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Abstract: The Valued Constraint Satisfaction Problem (VCSP) is a well-known combinatorial problem. An instance of VCSP is given by a finite set of variables, a finite domain of labels for the variables, and a sum of functions, each function depending on a subset of the variables. Each function can take finite rational values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. The case when all functions take only values 0 and infinity corresponds to the standard CSP. We study (assuming that $P\neq NP$) how the computational complexity of VCSP depends on the set of functions allowed in the instances, the so-called constraint language. Helped greatly by algebra, massive progress has been made in the last three years on this complexity classification question, and our work gives, in a way, the final answer to it, modulo the complexity of CSPs.

This is joint work with Vladimir Kolmogorov and Michal Rolinek (both from IST Austria).

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Abstract: The classical Analyst's Traveling Salesman Theorem of Peter Jones gives a condition for when a subset of Euclidean space can be contained in a curve of finite length (or in other words, when a "traveling salesman" can visit potentially infinitely many cities in space in a finite time). The length of this curve is given by a sum of quantities called beta-numbers that measure how non-flat the set is at each scale and location. Conversely, given such a curve, the sum of its beta-numbers is controlled by the total length of the curve, giving us quantitative information about how non-flat the curve is. This result and its subsequent variants have had applications to various subjects like harmonic analysis, complex analysis, and harmonic measure. In this talk, we will introduce a version of this theorem that holds for higher dimensional objects other than curves. This is joint work with Raanan Schul.

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Abstract: Given a problem from number theory a useful technique is to rephrase it in terms of a property of a dynamical system. One can then use the statistical/topological properties of the dynamical system to gain more insight and hopefully solve the original problem. This talk will be an exposition of this technique and will include many examples.

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Abstract: (Joint work with A. Le Boudec) The tree almost automorphism groups are non-discrete locally compact completions of the Higman-Thompson groups. The tree almost automorphism groups are independently interesting locally compact groups, and furthermore every group that almost acts on a sufficiently regular rooted tree embeds into one of these groups. We begin by introducing the almost automorphism groups and describing their relationship to the Higman-Thompson groups. We then consider the subgroups such that every element is contained in a compact subgroup; such groups are the topological analogue of torsion subgroups and are called periodic. We show every periodic subgroup is indeed locally elliptic - i.e. every finite set is contained in a compact subgroup. As applications, we recover a result for Thompson's group V as well as a new observation about the Röver group. We finally consider the commensurated subgroups of almost automorphism groups; these subgroups generalize normal subgroups. We show every commensurated closed subgroup of an almost automorphism group is either finite, compact and open, or equal to the entire group. As an application, we obtain new information on the possible lattice envelopes of Thompson's group T.

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Abstract: The number of irreducible polynomials over a finite field was first counted by Gauss. We will explain a connection between counting the number of irreducible polynomials over F_q with certain properties, and the number of rational points on some related algebraic curves. This idea can be used to count the number of irreducible polynomials with certain coefficients being 0. The appearance of supersingular curves explains the interesting periodic behaviour in the formulae, and new formulae are also obtained.

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Abstract: Majorana theory is an axiomatic framework in which to study objects related to the Monster group and its 196,884 dimensional representation, the Griess algebra. The theory was first developed in 2009 and was inspired by results by mathematicians such as M. Miyamoto and S. Sakuma who studied the Griess algebra using vertex operator algbras, objects used in the proof of Monstrous moonshine. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right, or as Majorana representations of certain groups. I will be discussing my work, which builds on that of A. Seress, developing an algorithm to construct Majorana representations.

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Abstract: Roughly speaking, a set is called a "Salem set" if it carries a measure whose Fourier transform decays polynomially with degree -s/2 where s is the Hausdorff dimension of the set (this is the fastest possible decay). Salem sets are often found via random processes, such as the random distortion of a Cantor set under Brownian motion. An old question of Kahane going back to the 1960s was whether or not the graph of classical Brownian motion is almost surely a Salem set. I will discuss this question in some detail: first I will show that the answer is 'no', and secondly I will show how to compute the optimal almost sure decay rate of the Fourier transform of measures supported on the graph. I will keep technical detail to a minimum and will not assume (much) a priori knowledge of Fourier analysis, or probability theory. This talk will include joint work with Tuomas Orponen (University of Helsinki) and Tuomas Sahlsten (University of Manchester).

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Abstract: Two fundamental finiteness properties in group theory are those of being finitely generated and of being finitely presented. These two properties were generalised to higher dimensions by C. T. C. Wall in 1965. A group G is said to be of type Fn if it has an Eilenberg-MacLane complex K(G,1) with finite n-skeleton. (Here K(G,1) is a certain nice topological space with fundamental group G.) It may be shown that a group is finitely generated if and only if it is of type F1 and is finitely presented if and only if it is of type F2. Related to this is a certain homological finiteness property called FPn. This property is defined for a monoid S in terms of the existence of certain resolutions of free left ZS-modules. The property FPn was introduced for groups by Bieri in (1976). In monoid and semigroup theory the property FPn arises naturally in the study of string rewriting systems. The connection between complete rewriting systems and homological finiteness properties is given by a result of Anick (1986) which shows that a monoid that admits such a presentation must be of type FPn for all n. The properties Fn and FPn are closely related. In particular, for finitely presented groups they are equivalent.

Because of the connection with rewriting systems, the finiteness property FPn for monoids has received a great deal of attention in the literature. It is sometimes easier to establish the topological finiteness properties Fn for a group than the homological finiteness properties FPn, especially if there is a suitable geometry/topological space on which the group acts in a nice way. Currently no theory of Fn exists for monoids. In this talk I will describe some recent joint work with Benjamin Steinberg (City College of New York) which was motivated by the question of developing a useful notion of Fn for monoids. This led us to develop a theory of monoids acting on CW complexes. I will explain the ideas we have developed and some of their applications.

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Abstract: The notion of an amenable group dates back to von Neumann in 1929, and appears in many different guises; such as in the Banach-Tarski paradox, and in the spectral geometry of manifolds. We discuss a more dynamical setting where amenability appears. Namely, we are interested in the growth of periodic orbits of the geodesic flow arising from negative curvature. No prior knowledge of these objects is assumed! This is joint work with R. Sharp.

Past colloquia can be found here.

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