Flat surfaces and stability structures

We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear.Comment: 67 pages, 7 figures, v2: expanded from previous version following feedback, new titl

3-d Calabi--Yau categories for Teichm\"uller theory

For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic differentials on a genus $g$ surface with simple zeros and $n$ simple poles. For a generic point in the moduli space the corresponding quantum/refined Donaldson--Thomas invariants are computed in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, these counts satisfy wall-crossing formulas.Comment: v2: re-written for streamlined exposition, results unchange

Why do wasp induced galls of Acacia longifolia photosynthesise?

While many stem and bud galls contain chlorophyll, and have the potential to photosynthesise, these insect-induced growths are generally thought to act as strong carbon sinks, manipulating the normal phloem transport of the host plant in order to serve the demands of the galling herbivore. This study investigated the photosynthetic capacity of bud galls induced by the wasp, Trichilogaster acaciae longifoliae (Pteromalidae) in the invasive Acacia longifolia. The role of this photosynthetic activity was examined in terms of its ability to subsidise carbon budgets, as well as to provide O₂ to the larvae and consume CO₂ in the dense gall tissue, thereby maintaining O₂ and CO₂ concentrations within the range of larval tolerance. Galls were found to contain an overall chlorophyll concentration that was less than half that of subtending phyllodes and a maximum stomata! conductance only 16% that of phyllodes. Gas exchange measurements indicated that while photosynthesis never fully compensated for the respiratory costs of the galls, light-induced carboxylation within galls contributed substantially to the maintenance and growth of galls, especially in the early stages of their development. Very low levels of O₂ were found within the larval chamber and internal tissues of galls, and these levels responded only marginally, if at all, to light, suggesting that the photosynthetic activity of galls does not play a critical role in providing 0 2 to the larvae. The percentage mortality and metabolic response of larvae in reaction to various atmospheres of reduced O₂ and elevated CO₂ indicated that larvae were tolerant of hypoxia and capable of rapidly reducing their respiratory rates to cope with hypercarbia, at least over the short term. Sustained metabolic arrest may, however, have toxic consequences for insects, causing cell damage or even death. The photosynthetic activity of galls substantially reduced internal CO₂ concentrations, thus preventing CO₂ from accumulating within galls over prolonged periods. Hence, the capacity of galls to photosynthesise has significant implications for the survival of the developing larvae by reducing the risk of hypercarbic_toxicity and supplying additional carbohydrates to the gall and its inhabitants, thereby creating a favourable microhabitat in which to live

Iterated logarithms and gradient flows

We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang--Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argument is a monotonicity property of the flow which holds in arbitrary dimension. The A-side analog is a modified curve shortening flow for which we provide a heuristic calculation in support of a detailed conjectural picture.Comment: 29 pages, comments encourage

Semistability, modular lattices, and iterated logarithms

We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings.Comment: v2: new introduction, typos correcte

Dynamical systems and categories

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.Comment: 35 page

Preserving Han-Ok: Reimaging the Korean Traditional House for Today using 3D Design

The purpose of this thesis project was to preserve the traditional elements of Han-Ok in the modern architecture in Korea. I have integrated traditional Korean design elements into modern architecture. I also solved the problems that the Han-Ok currently faces. This topic is important because a lot of the traditional culture has been disappearing within the current Korean society. Of all the aspects of Korean traditional culture, the Han-Ok, the Korean traditional house itself, is very important. A lot of Korea\u27s culture is represented in the Han-Ok, but people think this type of architecture is unnecessary and uncomfortable, and because of this people are pursuing another direction. It is important in our society to preserve the traditional culture from the past to pass on to future generations. I researched every component of the Han-Ok itself; identifying the weaknesses of the house, Korean traditional culture, and modern architecture. Finally integrated the Korean traditional style into modern architecture. The scope of my thesis focused heavily on the areas of architecture, camera movement, and 3D modeling within this industry

Flags and tangles

We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.Comment: v2: added discussion of dualities, more detailed proofs, typos correcte

Large-Scale Influences on Atmospheric River Induced Extreme Precipitation Events Along the Coast of Washington State

Atmospheric Rivers (ARs) are responsible for much of the precipitation along the west coast of the United States. In order to accurately predict AR events in numerical weather prediction, subseasonal and seasonal timescales, it is important to understand the large-scale meteorological influence on extreme AR events.Here, characteristics of ARs that result in an extreme precipitation event are compared to typical ARs on the coast of WashingtonState. In addition to more intense water vapor transport, notable differences in the synoptic forcing are present during extreme precipitation events that are not present during typical AR events.In particular, a negatively tilted low pressure system is positioned to the west in the Gulf of Alaska, alongside an upper level jet streak. Subseasonal and seasonal teleconnection patterns are known to influence the weather in the Pacific Northwest. The Madden JulianOscillation (MJO) is shown to be particularly important in determining the strength of precipitation associated with in AR ont he Washington coast