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Abstract: In the first part of the talk I recall the notion of renewal sequences associated with Markov chains and explain the connection with mixing. In the second part of the talk I discuss how renewal sequences can be understood in the context of (deterministic) dynamical systems, including dynamical systems with infinite measure, and summarise some recent result on mixing.

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Abstract: Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this talk, I will introduce and discuss the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. I will focus on partial well-order and antichains, exploring how the homomorphic image order behaves in the context of graphs and graph-like structures. In particular, I will discuss a near-complete characterization of partially well-ordered avoidance classes with one obstruction. This is joint work with Nik Ruskuc.

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Abstract: The mapping class group of an orientable surface (of finite type) is the group or orientation preserving diffeormorphisms of the the surface modulo isotopy. There are three types of mapping classes (Thurston classification): finite order, reducible and pseudo-Anosov genearlising the classification for modular group $SL(2,\mathbb{Z})$. From multiple perspectives, pseudo-Anosov maps are the most interesting type. This talk will survey the theory of pseudo-Anosov maps with small entropy. It will subsequently focus on deriving bounds in terms of genus for a particular notion of entropy: "translation distance in the curve complex". The main result is joint work with Chia-yen Tsai.

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Abstract: The word problem of a semigroup is the problem of deciding whether a pair of words over some generating set both represent the same element. I will discuss the word problems of some semigroups - namely the free inverse and free left ample monoids of rank 1 - in which the elements can be viewed as intervals of integers containing a marked point.

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Abstract: The symmetry of graphs is a widely studied topic, but less has been done on the symmetry of digraphs. In this talk I will outline some of the fundamental differences between the two topics and outline some recent research with Glasby, Li, Verret and Xia.

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Abstract: A quasisymmetric map is one that changes angles in a controlled way. As such they are generalizations of conformal maps and appear naturally in many areas, including Complex Analysis and Geometric group theory. A quasisphere is a metric sphere that is quasisymmetrically equivalent to the standard $2$-sphere. An important open question is to give a characterization of quasispheres. This is closely related to Cannon's conjecture. This conjecture may be formulated as stipulating that a group that ``behaves topologically'' as a Kleinian group ``is geometrically'' such a group. Equivalently, it stipulates that the ``boundary at infinity'' of such groups is a quasisphere. A Thurston map is a map that behaves ``topologically'' as a rational map, i.e., a branched covering of the $2$-sphere that is postcritically finite. A question that is analog to Cannon's conjecture is whether a Thurston map ``is'' a rational map. This is answered by Thurston's classification of rational maps. For Thurston maps that are expanding in a suitable sense, we may define ``visual metrics''. The map then is (topologically conjugate) to a rational map if and only if the sphere equipped with such a metric is a quasisphere. This talk is based on joint work with Mario Bonk.

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Abstract: A subset $S$ of a group $G$ is product-free if for all $x$ and $y$ in $S$, the product $xy$ is not in $S$. This definition generalises the notion of sum-free sets of integers, where these sets were first studied. In this talk I'll: give an overview of what's known about sum-free and product-free sets in groups; introduce the related concept of filled groups; describe some joint work in this area with Chimere Anabanti and Grahame Erskine.

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Abstract: In the talk I will discuss the following subjects: Introduction to the classical moment problem; The work of Thomas Jan Stieltjes; The work of Hans Hamburger and Marcel Riesz; Determinacy versus indeterminacy; The Nevanlinna parametrization of the indeterminate case; Order and type of entire functions; Order of indeterminante moment problems calculated from the recurrence coefficients.

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Abstract: It is a classical problem to try to count the number of closed curves on (hyperbolic) surfaces with bounded length. Due to people such as Delsart, Huber, and Margulis it is known that the asymptotic growth of the number of curves is exponential in the length. On the other hand, if one only looks at simple curves the growth is polynomial. Mirzakhani proved that the number of simple curves on a hyperbolic surface of genus $g$ of length at most $L$ is asymptotic to $L^{6g-6}$. Recently, she extended her result to also hold for curves with bounded self intersection, showing that the same polynomial growth holds. In this talk I will discuss her results and some recent generalizations.

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Abstract: Stemming from the Burnside problem, branch groups have delivered lots of exotic examples over the past 30 years. Among them are easily describable finitely generated torsion groups, as well as the first example of a finitely generated group with intermediate word growth. We will investigate a generalisation of the Grigorchuk-Gupta-Sidki branch groups and talk about their maximal subgroups and about their profinite completion. Additionally, we demonstrate a link to a conjecture of Passman on group rings.

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Abstract: Suppose that we have a framework consisting of finitely many fixed-length bars connected at universal joints. Such frameworks (and variants) arise in many guises, with applications to the study of sensor networks, the matrix completion problem in statistics, robotics and protein folding. The fundamental question in rigidity theory is to determine if a framework is rigid or flexible. The standard approach in combinatorial rigidity theory is to differentiate the quadratic equations constraining the distances between joints, and work with these linear equations to determine if the framework is infinitesimally rigid or flexible. In this talk I will discuss recent progress using algebraic matroids that gives further insight into the infinitesimal theory and also provides methods for identifying special bar lengths for which a generically rigid framework is flexible. We use circuit polynomials to identify stresses, or dependence relations among the linearized distance equations and to find bar lengths which give rise to motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant.

Past colloquia can be found here.

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