dynamical nature of the system, the patterned state displays Goldstone modes that are identical to those of supersolids, which are equilibrium superfluids with spontaneously arising periodic ordering.
We first provide a brief overview of the theoretical concepts underpinning the spontaneous emergence of the pattern, as well as its stabilization. We then present the experimental techniques used to observe the pattern, focusing mainly on tunable interactions and local control over the cloud using a digital micromirror device. Experimental results on the emergence of the structure are discussed, demonstrating that the pattern is truly a result of nonlinear phenomena far from equilibrium. We then turn towards explicit imprinting of lattices, which enables us to probe the phonon-phonon interactions that explain pattern stabilization. Beyond imprinting ideal lattices, we also explicitly instigate lattice and superfluid defects, observing their propagation. We identify two distinct speeds of sound for longitudinal excitations and a diffusive mode for transverse lattice deformations. We compare the extracted mode structure to a generic framework of superfluid smectics, extracting relevant hydrodynamic parameters of the system. Finally, we compare the dynamics of wavepackets to collective modes, finding good agreement
Die Unterbringung in einem psychiatrischen Krankenhaus gemäß § 63 StGB: Eine Untersuchung der Anordnungs- und Vollstreckungsverfahren in den Landgerichtsbezirken Heidelberg und Berlin
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Generative Machine Learning for Simulation-based Inference in High Energy Physics
Full text linkWith the upcoming High-Luminosity LHC the volume of collider data will increase dramatically, leading to a new era of precision measurements. However, this also creates computational and methodological challenges. Established simulation and inference pipelines require significant upgrades to prevent them from becoming bottlenecks. This thesis investigates how generative machine learning can address these challenges. First, we investigate modern generative architectures, diffusion models and autoregressive transformers, for fast and accurate LHC event generation. We find that they can learn complex phase space distributions to percent-level precision, demonstrating their potential as surrogate simulators. Second, we advance the use of machine learning for the matrix element method, showing how generative networks can be used to encode the transfer probability and keep the phase space integration tractable. Finally, we explore high-dimensional, unbinned unfolding using generative models. We benchmark the performance of a range of methods on the same datasets and contribute several methodological advancements, including a transformer-enhanced diffusion model that achieves state-of-the-art precision
The Emergent Properties of PC12 Populations in Confinement
Full text linkThis study explores neuron-like PC12 behavior in vitro with a focus on neuronal interactions within confined microenvironments, addressing a gap in neurobiological research related to spatially constrained neuron behavior. While previous studies have examined neuronal interactions with synthetic structures, how neuron populations selectively navigate confined areas has been underinvestigated. This work interrogates neuronal responses to spatial constraints by confining neuron-like PC12 cell populations inside 3D-printed microenvironments on glass substrates. The microenvironments enable localized interaction with neighboring cells and the printed structure. Key parameters analyzed include neurite outgrowth, migration, and cell proliferation in relation to confinement. The results reveal that PC12 cells in contact with structural features exhibit longer neurite outgrowth and a preference for maintaining contact rather than occupying open spaces within the confinement. Time-lapse imaging confirms PC12 cells actively seek structural interactions, a behavior not widely investigated up to now. These findings underscore the role played by spatial constraints in neuronal network formation and highlight how structural contact could guide neuronal behavior in confinement. The insights gained could enhance neurobiological modeling and tissue engineering, and suggest that a better understanding of spatially structured environments could improve neural tissue models. Future research could expand on these findings by varying confinement geometries and materials and investigating additional aspects of cellular behavior, to possibly advance bioengineering efforts that aim to create brain-like tissue structures and neural interfaces
Rapid 3D passive needle localization for automatic slice alignment in MR-guided interventions
Full text linkIn Magnetic Resonance Imaging (MRI)-guided needle interventions, typically 2D real-time imaging is used for the visual tracking of the needle. For imaging planes insufficiently aligned with the needle, realignment can become necessary for better needle display. In this work, techniques for the visualization of metallic needles with MRI and rapid 3D needle localization for automatic imaging plane alignment were developed and investigated.
First, the image artifact of a needle was characterized quantitatively for conventional gradient-echo (GRE) imaging with the fast low-angle shot (FLASH) technique, as well as for a positive, susceptibility-based imaging technique (dephased GRE). A model-based method for needle localization, using a baseline-subtracted radial FLASH k-space acquisition, was then developed and evaluated in phantom measurements, as well as demonstrated in an animal experiment. Additionally, a localization method based on undersampled dephased FLASH imaging and Convolutional Neural Network-based postprocessing was developed, investigated, and compared with the model-based method.
The needle artifacts were well described by the developed quantitative models (measured deviation ≤ 2 pixels). Both developed localization methods allowed for fast 3D needle localization (under 1 s). Feasibility of automatic slice alignment of 2D imaging planes with the detected needles was demonstrated retrospectively.
The developed methods for rapid, passive 3D needle localization can accelerate the workflow of
MRI-guided interventions and facilitate clinical applicability
Transcriptome evolution of the mammalian germline
Full text linkSexual reproduction plays a central part in life and evolution, as it allows the transmission of
genetic information across generations. As the offspring is generated from gametes of both
parents, sexual reproduction is a source for diversity within a population. The gametes are
formed through gametogenesis of the germ line, in a process which can generally be divided
into three phases: 1) Mitotic expansion of the progenitor pool. 2) Meiotic reductive divisions.
3) Post-meiotic changes. For males this takes place in the testis and generates sperm, while in
females this process takes place in the ovary and generates oocytes. From an evolutionary
perspective, especially the testis is an interesting tissue, as it has been shown to evolve rapidly
both morphologically as well as molecularly. Yet, so far it was not fully dissected, which cells
are the driver of this observation. In my thesis work, I address this question in adult testis on
the basis of single cell level transcriptional data from 11 species covering the three main
mammalian lineages (monotremes, marsupials, and eutherians) and a bird. Among the
mammalian species are seven primates, including all extant great apes, except for Orangutan.
To also further our understanding of the female germline evolution, I also analyzed a dataset
of single cell level transcriptional data of developing ovaries for three glires and a bird.
In these analyses, I show how we dissected the testicular cell type transcriptomes. We found
that the fast divergence of the testis is especially driven by the post-meiotic cell types.
Furthermore, we showed that this fast divergence is driven by both a relaxation of negative
selection, as well as an increase of positive selection. In addition, we found support for the
notion of the “out of the testis” hypothesis, according to which the testis plays an important
role in the origin of new genes.
Through assessing the expression dynamics of individual genes, we found shared and lineage
specific aspects of the transcriptome. In this analysis we identified a core set of genes, which
maintained their expression across mammalian evolution and is likely connected to central
testicular function and furthermore connected to maintenance of fertility. In the analysis of
lineage specific expression patterns, we also found genes with relevance for fertility. Tracing
the functional impact of individual genes, we also found conserved aspects of Sertoli cell to
germ cell communication.
Specifically focusing on the analysis of sex chromosome expression, we found an accumulation
of testis specific X-linked genes in spermatogonia. Furthermore, we dissected the transcriptomal differences between the X- and Y-carrying spermatids. In addition to this, we
assessed the silencing of sex chromosomes upon meiotic entry and showed, for the first time
transcriptomal evidence for meiotic sex chromosome inactivation (MSCI) in platypus.
In the analysis of the developmental ovary dataset I generated, I identified the basic somatic
cell types. In the assessment of the germ cells I found, that I could not clearly distinguish
multiple subtypes of germ cells in the mammalian data. In the bird data I was able to
distinguish germ cell subtypes.
Analyzing the cell type transcriptomes in this dataset, I found that germ cells diverge faster
than somatic cells.
In the dedicated analysis of the bird germ line transcriptomes in regards to sex chromosome
expression, I found support for the presence of MSCI in birds.
Taken together my work helps to explain the observed diversity in testicular phenotypes and
explain their molecular sources. For the ovary, my data provides an exciting starting step for
further exploration
On the Variation of Interstellar Dust Extinction
Full text linkDust obscures astronomical observations and plays a important role in the evolution of galaxies. Mapping dust in three dimensions not only allows accurate correction of observations but also constrains the evolution of the interstellar medium (ISM). Extinction is the central observable to trace the density and properties of dust, and is widely employed in dust mapping. Most existing dust maps assume a universal extinction curve, treating the properties of dust as uniform. This thesis
introduces variations in the slope of extinction curves, parameterized by R(V). I construct a data-driven forward model that predicts low-resolution spectra as a function of stellar parameters (effective temperature, surface gravity, and metallicity), parallax, and extinction properties. This model is applied to all 220 million Gaia XP spectra, yielding precise extinction curves for 130 million stars in the Milky Way, LMC, and SMC. From these, I construct the first 3D all-sky map of extinction curve variations. Unexpectedly, I find extinction curves steepen, rather than flatten, with increasing dust density in translucent dust clouds. I propose a theoretical explanation, attributing the steepening to the growth of polycyclic aromatic hydrocarbons (PAHs). The results also provide implications for future observations in the era of the JWST