8 problems

Let $X$ be a variety in characteristic $0$. The uniruledness and Iitaka-fibration conjecture. Either $X$ is uniruled, or $\operatorname{Kod}(X)\geq 0$. If…

Let $T$) be a first-order countable complete theory over a countable language, and let $I(\kappa,T)$ denote the number of models of $T$ of cardinality $\kappa$ up to isomorphism. F…

Let $A$ be a simple, separable, nuclear, infinite-dimensional $C^*$-algebra. The algebra $A$ has strict comparison if positive elements are ordered, in the Cuntz semigroup, accordi…

Let $T$ be a theory. It has weak $k$-$\mathrm{TP}_{1}$ if there exist a formula $\varphi(x,y)$ and tuples $\{b_{\eta}\}_{\eta\in\omega^{<\omega}}$ such that, for every…

Bigstar property conjecture. There is a classification-theoretic property $\bigstar$ such that: (1) there is a property $P$ of set systems for which $T$ has $\bigstar$ if and only…

In the setting of first-order theories, let $\mathrm{NSOP}_1$ denote the property of not having the strict order property of the first kind, and let $\mathrm{NTP}_1$ denote the pro…

Let $K$ be a class from the family under consideration, and let $I(\lambda,K)$ denote the relevant function counting models of cardinality $\lambda$. The main gap conjecture. The f…

Dichotomy conjecture. There is a dichotomy: either the model is similar to categorical theories, or there are “many complicated models”.