James Holland

jch258@scarletmail.rutgers.edu

Research

I am a student of Grigor Sargsyan, and am interested in set theory and logic. Logic is a kind of meta-mathematics, asking questions about the work that mathematicians do like whether certain statements are provable or not. One way to frame these sorts of questions is through the existence or non-existence of certain kinds of objects: is there a counter-example? Mathematics makes things precise using sets, and so these questions can be reduced to statements about what kinds of sets exist. Sometimes the existence or non-existence of such sets can't be proven (or worse, we can't even show their existence is possible). But such abstract concepts can provide a deeper understanding about the nature of sets and the mathematical universe. Their existence often tells us what the universe should look like and what statements should be true even if we have no hope of proving them. Investigating these sets whose existence cannot be proven, often called large cardinals, is a subject of much in research set theory from a variety of different perspectives.

I studied large cardinals primarily through the use of forcing, and to a lesser extent inner model theory, two major areas in set theory.

Research Statement

[Research Statement] This goes more in depth into my research.

Papers

Weak Indestructibility and Reflection (Part of thesis work, Submitted, arχiv)

Forcing More DC Over the Chang Model Using the Thorn Sequence (In preparation with Grigor Sargsyan)

Three Measures in HOD (In preparation with Navin Aksornthong, Takehiko Gappo, and Grigor Sargsyan)