- Danny Krashen (Rutgers University): Adventures in Constructive Galois Theory
- Nathan Broaddus (Ohio State Univerisity): Steinberg Modules of Braid Groups
- Emma Benn (Mount Sinai University): Racial/Ethnic Disparities in Health
- Julie Ivy (North Carolina State University): Using Decision Modeling to Personalize Policy
Understanding Galois extensions of fields is a central problem in algebra, with a number of open questions, accessible at a number of levels. At the core, Galois theory is an attempt to understand the arithmetic of fields, by studying the types of equations one can set up over a given field, and the structure and symmetries of their sets of solutions.
In this project we will explore some topics along the edges of "explicit inverse Galois theory," which tries to understand which groups arise as Galois groups for a given field, and how. Our goal will be to take constructive approaches to work in a less explored direction with these Galois extensions to understand richer algebraic structures and properties that collections of Galois extensions exhibit as a whole, in particular looking for reflections of the kinds of structure one seems in Kummer theory. ...
Many important groups of interest in topology are duality groups. As such they have an associated group cohomological object which we call the “Steinberg Module” of the group. We will begin with an introduction to the braid group and discuss a number of elementary descriptions of its Steinberg Module. Our first research goal will be to unify as many of these disparate descriptions as possible. While the motivations behind this first goal relate to group cohomology of the braid group and classification of pointed disk bundles. I expect that the techniques we will use to attack this portion of the project will mostly involve combinatorial surface topology. Concurrently, we will learn the basics of group cohomology and more general duality groups with an eye towards our second research goal--that of taking known results about other duality groups and applying our unified picture of the Steinberg Module of the braid group to proving similar results for the braid group.
Emma K. T. Benn (Mount Sinai University)
Racial/Ethnic Disparities in Health: Applying a More Nuanced Inferential Framework
Emma K. T. Benn, DrPH, MPH (she/her/hers)
Founding Director, Center for Scientific Diversity
Director of Data Science Training and Enrichment, Graduate School of Biomedical Science
Center for Biostatistics & Department of Population Health Science and Policy
Icahn School of Medicine at Mount Sinai
Reducing and eliminating health disparities is of utmost concern for many public health and biomedical researchers and has been a stated goal for Healthy People 2000, 2010, and 2020. However, when it comes to racial disparities in health, researchers have done well at describing differences, but have often struggled to identify mutable targets for intervention. This problem exists for a host of reasons, including the complex contextual factors surrounding racial disparities, however, this may also stem from the way in which we operationalize race in research.
For the proposed project, we will first explore the operationalization of race as a “cause” when examining racial disparities in health based on multidisciplinary discourse around this topic from statisticians informed by the potential outcomes framework, epidemiologists, clinical investigators, and others. Thus, we will collectively build a strong understanding of the methodologic implications of centering “race” as a cause of health disparities. Subsequently, we will critically scrutinize the traditional approaches to investigating disparities in health and apply a more nuanced inferential, rather than descriptive, approach to the statistical analysis of real-world biomedical data with an underlying objective to find efficacious interventions for eradicating health disparities.
Prerequisites for this project include having an introductory to intermediate foundation in statistics or biostatistics and introductory to intermediate proficiency in a statistical programming language (e.g., R, SAS, etc.). I very much look forward to working with a multidisciplinary cohort of ADJOINT scholars interested in applying their expertise, as well as the knowledge we will gain together, to address timely biomedical challenges that disproportionately burden racial/ethnic minorities.
Julie Ivy (North Carolina State University)
Using Decision Modeling to Personalize Policy in Complex Human-Centered Problems
The COVID-19 pandemic highlights the importance of sequential decision making under conditions of uncertainty, learning as the future evolves, and effectively using data to inform decision making. The pandemic further highlights the significant role that mathematical modeling can and should play in addressing complex human-centered problems. This research project will consider these types of problems from a systems modeling perspective. The focus of this project will be decision making under conditions of uncertainty with the goal of modeling complex interactions and quantitatively capturing the impact of different factors, objectives,system dynamics, intervention options and policies on outcomes with the goal of improving decision quality. We will explore Markov decision processes (MDPs), semi-Markov decision process (SMDPs) and partially observable Markov decision process (POMDP) modeling frameworks for structuring complex decision problems at the intersection of health, wealth, and education. We will explore how to use data integrated with stakeholder and subject matter expertise to formulate meaningful models to inform decision making. We will consider complimentary methods including simulation, deterministic optimization, robust optimization and stochastic programming methods. Some of the modeling and analysis challenges we will explore include: (i) issues around capturing natural history of a disease or process; (ii) defining metrics for “good”, i.e., metrics for quantifying a “good” decision; (iii) considering multiple objectives; (iv) measuring the value of information; (v) estimating the quality of information,e.g., estimating likelihood functions under limited information; (vi) estimating/quantifying preference; and (vii) dealing with messiness in data (e.g., missing with information. In this project, we will also consider how our research can best influence practitioners and how they address complex societal issues, such as health disparities, public health preparedness, and personalized medical decision-making.