X Sketch of a general result on quasiconformal surgery, which turns combinatorial data into homeomorphisms. Examples. Definition of non-trivial homeomorphism groups. These are totally disconnected and they have the cardinality of R. Generalization to other parameter spaces, e.g., cubic Newton maps.

X Review of holomorphic motions, transversal sections, and simple renormalization. Combinatorial construction and geometric structure of embedded Julia sets. Relation to asymptotic and local similarity.

X Review of holomorphic motions, transversal sections, straightening of quadratic-like maps. Self-contained presentation of primitive and satellite renormalization, including combinatorics, loci, and landing properties.

X Straightening of quasi-regular quadratic-like maps. General construction of homeomorphisms of the Mandelbrot set from combinatorial assumptions. Description and alternative construction by mapping external angles. Examples of homeomorphisms on generalized edges.

X Combinatorial description of fundamental domains at Misiurewicz points. Construction of corresponding homeomorphisms. Review of asymptotic self-similarity.

X Local similarity between the decorations of small Mandelbrot sets and Julia sets. Comparison to asymptotic similarity on multiple scales. Non-hairiness of decorations. Generalization to other parameter spaces.

X For formal matings of certain classes of geometrically finite and infinite polynomials, the structure and diameter of ray-equivalence classes is described explicitly, and the topological mating can be constructed without employing the Rees–Shishikura–Tan Theorem to exclude cyclic ray connections. On the other hand, unbounded cyclic ray connections are found when P is primitive renormalizable and Q is chosen appropriately; then the topological mating is not even defined on a pinched sphere, but there is no Hausdorff space at all. Moreover, matings with long ray connections are found alogithmically.

X Based on combinatorics of ray connections, simple examples of mating discontinuity and of unboundedly shared matings are given. Here the multiplicity grows linearly with preperiod and period. In some cases, upper bounds on the multiplicity are obtained as well.

X Lattès maps of type (2, 2, 2, 2) or (2, 4, 4) are represented by matings in basically nine, respectively three, different ways. The proof combines combinatorics of polynomials and ray-equivalence classes with the Shishikura Algorithm, which relates the topology of the formal mating to the multiplier of the corresponding affine map on a torus. This shows that all matings from non-conjugate limbs exist, which does not follow from the well-known absence of obstructions, since the orbifold (2, 2, 2, 2) is not hyperbolic.

X The Hurwitz equivalence between quadratic rational maps with the same ramification portrait is constructed explicitly, complementing the approach related to the moduli space map by Sarah Koch. Twisted Lattès maps, the pullback relation of curves, and the virtual endomorphism of Lattès maps are discussed, using the lift to a real affine map. Questions of equivalence are related to reduction of quadratic forms (joint work with Michael H. Mertens).

X When the Thurston Algorithm for the formal mating diverges in ordinary Teichmueller space due to postcritical identifications, it still converges on the level of rational maps and colliding marked points — it is not necessary to implement the essential mating by encoding ray-equivalence classes numerically. The proof is based on the extended pullback map on augmented Teichmueller space constructed by Selinger. An informal introduction to his results is included, and a strategy for proving the Thurston Theorems from the Canonical Obstruction Theorem is outlined.

X Equipotential gluing is an alternative definition of mating, which is not based on the Thurston Algorithm. Equipotential lines of the two polynomials are glued to define maps between spheres, and the limit of potential 0 is considered. The initialization of the slow mating algorithm depends on an initial radius R; when R goes to infinity, slow mating is shown to approximate equipotential gluing. The visualization in terms of holomorphically moving Julia sets and their convergence is discussed as well, and in the periodic case a conformal mating in the strongest sense is obtained: the semi-conjugation is the limit of a holomorphic motion. On the other hand, for matings of Lattès type (2, 2, 2, 2) the slow mating algorithm diverges in general: while the expected collisions are happening, a neutral eigenvalue from the one-dimensional Thurston Algorithm persists, producing an attracting center manifold in configuration space.

X Various Thurston maps are defined by moving a critical value along a path, which is simple to implement numerically and to visualize by progressive identifications. This includes a spider algorithm with a path instead of legs, Dehn twisted polynomials, moving the critical value by capture or recapture. The spider algorithm is shown to converge in the obstructed case of satellite Misiurewicz points as well. An alternative construction of quadratic matings by a repelling capture is due to Mary Rees, which can be used to obtain shared matings. Moreover, regluing at capture components and examples from V3 are discussed, including one-sided components and indirect captures.

X Anti-mating means that two planes or half-spheres are mapped to each other by quadratic polynomials, and the filled Julia sets of two quartic polynomials are glued together. An existence criterion due to Ahmadi Dastjerdi is discussed, which is analogous to the non-conjugate limbs condition for matings. An anti-equator is sufficient, and in the hyperbolic case necessary, for an anti-mating. The loci of mating and anti-mating are obtained conjecturally for specific families of quadratic rational maps.

X Markov matrices for postcritically finite Hubbard trees are combined with continuity results to discuss core entropy and biaccessibility dimension of quadratic polynomials. Specifically, results on monotonicity, level sets, renormalization, Hölder asymptotics and self-similarity are obtained.
—Meanwhile, continuity has been shown by Tiozzo and Dudko–Schleicher, as well as maximality.

X Correspondence between puzzle pieces and para-puzzle pieces. Stepwise construction of new para-puzzle pieces corresponding to preimages of a puzzle piece that corresponds to a known para-puzzle piece. Examples: limbs, edges, and frames.

X Review of combinatorial descriptions by external angles, Hubbard trees, kneading sequences, and internal addresses. Discussion and proof of the Stripping Algorithm, which finds external angles by iterating strips or intervals backwards according to a given kneading sequence. Early returns to the characteristic wake, and implementation details.

X Processwork (or Process Oriented Psychology) is a method of awareness training founded by Arnold Mindell in the 1980s. It is applied in various areas, including personal therapy and facilitation. A basic idea is to help the client to become aware of secondary processes and to integrate them. For example, they may identify as being peaceful and be disturbed by violent fantasies; after exploring both violent inner figures and the opposing belief system, the client shall be able to express their wishes and needs, as well as their boundaries, more directly. Alternatively, or in addition, the therapist may help the client to access the quality of being more direct as an essence of the violent fantasies, without exploring and amplifying the latter explicitly. Arnold Mindell developed and explained essence work from his background as a physicist, as well as his experience with Jungian psychology, Daoism and shamanism. Here I aim to present ideas from both quantum physics and Processwork to a general audience, giving examples of concepts seen as analogous. And I emphasize that in my opinion, these connections between physics and psychology need to be framed clearly as analogies: there is no claim whatsoever, that quantum phenomena guide cognitive processes in the brain.

X Processwork (or Process Oriented Psychology) is a method of awareness training with various areas of application, including psychotherapy and facilitation. Here tools and attitudes from Processwork are illustrated with my personal process of dealing with a traumatic experience: On July 29, 2019, a boy was shoved onto the tracks in Frankfurt main station, and killed by the incoming train. Witnessing this, and hearing the mother cry out in agony, I felt horrified like never before in my life. At the same time, I felt determined to work through it as deeply and consciously as possible; I came to find a deeper sense of love and compassion for my fellow human beings, as well as self-compassion and empowerment, and I am still learning to speak about this process.