Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra

Let \g be any simple Lie algebra over $\mathbb{C}$. Recall that there exists an embedding of $\mathfrak{sl}_2$ into \g, called a principal TDS, passing through a principal nilpotent element of \g and uniquely determined up to conjugation. Moreover, \wedge (\g^*)^\g is freely generated (in the super-graded sense) by primitive elements $\omega_1, \dots, \omega_\ell$, where $\ell$ is the rank of \g. N. Hitchin conjectured that for any primitive element \omega \in \wedge^d (\g^*)^\g, there exists an irreducible $\mathfrak{sl}_2$-submodule V_\omega \subset \g of dimension $d$ such that $\omega$ is non-zero on the line $\wedge^d (V_\omega)$. We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra. Let G be a connected, simply-connected, simple, simply-laced algebraic group and let $\sigma$ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G) \to R(K) is surjective, where R denotes the representation ring over $\mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology H^*(G)\to H^*(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.Comment: 14 page

Geometry of Centroaffine Surfaces in R5\mathbb{R}^5

We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in $\mathbb{R}^5 \setminus \{0\}$ with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class

S. O. S. Save Our Squirrels

This project focused on the issue of squirrel safety and environmental harmony on the Western campus. Through data research, staff interviews, and field monitoring I was able to define my thesis. Squirrels are a vital part of our campus ecosystem and we should not harm them with unnatural food sources. I found that squirrels on campus were becoming dependent on the human population for food via our trash. In order to help solve the problem, my solution was to attempt to alter human behavior by spreading awareness. I worked with Western\u27s Communications and Social Media Department as well as the Marketing Department to get word out. An interview was released detailing my project and providing informative tips on how changing our trash habits would help the squirrels and contribute to a healthier environment for all

Descending G-equivariant Line Bundles to GIT Quotients

In part one, we consider descent of line bundles to GIT quotients of products of flag varieties. Let $G$ be a simple, connected, algebraic group over $mathbb{C}$, $B$ a Borel subgroup, and $T subset B$ a maximal torus. Consider the diagonal action of $G$ on the projective variety $(G/B)^3 = G/B times G/B times G/B$. For any triple $(chi_1, chi_2, chi_3)$ of regular characters there is a $G$-equivariant line bundle $mathcal{L}$ on $(G/B)^3$. Then, $mathcal{L}$ is said to descend to the GIT quotient $pi:[(G/B)^3(mathcal{L})]^{ss} rightarrow (G/B)^3(mathcal{L})//G$ if there exists a line bundle $hat{mathcal{L}}$ on $(G/B)^3(mathcal{L})//G$ such that $mathcal{L}mid_{[(G/B)^3(mathcal{L})]^{ss}} cong pi^*hat{mathcal{L}}$. Let $Q$ be the root lattice, $Lambda$ the weight lattice, and $d$ the least common multiple of the coefficients of the highest root $theta$ of $mathfrak{g}$, the Lie algebra of $G$, written in terms of simple roots. We show that $mathcal{L}$ descends if $chi_1, chi_2, chi_3 in d Lambda$ and $chi_1 + chi_2 + chi_3 in Gamma$, where $Gamma$ is the intersection over root lattices $Q_mathfrak{s}$ of all semisimple Lie subalgebras $mathfrak{s} subset mathfrak{g}$ of maximal rank. Moreover, we show that $mathcal{L}$ never descends if $chi_1 + chi_2 + chi_3 notin Q$. In part two, we discuss joint work with Shrawan Kumar. Let $mathfrak{g}$ be any simple Lie algebra over $mathbb{C}$. Recall that there exists a principal TDS embedding of $mathfrak{sl}_2$ into $mathfrak{g}$ passing through a principal nilpotent element of $mathfrak{g}$. Moreover, $wedge (mathfrak{g}^*)^mathfrak{g}$ is generated by primitive elements $omega_1, dots, omega_ell$, where $ell$ is the rank of $mathfrak{g}$. N. Hitchin conjectured that for any primitive element $omega in wedge^d (mathfrak{g}^*)^mathfrak{g}$, there exists an irreducible $mathfrak{sl}_2$-submodule $V_omega subset mathfrak{g}$ of dimension $d$ such that $omega$ is non-zero on the line $wedge^d (V_omega)$. We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra. Let $G$ be a connected, simply-connected, simple, simply-laced algebraic group and let $sigma$ be a diagram automorphism of $G$ with fixed subgroup $K$. Then, we show that the restriction map $R(G) to R(K)$ is surjective, where $R$ denotes the representation ring over $mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology $H^*(G)to H^*(K)$ is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.Doctor of Philosoph

Development of resistance to an introduced marine pathogen by a native host

In 1957–1959, the introduced protistan parasite, Haplosporidium nelsoni, killed 90–95% of the oysters (Crassostrea virginica) in lower Delaware Bay and about half of those in the upper bay. Shortly thereafter, H. nelsoni-caused mortality in the wild population of the lower bay declined, approximating that of first-generation selectively bred oysters. For nearly three decades thereafter no further change in survival of the wild population was evident, although steady improvement was achieved by continued selective breeding. Survival of the wild population is thought to have plateaued because the great majority of oysters inhabited the upper bay where they were protected from H. nelsoni infection and selective mortality by low salinity. Consequently, they contributed most of the offspring to the bay population. From 1957 through 1987, H. nelsoni prevalence was cyclic, but overall high (annual maxima of 60 to 85%) in the lower bay. Since 1988, however, prevalence in wild oysters has rarely exceeded 30% anywhere in the bay, even though unselected oysters continue to become heavily infected when exposed, and molecular evidence indicates that the parasite remains present throughout the bay. This apparent second step in the development of resistance in the wild oysters occurred after a drought-associated incursion of H. nelsoni into the upper bay in the mid-1980s. Mortalities were widespread, heavy and more extreme than during the 1957–59 epizootic. Resistant survivors of the second epizootic have apparently repopulated the bay. When compared to unselected stocks, common-garden exposure to H. nelsoni of oysters from both upbay and downbay sites indicates that a high degree of resistance to the development of MSX disease has become widespread in the wild oyster population of Delaware Bay after two major selection events separated by nearly 30 years

Managing marine mollusc diseases in the context of regional and international commerce: policy issues and emerging concerns

Marine mollusc production contributes to food and economic security worldwide and provides valuable ecological services, yet diseases threaten these industries and wild populations. Although the infrastructure for mollusc aquaculture health management is well characterized, its foundations are not without flaws. Use of notifiable pathogen lists can leave blind spots with regard to detection of unlisted and emerging pathogens. Increased reliance on molecular tools has come without similar attention to diagnostic validation, raising questions about assay performance, and has been accompanied by a reduced emphasis on microscopic diagnostic expertise that could weaken pathogen detection capabilities. Persistent questions concerning pathogen biology and ecology promote regulatory paralysis that impedes trade and which could weaken biosecurity by driving commerce to surreptitious channels. Solutions that might be pursued to improve shellfish aquaculture health management include the establishment of more broad-based surveillance programmes, wider training and use of general methods like histopathology to ensure alertness to emerging diseases, an increased focus on assay assessment and validation as fundamental to assay development, investment in basic research, and application of risk analyses to improve regulation. A continual sharpening of diagnostic tools and approaches and deepening of scientific knowledge is necessary to manage diseases and promote sustainable molluscan shellfish industries

Effects Of Clam Aquaculture On Nektonic And Benthic Assemblages In Two Shallow-Water Estuaries

Aquaculture of the northern quahog (=hard clam) Mercenaria mercenaria (Linnaeus, 1758) is widespread in shallow waters of the United States from Cape Cod to the eastern Gulf of Mexico. Grow-out practices generally involve bottom planting and the use of predator exclusion mesh. Both the extent and scale of clam farms have increased in recent decades resulting in concerns regarding the impacts of these practices on estuarine fauna. Seasonal distribution, abundance, biomass, species richness, and community composition of nektonic, demersal, epibenthic, and infaunal organisms were examined in cultivated and uncultivated shallow-water habitats in Virginia and New Jersey. The results reveal that clam aquaculture, as practiced in both Virginia and New Jersey, has remarkably few quantifiable impacts on estuarine fauna. Seasonal variations were observed in the biota, but of the 39 population and community metrics tested, mean values associated with 26 did not differ between cultivated and uncultivated areas, 5 had decreases and 8 had increases. For recently harvested areas, 32 of 39 variables were not different from uncultivated areas, 6 decreased in at least one season, and 1, blue crab biomass, had a marginal increase. Decreases were observed in the abundance and biomass of infauna (exclusive of the cultured clams) on clam farms, including in harvested areas, relative to natural uncultivated areas. This was accompanied, however, by substantial increases in epibenthic macroalgae, which in some cases supported increased epifaunal species richness and abundance relative to uncultivated areas. Habitat use by finfish, crustaceans, and terrapins was largely unaffected by the presence of clam farms

Spatial and temporal variability of disease refuges in an estuary: Implications for the development of resistance

Although the concept of genetic refuge has long been employed in ecological and paleoecological context, it has only rarely been used to identify regions where organisms are protected from diseases that affect the rest of a population. The refuges harbor individuals that have not been exposed to selective mortality and remain susceptible to the disease. They represent a reservoir of susceptibility alleles that can mix with those from resistant survivors of disease and can retard the development of resistance in the population as a whole. Two water-borne protistan parasites affect oysters along the east coast of the United States: Haplosporidium nelsoni (MSX disease) and Perkinsus marinus (dermo disease). Both are sensitive to low salinity and their prevalence is reduced in the upper reaches of estuaries. We investigated the temporal and spatial structure and extent of putative refuges from these diseases in the upper Delaware Bay, USA and their potential to affect the development of resistance in the oyster population. Our results showed that refuges occurred as a continuum of zones, regions where a pathogen (1) was not present; (2) was present, but did not cause observable infections; and (3) caused infection, but neither disease nor mortality. The zones were transient, driven only partly by short-term climatic conditions, and differed according to parasite: H. nelsoni was often not present in the refuges, as inferred by the absence of polymerase chain reaction (PCR) – positive signals on the gills, and when it was present, it did not always cause lethal, or even histologically detectable, infections. In contrast, P. marinus was present in all upper estuary areas sampled, where it caused detectable, although not necessarily lethal-level, infections. Thus, a significant fraction of the oyster population is protected from selective mortality in these refuges even when the parasites are present. An incursion of H. nelsoni into the upper Bay in the 1980s left most of the surviving population highly resistant to MSX disease, although populations in the upper-most reaches are still susceptible. The lack of selection pressure in the refuges likely helps to retard the development of resistance to dermo disease, and theoretically could cause resistance to MSX disease to regress although there is no evidence to date that this has occurred

Long-term patterns of an estuarine pathogen along a salinity gradient

Parasitic, disease-causing pathogens can exert strong control over marine populations yet few long-term studies exist that describe these relationships. Understanding the connections to long-term large-scale processes relative to local short-term processes should facilitate better planning for disease impacts in the management of marine resources. We describe a 21-yr dataset of dermo disease (Perkinsus marinus) in eastern oysters (Crassostrea virginica) in Delaware Bay, USA. Analyses indicated (1) a strong positive association between disease and mortality that was non-linear and defined by thresholds, (2) a clear spatial gradient of increasing disease and mortality with increasing salinity, (3) an apparent 7-year cycle in which peaks were associated with strong positive anomalies of the North Atlantic Oscillation (NAO), (4) an inverse relationship with freshwater inflow, and (5) no obvious response to natural selection from persistent disease pressure. These data quantify the impact of environmental variables on the disease in a wild population and provide new insight into how disease interacts with host populations by linking disease patterns with larger climate controlling processes. Understanding these connections will facilitate prediction of and response to disease outbreaks