We formulate the Kurepa hypothesis (KH) and its generalizations in determinacy and prove KH fails there. More generally, we show that if $\kappa < \Theta$ and $cof(\kappa) > \omega$, then the set of branches through a  $\kappa$-tree must be well-orderable and has cardinality less than or equal to \kappa. We also show there are no maximal almost disjoint families at $\omega_1$ and its appropriate generalizations at regular cardinals $\kappa < \Theta$. This is joint work with William Chan and Steve Jackson.

1-dependent theories better known as NIP theories are the first class of the hierarchy of n-dependent structures. The random n-hypergraph is the canonical object which is n-dependent but not (n-1)-dependent. Thus the hierarchy is strict. Recently, in a joint work with Chernikov, we proved the existence of strictly n-dependent groups for all natural numbers n and we started studying their properties. The connected component over A, inspired by the definition of the connected component of algebraic group, is the intersection of all A-type definable subgroups of bounded index. A crucial fact about (type)definable groups in 1-dependent theories is the absoluteness of their connected components: Namely given a definable group G and a small set of parameters A, we have that the connected component of G over A coincides with the one over the empty set. A
 
We will give examples of n-dependent groups and discuss a generalization of absoluteness of the connected component to n-dependent theories.

Fix an arbitrary bipartite finite graph H, and let's consider the family of all finite graphs not containing H as an induced subgraph. Recently a strong form of Szemeredi's regularity lemma for such families was established by Lovasz and Szegedy. In particular, up to an arbitrary small error, such graphs can be uniformly approximated by direct products of unary relations. Using a combination of analytic, combinatorial and model-theoretic methods, we establish a generalization of this result for hypergraphs, showing that k-ary hypergraphs omitting a fixed k-partite hypergraph can be uniformly approximated by relations of arity smaller than k. No prior knowledge of the aforementioned notions will be assumed. Joint work with Henry Towsner.

Suppose $I$ is an index set, $(L_i)_{i \in I}$ is a family of first order languages, and $L_\cap$ is a first order language such that $L_i\cap L_j = L_\cap$ for all distinct $i,j \in I$. Suppose $T_i$ is acomplete, consistent, model complete $L_i$-theory for all $i \in I$ and suppose $T_\cap$ is an $L_\cap$-theory such that $T_i \cap T_j = T_\cap$ for all distinct $i,j \in I$. Let $T_\cup = \bigcup_{i \in I} T_i$. We discuss the question: When does $T_\cup$ have a model companion? Time permitting we will also discuss a number of motivating examples. Joint work with Minh Chieu Tran and Alex Kruckman.