I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the ``hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper $mathrm{CAT}(0)$ space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on the Morse boundary of the mapping class group or briefly describe joint work with David Hume developing the metric Morse boundary.

# Seminar Geometric Group Theory -- Morse boundaries of geodesic spaces

# Lecture Club mathématique UdeM -- Les nombres congruents

Un nombre congruenUn nombre congruent est un nombre rationnel qui est l'aire d'un triangle rectangle dont les longueurs des côtés sont des nombres rationnels. Caractériser les nombres congruents est un défi pour les mathématiques actuelles. C'est l'un (sinon le) des plus vieux problèmes d'arithmétique non résolus. En 1983, Tunnel a donné une caractérisation des nombres congruents qui est conditionnée au fait que la conjecture de Birch et Swinnerton-Dyer soit vraie. Dans cet exposé, nous allons présenter les nombres congruents et leur liens avec les courbes elliptiques.t est un nombre rationnel qui est l'aire d'un triangle rectangle dont les longueurs des côtés sont des nombres rationnels. Caractériser les nombres congruents est un défi pour les mathématiques actuelles. C'est l'un (sinon le) des plus vieux problèmes d'arithmétique non résolus. En 1983, Tunnel a donné une caractérisation des nombres congruents qui est conditionnée au fait que la conjecture de Birch et Swinnerton-Dyer soit vraie. Dans cet exposé, nous allons présenter les nombres congruents et leur liens avec les courbes elliptiques.

# Seminar Physique Mathématique -- Hitchin systems and CFT's

This Liouville theory 4-point function is related to the Painlevé VI tau function.

# Séminaire de la topologie computationnelle du SAG -- Taut foliations on graph manifolds

An L-space is a rational homology sphere with simplest possible Heegaard Floer homology. Ozsváth and Szabó have shown that if a closed, connected, orientable three-manifold has a C^2 coorientable taut foliation then it is not an L-space, and it has been recently shown that the C^2 condition can be replaced by C^0 in work of Bowden/Kazez and Roberts. The converse to this statement holds when restricting to graph manifolds; this is part of a joint project with J. Hanselman, J. Rasmussen, and S. Rasmussen. I will explain the gluing theorem that goes into this by introducing a calculus for studying bordered Floer homology developed in joint work with J. Hanselman.

# CRM -- Walls in Random Groups

I will give an overview of Gromov's density model for random groups. These groups are hyperbolic and for large densities are exotic enough to have Kazhdan's property (T). I will focus on small densities and explain the techniques of Ollivier and Wise, and Mackay and myself to tame these groups by finding "walls" and hence an action on a CAT(0) cube complex.

# Reducts of Tree-like structures

Given some first order structure **M**, a reduct of **M** is a structure **N** on the same universe such that **N** is definable in **M**. Informally, a reduct of a structure is obtained by forgetting about some of the structure. Given this notion, the general problem is to take a structure **M** and classify all of its reducts up to interdefinability. This has been demonstrated to be feasible in a number of cases where **M** is ω-categorical and homogeneous in a finite (often binary) language. All such classifications have been found to confirm the conjecture of Simon Thomas, that every countably infinite structure which is (ultra)homogeneous in a finite relational language has only finitely many reducts up to interdefinability. Don't worry I will define all these concepts when they are introduced.

When **M** is an ω-categorical structure on a countably infinite set **X**, the problem of classifiying the reducts of **M** is the same as classifying the closed subgroups of Sym(**X**) which contain Aut(**M**); considering the natural Polish topology on Sym(**X**). So then we can consider using what we know about infinite permutation groups to classify reducts.

In this talk I will describe the classification of the reducts of a family of tree-like structures called semilinear orderings; these are partial orderings which are not linear, but for every element the elements below it are linearly ordered. My theorem concerns those countably infinite semilinear orderings which were called 2-homogenous, and described up to isomorphism, by Manfred Droste. Their automorphism groups are examples of Jordan groups, so I will explain what that means and how my proof makes use of many results from the study of infinite, primitive Jordan groups.

# Decorated cospans, corelations, and electrical circuits

Given a category C with finite limits, it is well-known that one may form a category with morphisms spans in C, and composition given by pullback. Furthermore, when an epi-mono factorisation system is available, one may form a category with morphisms jointly-monic spans in C. In the category Set this construction gives relations between sets. Dually, one may talk of categories of cospans and corelations.

Given a cospan or corelation X→N←Y, we may equip the so-called apex N of the cospan with additional structure, such as a finite graph with vertices N, or a real valued function on N. We call this additional structure a decoration, and the cospan a decorated cospan. We shall discuss the conditions under which this construction gives a category, and how to construct functors between such categories.

As an example, we shall show that a class of electrical circuit diagrams can be considered as morphisms in a decorated cospan category, and that the semantics of these circuit diagrams may be view as a functor from this category to the category of linear relations.

This is an expanded version of my Octoberfest 2015 talk. Joint work with John Baez (UC Riverside).

# Epidemiology, Biostatistics and Occupational Health -- Personalizing Medicine: New Ideas for Dynamic Treatment Regimes

# $L^q$ norms and nodal sets of Laplace eigenfunctions

We will discuss a recent result that exhibits a relation between the average local growth of a Laplace eigenfunction on a compact, smooth Riemannian surface and the global size of its nodal (zero) set. More precisely, we provide a lower and an upper bound for the Hausdorff measure of the nodal set in terms of the average of the growth exponents of an eigenfunction on disks of small radius. Combined with Yau's conjecture and the work of Donnelly-Fefferman, the result implies that the average local growth of eigenfunctions on an analytic manifold with analytic metric is bounded by constants in the semi-classical limit.

# Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets

In this talk, we shall give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds. This is joint work with Niky Kamran (McGill University) and Francois Nicoleau (Universite de Nantes).