@phdthesis{Schardt2023,
  author    = {Schardt, Simon},
  title     = {Agent-based modeling of cell differentiation in mouse ICM organoids},
  doi       = {10.25972/OPUS-30194},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-301940},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2023},
  abstract  = {Mammalian embryonic development is subject to complex biological relationships that need to be understood. However, before the whole structure of development can be put together, the individual building blocks must first be understood in more detail. One of these building blocks is the second cell fate decision and describes the differentiation of cells of the inner cell mass of the embryo into epiblast and primitive endoderm cells. These cells then spatially segregate and form the subsequent bases for the embryo and yolk sac, respectively. In organoids of the inner cell mass, these two types of progenitor cells are also observed to form, and to some extent to spatially separate. This work has been devoted to these phenomena over the past three years. Plenty of studies already provide some insights into the basic mechanics of this cell differentiation, such that the first signs of epiblast and primitive endoderm differentiation, are the expression levels of transcription factors NANOG and GATA6. Here, cells with low expression of GATA6 and high expression of NANOG adopt the epiblast fate. If the expressions are reversed, a primitive endoderm cell is formed. Regarding the spatial segregation of the two cell types, it is not yet clear what mechanism leads to this. A common hypothesis suggests the differential adhesion of cell as the cause for the spatial rearrangement of cells. In this thesis however, the possibility of a global cell-cell communication is investigated. The approach chosen to study these phenomena follows the motto "mathematics is biology's next microscope". Mathematical modeling is used to transform the central gene regulatory network at the heart of this work into a system of equations that allows us to describe the temporal evolution of NANOG and GATA6 under the influence of an external signal. Special attention is paid to the derivation of new models using methods of statistical mechanics, as well as the comparison with existing models. After a detailed stability analysis the advantages of the derived model become clear by the fact that an exact relationship of the model parameters and the formation of heterogeneous mixtures of two cell types was found. Thus, the model can be easily controlled and the proportions of the resulting cell types can be estimated in advance. This mathematical model is also combined with a mechanism for global cell-cell communication, as well as a model for the growth of an organoid. It is shown that the global cell-cell communication is able to unify the formation of checkerboard patterns as well as engulfing patterns based on differently propagating signals. In addition, the influence of cell division and thus organoid growth on pattern formation is studied in detail. It is shown that this is able to contribute to the formation of clusters and, as a consequence, to breathe some randomness into otherwise perfectly sorted patterns.},
  subject      = {Mathematische Modellierung},
  language  = {en}
}
@phdthesis{Sapozhnikova2018,
  author    = {Sapozhnikova, Kateryna},
  title     = {Robust Stability of Differential Equations with Maximum},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-173945},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2018},
  abstract  = {In this thesis stability and robustness properties of systems of functional differential equations which dynamics depends on the maximum of a solution over a prehistory time interval is studied. Max-operator is analyzed and it is proved that due to its presence such kind of systems are particular case of state dependent delay differential equations with piecewise continuous delay function. They are nonlinear, infinite-dimensional and may reduce to one-dimensional along its solution. Stability analysis with respect to input is accomplished by trajectory estimate and via averaging method. Numerical method is proposed.},
  subject      = {Differentialgleichung},
  language  = {en}
}
@phdthesis{Boehm2015,
  author    = {B{\"o}hm, Christoph},
  title     = {Loewner equations in multiply connected domains},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-129903},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2015},
  abstract  = {The first goal of this thesis is to generalize Loewner's famous differential equation to multiply connected domains. The resulting differential equations are known as Komatu--Loewner differential equations. We discuss Komatu--Loewner equations for canonical domains (circular slit disks, circular slit annuli and parallel slit half-planes). Additionally, we give a generalisation to several slits and discuss parametrisations that lead to constant coefficients. Moreover, we compare Komatu--Loewner equations with several slits to single slit Loewner equations. Finally we generalise Komatu--Loewner equations to hulls satisfying a local growth property.},
  subject      = {Biholomorphe Abbildung},
  language  = {en}
}
@phdthesis{Schleissinger2013,
  author    = {Schleißinger, Sebastian},
  title     = {Embedding Problems in Loewner Theory},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-96782},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2013},
  abstract  = {The work at hand studies problems from Loewner theory and is divided into two parts: In part 1 (chapter 2) we present the basic notions of Loewner theory. Here we use a modern form which was developed by F. Bracci, M. Contreras, S. D{\´i}az-Madrigal et al. and which can be applied to certain higher dimensional complex manifolds. We look at two domains in more detail: the Euclidean unit ball and the polydisc. Here we consider two classes of biholomorphic mappings which were introduced by T. Poreda and G. Kohr as generalizations of the class S. We prove a conjecture of G. Kohr about support points of these classes. The proof relies on the observation that the classes describe so called Runge domains, which follows from a result by L. Arosio, F. Bracci and E. F. Wold. Furthermore, we prove a conjecture of G. Kohr about support points of a class of biholomorphic mappings that comes from applying the Roper-Suffridge extension operator to the class S. In part 2 (chapter 3) we consider one special Loewner equation: the chordal multiple-slit equation in the upper half-plane. After describing basic properties of this equation we look at the problem, whether one can choose the coefficient functions in this equation to be constant. D. Prokhorov proved this statement under the assumption that the slits are piecewise analytic. We use a completely different idea to solve the problem in its general form. As the Loewner equation with constant coefficients holds everywhere (and not just almost everywhere), this result generalizes Loewner's original idea to the multiple-slit case. Moreover, we consider the following problems: • The "simple-curve problem" asks which driving functions describe the growth of simple curves (in contrast to curves that touch itself). We discuss necessary and sufficient conditions, generalize a theorem of J. Lind, D. Marshall and S. Rohde to the multiple-slit equation and we give an example of a set of driving functions which generate simple curves because of a certain self-similarity property. • We discuss properties of driving functions that generate slits which enclose a given angle with the real axis. • A theorem by O. Roth gives an explicit description of the reachable set of one point in the radial Loewner equation. We prove the analog for the chordal equation.},
  subject      = {Biholomorphe Abbildung},
  language  = {en}
}