Link invariants of long pieces of orbits of a volume-preserving flow can be used to define diffeomorphism invariants of the flow. The wrapping number of a link in the solid torus and the trunk of a link can be generalized and define invariants of links with respect to a fibration on a 3-manifold. Extending work of Dehornoy and Rechtman, we apply this to define diffeomorphism invariants wrappingness and trunkenness of volume-preserving flows on 3-manifolds and interpret these invariants as obstructions to the existence of a global surface of section for the flow. We construct flows and show that wrappingness and trunkenness are not functions of the helicity.

Coding theory has many tools that come from Algebra and Algebraic Geometry. We will explore some of the important families of error-correcting codes that are used today.

In this talk we will give two motivations for building theory and methodologies around systems of equations over idempotent semirings. We will show how a theory of equations over idempotent semirings could be used in both real world applications, such as creating a solar system wide internet, as well as to create new mathematical tools in areas such as commutative algebra. We will first briefly discuss how the computational problem of routing in a deep space satellite network can be reduced to solving a matrix equation over specific idempotent semirings and how this model allows one to solve secondary problems such as determining storage requirements in a network. We will then see how idempotent semirings can be used as a tool to study commutative algebra. Through the Giansiracusa's generalized valuation theory one can study the spectrum and structure of commutative rings through valuations into idempotent semirings and the maps between them. We will conclude by examining what it means to "solve a system of equations" and how these problems can be modelled categorically.

In this talk I'll give a brief introduction to modular forms and discuss an analytic method of constructing them, focusing on new examples related to class numbers of binary quadratic forms.

We consider the complete topological reconstruction of a geodesic subspace of Euclidean space from the Vietoris--Rips complexes of a finite, noisy Euclidean sample. Instead of the Euclidean metric, our reconstruction technique uses a path-based metric on the sample to construct the Vietoris--Rips complexes.

We consider the restricted distortion, alpha-distortion, convexity radius, and Alexandrov curvature of a geodesic space as our sampling parameters. With restricted distortion, we guarantee a homotopy-equivalent reconstruction from the sample. With alpha-distortion, we guarantee homology groups equivalence and fundamental group equivlance with reconstruction from the sample. This study provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and $\mu$--reach.

Pediatric population presents several barriers for clinical trial design and analysis, including ethical constraints on the sample size and slow accrual rate. Pediatric drug development lags adult development by about 8 years and children represent a large underserved population of "therapeutic orphans," as an estimated 80% of children are treated off-label (Mulugeta et al. in Pediatr Clin 64(6):1185-1196, 2017). It may be appropriate to extrapolate existing data when the course of the disease or condition and effects of the drugs are sufficiently similar between source and target populations. Pediatric extrapolation is an approach that the FDA and EMA are encouraging sponsors to adopt to minimize the number of children that are required to participate in clinical trials. With recent ICH harmonization on pediatric extrapolation framework, despite uncertainties, pediatric drug development programs should initially assume some degree of extrapolation. The degree to which extrapolation can be used lies along a continuum representing the uncertainties to be addressed through generation of new pediatric evidence (Gamalo et al. 2021). Represented by ASA Pediatric Working Group of Extrapolation Subgroup, this talk will help audience recognize when extrapolation is appropriate, understand the extrapolation landscape in pediatrics, how extrapolation was applied in a small pediatric trial through real examples, introduce recent statistical developments for extrapolation frameworks, and share our thoughts on the regulatory guideline of ICH E11.