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Title: Diagonal Actions of Groups Acting on Rooted Trees Speaker: Dima Savchuk Time: 2:00pm–3:00pm Place: CMC 118

For a group \(G\) acting on a regular rooted \(d\)-ary tree \(T_d\) and on its boundary \(\partial T_d\) we consider the diagonal actions of \(G\) on the powers of \(T_d\) and \(\partial T_d\). For the action of the full group \(\mathrm{Aut}\left(T_d\right)\) of automorphisms of \(T_d\) we describe the ergodic decomposition of its action on \(\left(\partial T_d\right)^n\) for all \(n\geq 1\). To achieve it we analyze the orbits of \(n\)-tuples of elements of vertices of any fixed finite level of \(T_d\). For a subgroup \(G\) of \(\mathrm{Aut}\left(T_d\right)\) the corresponding orbits may be smaller, but sometimes they coincide with the orbits of the full group of automorphisms for all levels. In this case we say that the action of \(G\) on \(\mathrm{Aut}\left(T_d\right)\) is maximally tree \(n\)-transitive. For example, maximal tree \(1\)-transitivity is equivalent to level transitivity of the action of \(G\) on \(T_d\). It follows from the results of \([1,2]\) that Grigorchuk group and Basilica group act maximally tree \(2\)-transitively on \(\partial T_2\). We show that the action of Grigorchuk group on \(\partial T_2\) is, in-fact, maximally tree \(4\)-transitive but not maximally \(5\)-transitive. The talk is based on a joint work with Rostislav Grigorchuk and Zoran Šunić.

Seminar is cancelled due to Hurricane Milton.

Title: Instanton Floer homology and Dehn surgery, Part II Speaker: Zhenkun Li Time: 2:00pm–3:00pm Place: CMC 118

Title: Instanton Floer homology and Dehn surgery Speaker: Zhenkun Li Time: 2:00pm–3:00pm Place: CMC 118

Dehn surgery is a fundamental tool in dimension three to construct new 3-manifolds out of the old and instanton Floer homology is a powerful tool to study the topology of 3-manifolds. In the first talk, I will present some background knowledge about these objects and in the second talk, I will present some results by my collaborators and I about how to use instanton Floer homology to study Dehn surgeries and related problems in 3-dimensional topology.

Title: Introduction to 4-manifolds and symplectic topology, Part II Speaker: Sümeyra Sakallı Time: 2:00pm—3:00pm Place: CMC 118

Title: Introduction to 4-manifolds and symplectic topology, Part I Speaker: Sümeyra Sakallı Time: 2:00pm—3:00pm Place: CMC 118

The topology and geometry of 4-manifolds is one of the main research areas in symplectic and low dimensional topology, and it is also related to topics such as complex surfaces and singularity theory in algebraic geometry. 4-manifolds admitting different structures in different categories are still not completely known and there are many other open problems related to this question. In this sequence of talks I will start with an introduction to 4-manifolds and symplectic topology, then discuss my results, and finish with some interesting open problems.