In this chapter we discuss advanced tools and techniques, which rely on additional concepts from algebraic geometry. These tools could be helpful for people who do research work in incidence theory and related topics. A reader who is new to this field might prefer to skip this chapter.

We sometimes wish to consider families of varieties, such as the set of circles in the plane or the set of planes in R^3 that not are incident to the origin. In this chapter, we rigorously define such families. We also generalize the idea of point-line duality to every family of varieties. We then see how these notions could be used to prove various results. In particular, we derive a new incidence bound and prove various properties of surfaces in R^3 and C^3.

This chapter presents an overview of heritage speakers’ linguistic outcomes with the goal of bringing some answers to longstanding debates in contact linguistics. Of special interest are mechanisms of contact-induced change such as convergence and grammaticalization, the role of transfer and priming in those processes, and whether or not these processes lead to simplification or complexification in heritage grammars. Overall, the evidence in the literature favors the argument that syntactic material rarely gets transferred, and that patterns of convergence toward the dominant language are a byproduct of other mechanisms. Contrary to proponents of the theory that social factors take a secondary role in contact-induced grammaticalization, it has also revealed that contact-induced grammaticalization processes in heritage grammars are dependent on the social dynamics of the community. While priming may lead to converge, such effects are also heavily dependent on the social dynamics of the heritage speakers. Results also indicate that patterns of simplification and complexification are intrinsically linked to usage. The discussion makes a call for more work considering intersubjective factors in the study of heritage grammars, in an attempt to better understand the relationship between linguistic systems and social structures.

In financial regulation, regulated entities are required to prepare disclosures that detail the risks of their transactions. However, these regulatory requirements sometimes neglect to ensure that financial entities have strong incentives to understand the risks themselves, much less ensure the disclosed risks are comprehensible to others. This chapter applies the concept of comprehension asymmetries to financial regulation and uncovers ways that the existing laws and rules tolerate and sometimes exacerbate wilfull ignorance and incomprehensible disclosures. The chapter closes with suggestions for reform.

Consumer protection law is notoriously imbalanced with respect to the superior ability of sellers to process information as compared to their customers. Yet despite the resulting comprehension asymmetries, the design of consumer contract law and disclosure requirements regularly fail to encourage sellers to communicate meaningfully with the target audience. This chapter explores how consumer protection law tacitly encourages incomprehensibility and proposes reforms which would provide increased incentives for meaningful communication between buyers and sellers.