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.

Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in probability theory as well as stochastic processes. Nonstandard analysis allows construction of a single object---a hyperfinite probability space---which satisfies all the first order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly in porting discrete results into their continuous settings. 

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In this talk, for every general-state-space discrete-time Markov process satisfying appropriate conditions, we construct a hyperfinite Markov process which has all the basic order logical properties of a finite Markov process to represent it.  We show that the mixing time and the hitting time agree with each other up to some multiplicative constants for discrete-time general-state-space reversible Markov processes satisfying certain condition. Finally, we show that our result is applicable to a large class of Gibbs samplers and Metropolis-Hasting algorithms.