HADWIGER INTEGRATION OF DEFINABLE FUNCTIONS Matthew L. Wright Robert Ghrist, Advisor This thesis defines and classifies valuations on definable functionals. The intrinsic volumes are valuations on “tame” subsets of R, and by easy extension, valuations on functionals on R with finitely many level sets, each a “tame” subset of R. We extend these valuations, which we call Hadwiger integrals, to de...

The notion of a cell and that of a cell decomposition has been a central one in the study of certain first-order theories. A cell is a particular kind of definable set. The notion of a cell was first explicitly considered in [8], in the context of the theory of real closed fields. Collins defined a class of cells in this context, and showed that every definable subset of a real closed field is ...

It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o-minimal structure is totally transcendental of finite Morley rank. Motivation for this paper is a problem concerning stable groups definable in ominimal structures. One theorem is that any defina...

We explore when finite epistemic models are definable up to simulation or bisimulation, either over the basic multi-agent epistemic language L or over its extension L with common knowledge operators. Our main results are the following: 1. Bisimulation to finite epistemic states (i.e., pointed models) is not definable in L. 2. Global bisimulation to finite epistemic models is definable using mod...

The concept of definability of physical fields in a set-theoretical foundation is introduced. A set theory is selected in which we get mathematics enough to produce a nonlinear sigma model. Quantization of the model requires only a null postulate and is then shown to be necessary and sufficient for definability in the theory. We obtain scale invariance and compactification of the spatial dimens...

We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in a...

We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}{\mathrm{alg}}$-definable analytic structure, and prove that any map associated to an admissible variation of Hodge structures is definable in $\mathbb{R}{\mathrm{an,exp}}$ respect this structure. As consequence we re-prove the zero loci normal functions are algebraic.

An answer set program with variables is first-order definable on finite structures if the set of its finite answer sets can be captured by a first-order sentence. Characterizing classes of programs that are first-order definable on finite structures is theoretically challenging and of practical relevance to Answer Set Programming. In this paper, we identify a non-trivial class of answer set pro...

In this article we survey a body of results about classes of finite first order structures in which definable sets have a rather uniform asymptotic behaviour. Non-principal ultraproducts of such classes may have unstable theory, but will be supersimple of finite rank. In addition, they inherit a definable measure on definable sets. The starting point for this work was the following result of [1...