Descriptive Inner Model Theory , Large Cardinals , and Combinatorics Research Statement Nam Trang

My research interest is in mathematical logic and set theory. My current research focuses on studying the connections between inner models, (determined) sets of reals, hybrid structures (such as HOD1 of determinacy models), forcing, and strong combinatorial principles (such as the Proper Forcing Axiom (PFA), (generalizations of) the tree property, the Unique Branch Hypothesis (UBH)). I’m also interested in applications of strong forcing axioms such as PFA and their connection with other combinatorial principles in set theory. The research described below mostly belongs to an area of set theory called descriptive inner model theory (DIMT). DIMT is an emerging field in set theory that explores deep connections between descriptive set theory (DST) and inner model theory (IMT). DST studies a certain class of well-behaved subsets of the reals and of Polish spaces (e.g. Borel sets, analytic sets) and has its roots in classical analysis, through work of Baire, Borel, Lebesgue, Lusin, Suslin and others. One way a collection Γ of subsets of a Polish space can be well-behaved is that they satisfy various regularity properties, e.g. they have the Baire property, every uncountable set in Γ contains a perfect subset, every set in Γ is Lebesgue measurable. A cornerstone in the history of the subject is the discovery of the Axiom of Determinacy (AD) by Mycielsky and Steinhaus in 1962. AD states that every infinite-length, two-person game of perfect information where players take turns play integers is determined, i.e. one of the players has a winning strategy. If every set in Γ ⊆ P(R) is determined, then they have all the regularity properties listed above (and more), and hence very well-behaved. AD contradicts the axiom of choice as the latter implies the existence of very irregular sets like the Vitali set; however, inside a universe of ZFC, there may be many interesting sub-universes (models) that satisfy AD, for instance, L(R) the minimal transitive class model of ZF that contains the reals may satisfy AD. One important and fruitful branch of descriptive set theory studies structure theory of models of AD. IMT forms one of the core subjects in modern set theory; its main objective is study “canonical” models of various extensions of ZFC, called large cardinal axioms (or simply large cardinals) and construct such models under various circumstances (e.g. see question (2) below). The large cardinal axioms form a linear hierarchy of axioms (in terms of consistency strength) extending ZFC and every known, natural axiom in mathematics/set theory is decided by one such axiom.2 The first “canonical model” of large cardinals is Gödel’s constructible universe L, the minimal model of ZFC. It is well-known that L cannot admit “very large” large cardinals; the Gödel’s inner model program, a major program in inner model theory, aims to construct and analyze L-like models that can accommodate larger large cardinals under various hypotheses. Benchmark properties that help determine the canonicity of these models include the Generalized Continuum Hypothesis (GCH), Jensen’s -principles (see Question (1), more details later).