Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals ($x$, $Tx$, $T^2 x$,...,$T^n x$) in front of the point $x$. We prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of $T$ that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields (forward) ergodic theorems for countable groups, in particular, ones for pmp actions of free groups and semigroups of finite rank. In each case, the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This is joint work with Anush Tserunyan.
Abstract: In this talk, I will discuss some of the work that Miriam Parnes and I did with four undergraduate students at an REU over the summer. This work was motivated by earlier work of myself and various other authors (Cameron Hill, Miriam Parnes, and Lynn Scow) investigating configurations as a means of classifying theories. We discovered that imposing some level of uniformity on the index class aided in understanding configurations and, while the Ramsey property was too restrictive, indivisibility seemed be sufficient. This talk will focus on indivisibility, especially viewed through the lens of classes of structures with a notion of strong substructure. (This work is joint with Felix Nusbaum, Zain Padamsee, Miriam Parnes, Christian Pippin, and Ava Zinman.)
Abstract: In this talk, I will discuss some of the work that Miriam Parnes and I did with four undergraduate students at an REU over the summer. This work was motivated by earlier work of myself and various other authors (Cameron Hill, Miriam Parnes, and Lynn Scow) investigating configurations as a means of classifying theories. We discovered that imposing some level of uniformity on the index class aided in understanding configurations and, while the Ramsey property was too restrictive, indivisibility seemed be sufficient. This talk will focus on indivisibility, especially viewed through the lens of classes of structures with a notion of strong substructure. (This work is joint with Felix Nusbaum, Zain Padamsee, Miriam Parnes, Christian Pippin, and Ava Zinman.)
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