Perverse Sheaves on the Nilpotent Cone and Lusztig's Generalized Springer Correspondence

In this note, we consider perverse sheaves on the nilpotent cone. We prove orthogonality relations for the equivariant category of sheaves on the nilpotent cone in a method similar to Lusztig's for character sheaves. We also consider cleanness for cuspidal perverse sheaves and the generalized Lusztig--Shoji algorithm.Comment: 18 pages, minor typos correcte

Graham\u27s variety and perverse sheaves on the nilpotent cone

In recent work, Graham has defined a variety which maps to the nilpotent cone, and which shares many properties with the Springer resolution. However, Graham\u27s map is not an isomorphism over the principal orbit, and for type A in particular, its fibers have a nice relationship with the fundamental groups of the nilpotent orbits. The goal of this dissertation is to determine which simple perverse sheaves appear when the Decomposition Theorem for perverse sheaves is applied in Graham\u27s setting for type A, and to begin to answer this question in the other types as well. In Chapter 1, we give some motivation and a brief description of this project. Then, Chapter 2 is a summary of several background topics. In Chapter 3, we review Graham\u27s construction of his variety. In Chapter 4, we use results of Tymozcko to study the fibers of Graham\u27s map in type A. Chapter 5 contains the conclusions in the perverse sheaf setting, and lastly, Chapter 6 contains results pertaining to Graham\u27s fibers in the other types

Environmental variability affects gut microbiota and experimental outcomes in mouse models

The gut microbiome of humans and animals is critical to host health. Mice are used to investigate the microbiome and its influences; however, the predictive value of such studies is hindered by cage effects due to coprophagy. Our objectives were to evaluate the influence of cage density on the statistical power to detect treatment-dependent effects of a selective pressure on microbiome composition. C57BL/6 mice were separated into groups of 2 or 4 mice per cage, and then assigned to groups receiving enrofloxacin, broad-spectrum antibiotics, or control. Fecal samples were collected at weeks 0, 1, and 4, along with contents of the jejunum and cecum. Bacterial DNA was analyzed for microbiome richness, diversity, and variability within and between cages. Statistical analyses revealed that reduced housing density consistently resulted in comparable susceptibility to antibiotics, reduced cage effect, and increased statistical power to detect treatment-associated effects, justifying the practice of reduced housing density. There is limited understanding of how the microbiota colonizing various maternal tissues contribute to the development of the neonatal gut microbiota (GM). To determine the contribution of various maternal microbiotic sites to the offspring microbiota in the upper and lower gastrointestinal tract (GIT) during early life, litters of mice were sacrificed at 7, 9, 10, 11, 12, 14, and 21 days of age, and fecal and ileal samples were collected. Dams were euthanized alongside their pups, and oral, vaginal, ileal, and fecal samples were collected. This was done in parallel using mice with either a low-richness or high-richness microbiota to assess the consistency of findings across multiple microbial compositions. Sample were analyzed using 16S rRNA amplicon sequencing. The similarities between matched pup and dam samples were used to determine the contribution of each maternal source to the pup fecal and ileal composition at each timepoint. As expected, similarity between pup and maternal feces increased significantly over time. During earlier time-points however, the offspring fecal and ileal microbiotas were closer in composition to the maternal oral microbiota than other maternal sites. Prominent taxa contributed by the maternal oral microbiota to the neonatal gut microbiota were supplier-dependent and included Lactobacillus spp., Streptococcus spp., and a member of the Pasteurellaceae family. These findings align with the microbial taxa reported in infant microbiotas, highlighting the translatability of mouse models in this regard, as well as the dynamic nature of the gut microbiota during early life. The Non-Obese diabetic model of type 1 diabetes (T1D) has been the most highly utilized animal model in T1D research, resulting in multiple studies that have furthered the understanding of T1D development. However, this model suffers from inconsistency in disease incidence between studies that has been attributed to environmental variability, resulting in compositional shift in the gut microbiota (GM). It is currently unclear how these variations in GM composition influence the overall T1D phenotype in NOD mice. Therefore, this study aims to determine how sex, GM composition, and exposure to specific pathobionts previously suspected to alter T1D incidence interact to influence the T1D phenotype in NOD mice. Two cohorts of NOD mice were utilized to analyze diabetic and pre-diabetic timepoints of disease development for T1D incidence and insulitis severity respectively. The GM composition of these mice were then analyzed to identify shifts in beta diversity and predicted metabolic function associated with pathobiont inoculation. Results show all three factors interact to effect T1D incidence, insulitis severity, and GM beta diversity. Most prevalent, H. hepaticus shifted beta diversity in all groups and tended to exacerbate disease onset and severity. This H. hepaticus driven shift in beta diversity was associated with predicted changes in metabolic functions related to T1D.Includes bibliographical references

Geometric and combinatorial properties of extended Springer fibers

We consider a generalization of the Springer resolution studied in earlier work of the authors, called the extended Springer resolution. In type $A$, this map plays a role in Lusztig's generalized Springer correspondence comparable to that of the Springer resolution in the Springer correspondence. The fibers of the Springer resolution play a key part in the latter story, and connect the combinatorics of tableaux to geometry. Our main results prove the same is true for fibers of the extended Springer resolution -- their geometry is governed by the combinatorics of tableaux. In particular, we prove that these fibers are paved by affines, up to the action of a finite group, and give combinatorial formulas for their Betti numbers. This yields, among other things, a simple formula for dimensions of stalks of the Lusztig sheaves arising in the study of the generalized Springer correspondence, and shows that there is a close resemblance between each Lusztig sheaf and the Springer sheaf for a smaller group.Comment: 37 pages, 2 figure

Positive Solutions to a Diffusive Logistic Equation with Constant Yield Harvesting

We consider a reaction diffusion equation which models the constant yield harvesting of a spatially heterogeneous population which satisfies a logistic growth. In particular, we study the existence of positive solutions subject to a class of nonlinear boundary conditions. We also provide results for the case of Neumann and Robin boundary conditions. We obtain our results via a quadrature method and Mathematica computations

TOMM40 ‘523 associations with baseline and longitudinal cognition in APOEε3 homozygotes

TOMM40 ‘523 is associated with Alzheimer’s disease (AD), but APOE linkage disequilibrium confounds this association. In 170 APOE ε3 homozygotes, we evaluated relationships between short and very long TOMM40 alleles and longitudinal declines in three cognitive domains (attention, verbal memory, and executive function). We used factor analysis to create composite scores from 10 individual cognitive tests, and latent growth curve modeling adjusting for clinical status (normal, amnestic mild cognitive impairment, or AD) to summarize initial performance and change over three years. Relative to individuals with two very long TOMM40 alleles, APOE ε3 homozygotes with one or two short alleles showed lower baseline cognitive performance regardless of clinical status. The number of short or very long TOMM40 alleles was not associated with longitudinal cognitive changes. In APOE ε3 homozygotes from the KUADC cohort, an association between TOMM40 ‘523 and cognition is consistent with the possibility that TOMM40 influences cognition independent of APOE.P30AG03598

A comparison of placebo and nocebo effects on objective and subjective postural stability: a double-edged sword?

Background: Positive expectations (i.e., placebo effect) can improve postural control during quiet standing. This raises an important question: if postural control is susceptible to positive expectations, is it possible to elicit the opposite, a decline in postural stability, simply by suggesting a performance impairment (i.e., nocebo) will take place? Yet no studies have examined the nocebo effect on balance performance. To better understand both phenomena, comparative studies, which include both placebo and nocebo conditions, are needed. Method: Forty-two healthy adults were initially assessed for objective (center of pressure movement) and subjective (perceived) postural stability and performance expectations. Participants were then randomly assigned in equal numbers to a placebo (positive expectation), nocebo (negative expectation) or control (no suggestion) group. Participants in the placebo/nocebo groups were deceptively administered an inert capsule described as a potent supplement which would either positively or negatively influence their balance performance. Objective and subjective postural stability, and performance expectations were reassessed 20 min later. Results: The nocebo procedure evoked an increase in COP sway movements and reduced perceived stability compared to a control group. The placebo group presented with reductions COP sway movements and increased perceived stability following expectation manipulation. Compared to the control group, the placebo group showed a significantly higher performance expectation whilst the nocebo group showed a significantly lower performance expectation. Regression analyses also revealed that performance expectations following the placebo/nocebo procedure significantly predicted perceptions of postural instability (i.e., perceived performance), accounting for around 50% of the variance. These results remained even when controlling for actual performance (i.e., objective postural stability). Conclusion: Our findings indicate that positive and negative performance expectations evoked by instructional manipulation can profoundly influence both objective and subjective postural stability. Postural control—and perceptions regarding such—are clearly susceptible to expectation manipulation, which could have important practical implications and repercussions on testing, training interventions and rehabilitation programs. Positive and negative expectancies are a double-edged sword for postural control

Concepts of Health in Older Urban African American Women with Chronic Health Conditions: A Focus Group Study

The purpose of this report is to relate the findings of the study ‘Concepts of Health in Older Urban African American Women with Chronic Health Conditions.’ This investigation, undertaken at the request of the Indiana Minority Health Coalition, Inc. (IMHC) is collaboration between IMHC, Butler University College of Pharmacy and Health Sciences, and the community partner, the Black Nurses Association of Indianapolis, Inc. (BNA). The purpose of the study is to gain an understanding of the ways older African American women from medically underserved areas of Indianapolis characterize the various parts of the concept of health; how they understand and interpret the determinants of health status; and to understand how they assess health. Researchers decided to hold focus groups of African American women from Indianapolis ages 50 years and older who had at least one diagnosis of a chronic illness. The Principal Investigator created a moderator’s guide for the focus group. All research partners met to review and redesign the guide to make sure that it was appropriate and would be effective in stimulating discussion to address the three specific aims. The groups, one of currently employed women, two of residents of senior/disabled housing, one group of members of a community church, and one mixed-recruitment group, were held in community settings. Participants received $25.00 gift cards for their involvement. Major findings are that women view ‘health’ as a concept that includes many elements: physical health, mental/emotional health, ability to function through day-to-day activities, and spiritual health. These components work together to produce a state of wholeness or well-being. ‘Health’ is determined through interplay between influences that operate on many levels: the personal, the interpersonal, and the immediate and social/political environment. Specific determinates include health behaviors, state of mind, stress, relationship with God, and the aging process. Women speak at length about their relationship with institutions of healthcare. They express problems in communicating with their healthcare providers, feeling that they are not treated as unique individuals, and they often express distrust of providers. Participants assess their own health through paying attention to their own physical symptoms, their state of mind, and their energy level. Medications are named as determinants of health and markers of health status. Women assess health of their families and friends through observation, intuition, and communication. These women act as facilitators of health status and behaviors for their family members and others in the community. Women are well aware of the importance of positive health behaviors, especially diet and exercise, and they feel strong senses of control over their health. As these women envision ‘health’ as involving many concepts and levels, interventions need to address issues beyond the physical, involving women actively through relationships within communities. Public health professionals need to work with these women in a respectful and collaborative manner. Issues that need further exploration are interplay between mental and physical health through the aging process, patient-provider communication, and women’s understanding of medications in maintaining and improving health