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, we will discuss the results of the 2023 Towson University REU on indivisibility of classes of finite graphs. We will first consider indivisibility for hereditarily sparse graphs. Then we will turn our attention to a number of classes of graphs which are characterized by forbidden induced subgraphs, including cographs and perfect graphs. This is joint work with Vince Guingona, Felix Nusbaum, Zain Padamsee, Christian Pippin, and Ava Zinman.
Abstract: We will discuss various strengthenings of Szemerédi's regularity lemma for hypergraphs that are tame from the model-theoretic point of view. Generalizing the case of stable graphs due to Malliaris and Shelah, we have shown the following in a joint work with Starchenko: if a 3-hypergraph E(x,y,z) on X x Y x Z is stable when viewed as a binary relation under any partition of its variables in two groups, then there are partitions X_i of X, Y_j or Y and Z_k of Z so that the density of E on any box X_i x Y_j x Z_k is either 0 or 1. Terry and Wolf raised the question if the assumption can be relaxed to slice-wise stability, i.e. for any z in Z, the corresponding fiber E_z is a stable relation on X x Y, and similarly for any permutation of the variables (analogous slice-wise assumption is known to be correct in the NIP case). We provide an example of a slice-wise stable 3-hypergraph which does not satisfy the stable regularity lemma above, and establish an optimal weaker partition result for slice-wise stable hypergraphs. Joint work with Henry Towsner.
Abstract: We will discuss various strengthenings of Szemerédi's regularity lemma for hypergraphs that are tame from the model-theoretic point of view. Generalizing the case of stable graphs due to Malliaris and Shelah, we have shown the following in a joint work with Starchenko: if a 3-hypergraph E(x,y,z) on X x Y x Z is stable when viewed as a binary relation under any partition of its variables in two groups, then there are partitions X_i of X, Y_j or Y and Z_k of Z so that the density of E on any box X_i x Y_j x Z_k is either 0 or 1. Terry and Wolf raised the question if the assumption can be relaxed to slice-wise stability, i.e. for any z in Z, the corresponding fiber E_z is a stable relation on X x Y, and similarly for any permutation of the variables (analogous slice-wise assumption is known to be correct in the NIP case). We provide an example of a slice-wise stable 3-hypergraph which does not satisfy the stable regularity lemma above, and establish an optimal weaker partition result for slice-wise stable hypergraphs. Joint work with Henry Towsner.
Abstract: We will discuss various strengthenings of Szemerédi's regularity lemma for hypergraphs that are tame from the model-theoretic point of view. Generalizing the case of stable graphs due to Malliaris and Shelah, we have shown the following in a joint work with Starchenko: if a 3-hypergraph E(x,y,z) on X x Y x Z is stable when viewed as a binary relation under any partition of its variables in two groups, then there are partitions X_i of X, Y_j or Y and Z_k of Z so that the density of E on any box X_i x Y_j x Z_k is either 0 or 1. Terry and Wolf raised the question if the assumption can be relaxed to slice-wise stability, i.e. for any z in Z, the corresponding fiber E_z is a stable relation on X x Y, and similarly for any permutation of the variables (analogous slice-wise assumption is known to be correct in the NIP case). We provide an example of a slice-wise stable 3-hypergraph which does not satisfy the stable regularity lemma above, and establish an optimal weaker partition result for slice-wise stable hypergraphs. Joint work with Henry Towsner.
Abstract: We will discuss a characterization of first-order theories realizing a certain combinatorial tree configuration in terms of special coheirs. One direction of the proof can be understood as a kind of forcing argument.
Abstract: Knaster continua are a class of compact, connected, metrizable spaces which are indecomposable in the sense that they cannot be written as the union of two proper non-trivial subcontinua. Let $K$ be the universal Knaster continuum (this is a unique Knaster continuum which continuously and openly surjects onto all other Knaster continua). The group $\textrm{Homeo}(K)$ of all homeomorphisms of the universal Knaster continuum is a non-locally compact Polish group. We prove that it contains an open subgroup which exhibits two ``large Polish group'' phenomena: the existence of a comeager conjugacy class and extreme amenability.
Abstract: The concept of a Schauder basis is central to Banach space theory, but occasionally too wide to be useful in Banach lattices, while too restrictive for general separable Banach spaces. In Banach lattices, new notions of bases incorporate different types of order convergence, whereas in Banach space, one can conversely weaken sequential convergence to filter convergence. Applying tools from descriptive set theory, we answer several fundamental questions due to Gumenchuk, Karlova and Popov (J. Funct. Anal. 2015), Taylor and Troitsky (J. Funct. Anal. 2020), and de Rancourt, Kania and Swaczyna (J. Funct. Anal. 2023) related to these concepts. This is a joint work with A. Àviles, M. A. Taylor and P. Tradacete.
Abstract: The concept of a Schauder basis is central to Banach space theory, but occasionally too wide to be useful in Banach lattices, while too restrictive for general separable Banach spaces. In Banach lattices, new notions of bases incorporate different types of order convergence, whereas in Banach space, one can conversely weaken sequential convergence to filter convergence. Applying tools from descriptive set theory, we answer several fundamental questions due to Gumenchuk, Karlova and Popov (J. Funct. Anal. 2015), Taylor and Troitsky (J. Funct. Anal. 2020), and de Rancourt, Kania and Swaczyna (J. Funct. Anal. 2023) related to these concepts. This is a joint work with A. Àviles, M. A. Taylor and P. Tradacete.
Abstract: Abstract: We introduce the notion of strongly $\binom{k}{2}$-free graphs, which contain dp-minimal graphs. We show that under some sparsity assumption, given a rainbow $\binom{k}{2}$-free blockade we can find a rainbow $\binom{k-1}{2}$-free blockade. This might serve as an intermediate step towards Erdős-Hajnal property for dp-minimal graphs.
Abstract: I will discuss recent work with Abd Aldaim and Terry on the model theory of a new notion of stability formulated for a (k+1)-partitioned formula (with classical stability corresponding to the k=1 case). The focus will be on basic properties, fundamental examples, and further open questions.
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