  The report describes the discussions from the Workshop on Mathematical
Opportunities in Digital Twins (MATH-DT) from December 11-13, 2023, George
Mason University.
  It illustrates that foundational Mathematical advances are required for
Digital Twins (DTs) that are different from traditional approaches. A
traditional model, in biology, physics, engineering or medicine, starts with a
generic physical law (e.g., equations) and is often a simplification of
reality. A DT starts with a specific ecosystem, object or person (e.g.,
personalized care) representing reality, requiring multi -scale, -physics
modeling and coupling. Thus, these processes begin at opposite ends of the
simulation and modeling pipeline, requiring different reliability criteria and
uncertainty assessments. Additionally, unlike existing approaches, a DT assists
humans to make decisions for the physical system, which (via sensors) in turn
feeds data into the DT, and operates for the life of the physical system.
  While some of the foundational mathematical research can be done without a
specific application context, one must also keep specific applications in mind
for DTs. E.g., modeling a bridge or a biological system (a patient), or a
socio-technical system (a city) is very different. The models range from
differential equations (deterministic/uncertain) in engineering, to stochastic
in biology, including agent-based. These are multi-scale hybrid models or large
scale (multi-objective) optimization problems under uncertainty. There are no
universal models or approaches. For e.g., Kalman filters for forecasting might
work in engineering, but can fail in biomedical domain. Ad hoc studies, with
limited systematic work, have shown that AI/ML methods can fail for simple
engineering systems and can work well for biomedical problems.
  A list of `Mathematical Opportunities and Challenges' concludes the report.
