PALS Seminar, University of Colorado, Boulder, Department of Mathematics, Boulder, CO
We propose an analogue definition of No Independence Property (NIP) for abstract elementary classes (AECs) that coincides with NIP when the class is elementary. We construct a forking-like relation on AECs with NIP, and show that its negation leads to being able to encode subsets.
Contributed Talk, 2023 North American Annual Meeting of the Association for Symbolic Logic, UC Irvine, Irvine, CA
We propose an analogue definition of No Independence Property (NIP) for abstract elementary classes (AECs) that coincides with NIP in first order model theory, where the class is elementary. As we have no formulas in this context we work with Galois types. We construct a forking-like relation on AECs with NIP. Our work can be viewed as a new chapter in the neo-stability of AECs, building on Armida-Mazari’s work.
HoTT Seminar, Carnegie Mellon University, Pittsburgh, PA
Deligne’s theorem states that any coherent topos has enough points. The theorem can be viewed as a completeness theorem when specialized to the classifying topos of a geometric theory. First we explain this connection to logic and then present a proof of Deligne’s theorem.
HoTT Seminar, Carnegie Mellon University, Pittsburgh, PA
We will investigate the relationship between μ-abstract elementary classes, which are generalizations of AECs by requiring only μ-directed unions to exist, and accessible categories, which are categories generated by certain “small” objects under μ-directed colimits.
Model Theory Seminar, Carnegie Mellon University, Pittsburgh, PA
We present Sebastien Vasey’s proof of Shelah’s categoricity conjecture in universal classes. First we change the substructure relation of the given class to obtain an AEC with better properties, except that the union axiom might not hold. To prove that the union axiom holds, we use the independence framework AxFr developed by Shelah, build an “independent” tree assuming the failure of union, and contradict stability.
Model Theory Seminar, Carnegie Mellon University, Pittsburgh, PA
We review a paper by Shelah and Vasey (2018), and obtain some properties of an AEC K assuming $\aleph_0$-stability. The main result is: under tameness, amalgamation and no maximal models in $\aleph_0$, there is a good $\aleph_0$-frame and a superlimit in $\aleph_1$.