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  1. ; ; ( , Revista Matemática Iberoamericana)
    In this paper, we generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in X by bi-Lipschitz curves. 
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    Free, publicly-accessible full text available May 6, 2025
  2. ; ; ( , Analysis and Geometry in Metric Spaces)
    We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^2 (P. Jones, 1990), in R^n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' beta numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R^n that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space. 
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    Free, publicly-accessible full text available November 22, 2024
  3. ; ; ; ( , Annales Academiae Scientiarum Fennicae Mathematica)
    Full Text Available 
  4. ; ( , Indiana University Mathematics Journal)
    Full Text Available 
  5. ; ; ; ; ; ; ; ; ( , Alzheimer's & Dementia)
    Abstract Introduction

    We evaluated whether competing risk of death or selective survival could explain the reported inverse association between cancer history and dementia incidence (incidence rate ratio [IRR] ≈ 0.62‐0.85).

     Methods

    A multistate simulation model of a cancer‐ and dementia‐free cohort of 65‐year‐olds was parameterized with real‐world data (cancer and dementia incidence, mortality), assuming no effect of cancer on dementia (true IRR = 1.00). To introduce competing risk of death, cancer history increased mortality. To introduce selective survival, we included a factor (prevalence ranging from 10% to 50%) that reduced cancer mortality and dementia incidence (IRRs ranged from 0.30 to 0.90). We calculated IRRs for cancer history on dementia incidence in the simulated cohorts.

     Results

    Competing risk of death yielded unbiased cancer‐dementia IRRs. With selective survival, bias was small (IRRs = 0.89 to 0.99), even under extreme scenarios.

     Discussion

    The bias induced by selective survival in simulations was too small to explain the observed inverse cancer‐dementia link, suggesting other mechanisms drive this association.

     
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