A product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials

Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated to the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood-Richardson coefficient have at most 3 parts. We also show that this result extends to Macdonald polynomials.Comment: 30 page

Divergence, thickness and hypergraph index for general Coxeter groups

We study divergence and thickness for general Coxeter groups $W$. We first characterise linear divergence, and show that if $W$ has superlinear divergence then its divergence is at least quadratic. We then formulate a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems $(W,S)$. This generalises Levcovitz's definition for the right-angled case. We prove that if $(W,S)$ has finite hypergraph index $h$, then $W$ is (strongly algebraically) thick of order at most $h$, hence has divergence bounded above by a polynomial of degree $h+1$. We conjecture that these upper bounds on the order of thickness and divergence are in fact equalities, and we prove our conjecture for certain families of Coxeter groups. These families are obtained by a new construction which, given any right-angled Coxeter group, produces infinitely many examples of non-right-angled Coxeter systems with the same hypergraph index. Finally, we give an upper bound on the hypergraph index of any Coxeter system $(W,S)$, and hence on the divergence of $W$, in terms of, unexpectedly, the topology of its associated Dynkin diagram.Comment: 43 pages, 11 figure

A Gallery Model for Affine Flag Varieties via Chimney Retractions

This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat–Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat–Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Sa\c{c}lio\~{g}lu, we present the 238 hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.Comment: J. Phys. A: Math. Theor (to appear

The global burden of adolescent and young adult cancer in 2019 : a systematic analysis for the Global Burden of Disease Study 2019

Background In estimating the global burden of cancer, adolescents and young adults with cancer are often overlooked, despite being a distinct subgroup with unique epidemiology, clinical care needs, and societal impact. Comprehensive estimates of the global cancer burden in adolescents and young adults (aged 15-39 years) are lacking. To address this gap, we analysed results from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019, with a focus on the outcome of disability-adjusted life-years (DALYs), to inform global cancer control measures in adolescents and young adults. Methods Using the GBD 2019 methodology, international mortality data were collected from vital registration systems, verbal autopsies, and population-based cancer registry inputs modelled with mortality-to-incidence ratios (MIRs). Incidence was computed with mortality estimates and corresponding MIRs. Prevalence estimates were calculated using modelled survival and multiplied by disability weights to obtain years lived with disability (YLDs). Years of life lost (YLLs) were calculated as age-specific cancer deaths multiplied by the standard life expectancy at the age of death. The main outcome was DALYs (the sum of YLLs and YLDs). Estimates were presented globally and by Socio-demographic Index (SDI) quintiles (countries ranked and divided into five equal SDI groups), and all estimates were presented with corresponding 95% uncertainty intervals (UIs). For this analysis, we used the age range of 15-39 years to define adolescents and young adults. Findings There were 1.19 million (95% UI 1.11-1.28) incident cancer cases and 396 000 (370 000-425 000) deaths due to cancer among people aged 15-39 years worldwide in 2019. The highest age-standardised incidence rates occurred in high SDI (59.6 [54.5-65.7] per 100 000 person-years) and high-middle SDI countries (53.2 [48.8-57.9] per 100 000 person-years), while the highest age-standardised mortality rates were in low-middle SDI (14.2 [12.9-15.6] per 100 000 person-years) and middle SDI (13.6 [12.6-14.8] per 100 000 person-years) countries. In 2019, adolescent and young adult cancers contributed 23.5 million (21.9-25.2) DALYs to the global burden of disease, of which 2.7% (1.9-3.6) came from YLDs and 97.3% (96.4-98.1) from YLLs. Cancer was the fourth leading cause of death and tenth leading cause of DALYs in adolescents and young adults globally. Interpretation Adolescent and young adult cancers contributed substantially to the overall adolescent and young adult disease burden globally in 2019. These results provide new insights into the distribution and magnitude of the adolescent and young adult cancer burden around the world. With notable differences observed across SDI settings, these estimates can inform global and country-level cancer control efforts. Copyright (C) 2021 The Author(s). Published by Elsevier Ltd.Peer reviewe

Global, regional, and national sex differences in the global burden of tuberculosis by HIV status, 1990–2019: results from the Global Burden of Disease Study 2019

Background Tuberculosis is a major contributor to the global burden of disease, causing more than a million deaths annually. Given an emphasis on equity in access to diagnosis and treatment of tuberculosis in global health targets, evaluations of differences in tuberculosis burden by sex are crucial. We aimed to assess the levels and trends of the global burden of tuberculosis, with an emphasis on investigating differences in sex by HIV status for 204 countries and territories from 1990 to 2019. Methods We used a Bayesian hierarchical Cause of Death Ensemble model (CODEm) platform to analyse 21 505 site-years of vital registration data, 705 site-years of verbal autopsy data, 825 site-years of sample-based vital registration data, and 680 site-years of mortality surveillance data to estimate mortality due to tuberculosis among HIV-negative individuals. We used a population attributable fraction approach to estimate mortality related to HIV and tuberculosis coinfection. A compartmental meta-regression tool (DisMod-MR 2.1) was then used to synthesise all available data sources, including prevalence surveys, annual case notifications, population-based tuberculin surveys, and tuberculosis cause-specific mortality, to produce estimates of incidence, prevalence, and mortality that were internally consistent. We further estimated the fraction of tuberculosis mortality that is attributable to independent effects of risk factors, including smoking, alcohol use, and diabetes, for HIV-negative individuals. For individuals with HIV and tuberculosis coinfection, we assessed mortality attributable to HIV risk factors including unsafe sex, intimate partner violence (only estimated among females), and injection drug use. We present 95% uncertainty intervals for all estimates. Findings Globally, in 2019, among HIV-negative individuals, there were 1.18 million (95% uncertainty interval 1.08-1.29) deaths due to tuberculosis and 8.50 million (7.45-9.73) incident cases of tuberculosis. Among HIV-positive individuals, there were 217 000 (153 000-279 000) deaths due to tuberculosis and 1.15 million (1.01-1.32) incident cases in 2019. More deaths and incident cases occurred in males than in females among HIV-negative individuals globally in 2019, with 342 000 (234 000-425 000) more deaths and 1.01 million (0.82-1.23) more incident cases in males than in females. Among HIV-positive individuals, 6250 (1820-11 400) more deaths and 81 100 (63 300-100 000) more incident cases occurred among females than among males in 2019. Age-standardised mortality rates among HIV-negative males were more than two times greater in 105 countries and age-standardised incidence rates were more than 1.5 times greater in 74 countries than among HIV-negative females in 2019. The fraction of global tuberculosis deaths among HIV-negative individuals attributable to alcohol use, smoking, and diabetes was 4.27 (3.69-5.02), 6.17 (5.48-7.02), and 1.17 (1.07-1.28) times higher, respectively, among males than among females in 2019. Among individuals with HIV and tuberculosis coinfection, the fraction of mortality attributable to injection drug use was 2.23 (2.03-2.44) times greater among males than females, whereas the fraction due to unsafe sex was 1.06 (1.05-1.08) times greater among females than males. Interpretation As countries refine national tuberculosis programmes and strategies to end the tuberculosis epidemic, the excess burden experienced by males is important. Interventions are needed to actively communicate, especially to men, the importance of early diagnosis and treatment. These interventions should occur in parallel with efforts to minimise excess HIV burden among women in the highest HIV burden countries that are contributing to excess HIV and tuberculosis coinfection burden for females. Placing a focus on tuberculosis burden among HIV-negative males and HIV and tuberculosis coinfection among females might help to diminish the overall burden of tuberculosis. This strategy will be crucial in reaching both equity and burden targets outlined by global health milestone

Cancer Incidence, Mortality, Years of Life Lost, Years Lived With Disability, and Disability-Adjusted Life Years for 29 Cancer Groups From 2010 to 2019: A Systematic Analysis for the Global Burden of Disease Study 2019.

The Global Burden of Diseases, Injuries, and Risk Factors Study 2019 (GBD 2019) provided systematic estimates of incidence, morbidity, and mortality to inform local and international efforts toward reducing cancer burden. To estimate cancer burden and trends globally for 204 countries and territories and by Sociodemographic Index (SDI) quintiles from 2010 to 2019. The GBD 2019 estimation methods were used to describe cancer incidence, mortality, years lived with disability, years of life lost, and disability-adjusted life years (DALYs) in 2019 and over the past decade. Estimates are also provided by quintiles of the SDI, a composite measure of educational attainment, income per capita, and total fertility rate for those younger than 25 years. Estimates include 95% uncertainty intervals (UIs). In 2019, there were an estimated 23.6 million (95% UI, 22.2-24.9 million) new cancer cases (17.2 million when excluding nonmelanoma skin cancer) and 10.0 million (95% UI, 9.36-10.6 million) cancer deaths globally, with an estimated 250 million (235-264 million) DALYs due to cancer. Since 2010, these represented a 26.3% (95% UI, 20.3%-32.3%) increase in new cases, a 20.9% (95% UI, 14.2%-27.6%) increase in deaths, and a 16.0% (95% UI, 9.3%-22.8%) increase in DALYs. Among 22 groups of diseases and injuries in the GBD 2019 study, cancer was second only to cardiovascular diseases for the number of deaths, years of life lost, and DALYs globally in 2019. Cancer burden differed across SDI quintiles. The proportion of years lived with disability that contributed to DALYs increased with SDI, ranging from 1.4% (1.1%-1.8%) in the low SDI quintile to 5.7% (4.2%-7.1%) in the high SDI quintile. While the high SDI quintile had the highest number of new cases in 2019, the middle SDI quintile had the highest number of cancer deaths and DALYs. From 2010 to 2019, the largest percentage increase in the numbers of cases and deaths occurred in the low and low-middle SDI quintiles. The results of this systematic analysis suggest that the global burden of cancer is substantial and growing, with burden differing by SDI. These results provide comprehensive and comparable estimates that can potentially inform efforts toward equitable cancer control around the world.Funding/Support: The Institute for Health Metrics and Evaluation received funding from the Bill & Melinda Gates Foundation and the American Lebanese Syrian Associated Charities. Dr Aljunid acknowledges the Department of Health Policy and Management of Kuwait University and the International Centre for Casemix and Clinical Coding, National University of Malaysia for the approval and support to participate in this research project. Dr Bhaskar acknowledges institutional support from the NSW Ministry of Health and NSW Health Pathology. Dr Bärnighausen was supported by the Alexander von Humboldt Foundation through the Alexander von Humboldt Professor award, which is funded by the German Federal Ministry of Education and Research. Dr Braithwaite acknowledges funding from the National Institutes of Health/ National Cancer Institute. Dr Conde acknowledges financial support from the European Research Council ERC Starting Grant agreement No 848325. Dr Costa acknowledges her grant (SFRH/BHD/110001/2015), received by Portuguese national funds through Fundação para a Ciência e Tecnologia, IP under the Norma Transitória grant DL57/2016/CP1334/CT0006. Dr Ghith acknowledges support from a grant from Novo Nordisk Foundation (NNF16OC0021856). Dr Glasbey is supported by a National Institute of Health Research Doctoral Research Fellowship. Dr Vivek Kumar Gupta acknowledges funding support from National Health and Medical Research Council Australia. Dr Haque thanks Jazan University, Saudi Arabia for providing access to the Saudi Digital Library for this research study. Drs Herteliu, Pana, and Ausloos are partially supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNDS-UEFISCDI, project number PN-III-P4-ID-PCCF-2016-0084. Dr Hugo received support from the Higher Education Improvement Coordination of the Brazilian Ministry of Education for a sabbatical period at the Institute for Health Metrics and Evaluation, between September 2019 and August 2020. Dr Sheikh Mohammed Shariful Islam acknowledges funding by a National Heart Foundation of Australia Fellowship and National Health and Medical Research Council Emerging Leadership Fellowship. Dr Jakovljevic acknowledges support through grant OI 175014 of the Ministry of Education Science and Technological Development of the Republic of Serbia. Dr Katikireddi acknowledges funding from a NHS Research Scotland Senior Clinical Fellowship (SCAF/15/02), the Medical Research Council (MC_UU_00022/2), and the Scottish Government Chief Scientist Office (SPHSU17). Dr Md Nuruzzaman Khan acknowledges the support of Jatiya Kabi Kazi Nazrul Islam University, Bangladesh. Dr Yun Jin Kim was supported by the Research Management Centre, Xiamen University Malaysia (XMUMRF/2020-C6/ITCM/0004). Dr Koulmane Laxminarayana acknowledges institutional support from Manipal Academy of Higher Education. Dr Landires is a member of the Sistema Nacional de Investigación, which is supported by Panama’s Secretaría Nacional de Ciencia, Tecnología e Innovación. Dr Loureiro was supported by national funds through Fundação para a Ciência e Tecnologia under the Scientific Employment Stimulus–Institutional Call (CEECINST/00049/2018). Dr Molokhia is supported by the National Institute for Health Research Biomedical Research Center at Guy’s and St Thomas’ National Health Service Foundation Trust and King’s College London. Dr Moosavi appreciates NIGEB's support. Dr Pati acknowledges support from the SIAN Institute, Association for Biodiversity Conservation & Research. Dr Rakovac acknowledges a grant from the government of the Russian Federation in the context of World Health Organization Noncommunicable Diseases Office. Dr Samy was supported by a fellowship from the Egyptian Fulbright Mission Program. Dr Sheikh acknowledges support from Health Data Research UK. Drs Adithi Shetty and Unnikrishnan acknowledge support given by Kasturba Medical College, Mangalore, Manipal Academy of Higher Education. Dr Pavanchand H. Shetty acknowledges Manipal Academy of Higher Education for their research support. Dr Diego Augusto Santos Silva was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil Finance Code 001 and is supported in part by CNPq (302028/2018-8). Dr Zhu acknowledges the Cancer Prevention and Research Institute of Texas grant RP210042

HYPERBOLIC KAC-MOODY WEYL GROUPS, BILLIARD TABLES AND ACTIONS OF LATTICES ON BUILDINGS

Abstract. Let G be a locally compact Kac-Moody group associated to a symmetrizable rank 3 Kac-Moody algebra of noncompact hyperbolic type. It is known that the fundamental chambers of Weyl groups of certain algebras in this class serve as billiard tables for a physical phenomenon known as cosmological billiards. We discuss the classification of Weyl groups in this class, we determine their group presentations and exhibit their tessellations on the Poincaré disk. We show that each Weyl group W is an amalgam of finite Coxeter groups by constructing an action of W on a tree Y. It follows that among all Kac-Moody Weyl groups of affine or hyperbolic type, those of rank 2 and of rank 3 noncompact hyperbolic type are the only ones that have nontrivial amalgam decompositions. We show that the degrees of Y can be determined from the Dynkin diagram of G. We construct a bihomogeneous bipartite tree X embedded in the Tits building of G, a rank 3 locally finite hyperbolic building, on which the full Kac-Mooody group G acts and we determine the degrees of X. We show that there is a nonuniform lattice subgroup of G which acts properly on this tree, that is, with finite vertex stabilizers. 1