Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Alternative reproductive tactics, sexual selection and the effects of inbreeding in the ant Hypoponera opacior

Inbreeding can lead to a fitness reduction due to the unmasking of deleterious recessive alleles and the loss of heterosis. Therefore, most sexually reproducing organisms avoid inbreeding, often by disperal. Besides the avoidance of inbreeding, dispersal lowers intraspecific competition on a local scale and leads to a spreading of genotypes into new habitats. In social insects, winged reproductives disperse and mate during nuptial flights. Therafter, queens independently found a new colony. However, some species also produce wingless sexuals as an alternative reproductive tactic. Wingless sexuals mate within or close to their colony and queens either stay in the nest or they found a new colony by budding. During this dependent colony foundation, wingless queens are accompanied by a fraction of nestmate workers. The production of wingless reproductives therefore circumvents the risks associated with dispersal and independent colony foundation. However, the absence of dispersal can lead to inbreeding and local competition.rnIn my PhD-project, I investigated the mating biology of Hypoponera opacior, an ant that produces winged and wingless reproductives in a population in Arizona. Besides the investigation of the annual reproductive cycle, I particularly focused on the consequences of wingless reproduction. An analysis of sex ratios in wingless sexuals should reveal the relative importance of local resource competition among queens (that mainly compete for the help of workers) and local mate competition among males. Further, sexual selection was expected to act on wingless males that were previously found to mate with and mate-guard pupal queens in response to local mate competition. We studied whether males are able to adapt their mating behaviour to the current competitive situation in the nest and which traits are under selection in this mating situation. Last, we investigated the extent and effects of inbreeding. As the species appeared to produce non-dispersive males and queens quite frequently, we assumed to find no or only weak negative effects of inbreeding and potentially mechanisms that moderate inbreeding levels despite frequent nest-matings.rnWe found that winged and wingless males and queens are produced during two separate seasons of the year. Winged sexuals emerge in early summer and conduct nuptial flights in July, when climate conditions due to frequent rainfalls lower the risks of dispersal and independent colony foundation. In fall, wingless sexuals are produced that reproduce within the colonies leading to an expansion on the local scale. The absence of dispersal during this second reproductive season resulted in a local genetic population viscosity and high levels of inbreeding within the colonies. Male-biased sex ratios in fall indicated a greater importance of local resource competition among queens than local mate competition among males. Males were observed to adjust mate-guarding durations to the competitive situation (i.e. the number of competing males and pupae) in the nest, an adaptation that helps maximising their reproductive success. Further, sexual selection was found to act on the timing of emergence as well as on body size in these males, i.e. earlier emerging and larger males show a higher mating success. Genetic analyses revealed that wingless males do not actively avoid inbreeding by choosing less related queens as mating partners. Further, we detected diploid males, a male type that is produced instead of diploid females if close relatives mate. In contrast to many other Hymenopteran species, diploid males were here viable and able to sire sterile triploid offspring. They did not differ in lifespan, body size and mating success from “normal” haploid males. Hence, diploid male production in H. opacior is less costly than in other social Hymenopteran species. No evidence of inbreeding depression was found on the colony level but more inbred colonies invested more resources into the production of sexuals. This effect was more pronounced in the dispersive summer generation. The increased investment in outbreeding sexuals can be regarded as an active strategy to moderate the extent and effects of inbreeding. rnIn summary, my thesis describes an ant species that has evolved alternative reproductive tactics as an adaptation to seasonal environmental variations. Hereby, the species is able to maintain its adaptive mating system without suffering from negative effects due to the absence of dispersal flights in fall.rn