The Delta Conjecture

International audienceWe conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author

Asymmetric function theory

The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to the larger ring of quasisymmetric functions, with corresponding applications. Here, we survey recent work extending this theory further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the Schubert calculus conference in Guangzhou, Nov. 201

Frequent Promoter Hypermethylation of the APC and RASSF1A Tumour Suppressors in Parathyroid Tumours

BACKGROUND: Parathyroid adenomas constitute the most common entity in primary hyperparathyroidism, and although recent advances have been made regarding the underlying genetic cause of these lesions, very little data on epigenetic alterations in this tumour type exists. In this study, we have determined the levels of promoter methylation regarding the four tumour suppressor genes APC, RASSF1A, p16(INK4A) and RAR-beta in parathyroid adenomas. In addition, the levels of global methylation were assessed by analyzing LINE-1 repeats. METHODOLOGY/PRINCIPAL FINDINGS: The sample collection consisted of 55 parathyroid tumours with known HRPT2 and/or MEN1 genotypes. Using Pyrosequencing analysis, we demonstrate APC promoter 1A and RASSF1A promoter hypermethylation in the majority of parathyroid tumours (71% and 98%, respectively). Using TaqMan qRT-PCR, all tumours analyzed displayed lower RASSF1A mRNA expression and higher levels of total APC mRNA than normal parathyroid, the latter of which was largely conferred by augmented APC 1B transcription levels. Hypermethylation of p16(INK4A) was demonstrated in a single adenoma, whereas RAR-beta hypermethylation was not observed in any sample. Moreover, based on LINE-1 analyses, parathyroid tumours exhibited global methylation levels within the range of non-neoplastic parathyroid tissues. CONCLUSIONS/SIGNIFICANCE: The results demonstrate that APC and RASSF1A promoter hypermethylation are common events in parathyroid tumours. While RASSF1A mRNA levels were found downregulated in all tumours investigated, APC gene expression was retained through APC 1B mRNA levels. These findings suggest the involvement of the Ras signaling pathway in parathyroid tumorigenesis. Additionally, in contrast to most other human cancers, parathyroid tumours were not characterized by global hypomethylation, as parathyroid tumours exhibited LINE-1 methylation levels similar to that of normal parathyroid tissues

Lived Experiences of American Adults who Survive COVID-19: Implications for Physical Activity and Interpersonal Stress

Introduction: The cumulative number of COVID-19 cases has surpassed 579 million globally. Symptoms during and after COVID-19 infection vary from mild cold symptoms to severe multisystem illness. Given the wide range of symptom presentations and complications post COVID-19, the purpose of this study was to describe the lived experience of American adults surviving COVID-19. Method: This study employed an exploratory qualitative description design. Semi-structured interviews were conducted with a sample of 35 individuals, [white (94%), female (71%), mean age = 43.7 years], with proximity to a university in an urban Midwest American city. Interviews occurred between May and August 2021, three or more months after participants tested positive for COVID-19. Results: Forty percent of the 35 participants experienced prolonged COVID-19 symptoms impacting their lifestyle. Four themes characterized the impacts of the post COVID-19 condition on the lives of the participants within the context of a global pandemic: (a) disruptions in health & well-being, (b) persistent uncertainty, (c) disruptions in interpersonal relationships, (d) beneficent outcomes and adaptation. Discussion: This study of COVID-19 has identified important implications for physical activity and interpersonal stress. Prolonged COVID-19 symptoms led to disruptions in the health, well-being, and interpersonal relationships of participants. Healthcare professionals need to attend to symptoms post COVID-19, assess interpersonal functioning, and provide guidance on physical activity. Future studies are recommended to track consequences of COVID-19’s impact on long-term health and well-being