Minicourses

Viveka Erlandsson: Counting curves on surfaces 

In her thesis Maryam Mirzakhani established the asymptotic growth, as L tends to infinity, of the number of simple closed geodesics of length at most L in a hyperbolic surface. More specifically, for any fixed simple geodesic c, she proved that the number of closed geodesics in the mapping class group orbit of c and of length bounded by L is asymptotic to a constant times L^{6g-6+2r}. Here g and r are the genus and number of punctures of the surface. Some years later, using very different methods, she generalized this result to hold also for the case when Γ_0 is non-simple.
In this mini-course I will give an idea of how to prove these results. We will concentrate on the (punctured) torus, although the ideas hold in general. We will in fact discuss a different proof from the original ones, using simpler methods, which allows us to treat both cases at once. Our main tools will be geodesic currents (which can be seen as the completion of the set of closed geodesics) and train tracks (combinatorial models of simple geodesics on surfaces).

Anne Parreau: Compactifications of character varieties of finitely generated groups in semisimple Lie groups

The character variety of a finitely generated group Γ into a semisimple real Lie group G is the quotient by G-conjugation of the space of completely reducible representations of Γ in G. They contains many subspaces of particular interest, such as the higher Teichmüller spaces when Γ is a surface group, that consists of Hitchin representations when G=PSL(n,R) or maximal representations when G=Sp(2m,R), and spaces of Anosov representations for more general hyperbolic groups Γ. In this mini-course, we will describe two different ways of compactifying general character varieties: the length(s) compactifications, that generalize Thurston's compactification for Teichmüller space, and the real spectrum compactification, which is based on tools from real algebraic geometry. We will explain how boundary points can be seen as representations over non-Archimedean real closed fields, acting on real euclidean buildings, and the links between these two compactifications.

Yulan Qing: Geometric boundaries of groups

In this minicourse we introduce a couple of geometric boundaries for proper geodesic spaces and finitely generated groups. In particular we will study the sublinearly Morse boundaries and the quasi-redirecting boundary. We will discuss the construction of these spaces and the key tools needed in working with them. We will also discuss the applications and accessible questions in this field.

Research Talks

Johanna Bimmermann: Hamiltonian circle actions and symplectic cuts

If a non compact symplectic manifold arises as regular sublevel set of a Hamiltonian that generates a circle action, one can compactify it using a symplectic cut. We use this technique to compactify the disc cotangent bundles of spheres, real and complex projective spaces. If time permits we will see how this compactification can be used to compute certain symplectic invariants (Gromov width & Hofer-Zehnder Capacity) of these disc cotangent bundles.

María Cumplido Cabello: An algorithm to solve the word problem for 3-free Artin groups in quadratic time

Artin groups are very easy to define, they are just finitely generated groups with finite relations having the form stst...=tsts..., where s and t are generators of the group and both parts of the equality have the same word length. However, classic problems remain open in these groups, being a very important one the word problem. We will explain how to construct a quadratic-time explicit and computable algorithm to solve the word problem for Artin groups that do not contain any relations of length 3. This proof is highly combinatorial and only uses rules to do rewritings in a word in order to obtain a geodesic word representing the same element.  

Arielle Leitner: An Advertisement for Coarse Groups and Coarse Geometry

Coarse structures are used to study the large scale geometry of a space.  For example, although the integers and the real line are different topologically, they look the same from "far away", in the sense that any geometric configuration in the real line can be approximated by one in the integers, up to some uniformly bounded error.  A coarse group is a group object in the category of coarse spaces, for example, this means the group operation is only "coarsely associative," etc. In joint work with Federico Vigolo we study coarse groups. This talk will be an advertisement for our work, as we walk through examples that illustrate some of our main results, and connections to other subjects. 

Rylee Lyman: Dynamics and Geometry of Outer Automorphisms of Virtually Free Groups

A theorem of Karrass, Pietrowski and Solitar says that a finitely generated group is virtually free just when it acts on a tree with finite stabilizers and finite (i.e. compact) quotient. In the language of Bass and Serre, finitely generated virtually free groups are exactly the fundamental groups of finite graphs of finite groups. One goal of this talk is to introduce the canonical deformation space, called Outer Space, of these groups. Another is to connect its geometry to the dynamics of the action of the outer automorphism group of the virtually free group. I will attempt to state a few theorems and open questions.

Marta Magnani: Arc coordinates for maximal representations

Given a hyperbolic surface with boundary, arc coordinates provide a parametrization of the Teichmüller space. They rely on the choice of a family of arcs which start and end at boundary components and are orthogonal to them. Higher rank Teichmüller theories are a generalization of classical Teichmüller theory and are concerned with the study of representations of the fundamental group of an oriented surface S of negative Euler characteristic into simple real Lie groups G of higher rank. It is well known that maximal representations are a higher rank Teichmüller theory for G Hermitian. In the talk we will discuss how to generalize arc coordinates for maximal representations, focusing on the case where S is a pair of pants and G=PSp(4,R).

Sara Maloni: Pleated surfaces in PSL_d(C)

Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds, and can be described as piece-wise totally geodesic surfaces, bent along a geodesic lamination lambda. Bonahon generalized this notion to representations of surface groups in PSL_2(C), and described a holomorphic parametrization of the resulting open charts of the character variety in term of shear-bend cocycles.
In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to representations in PSL_d(C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.

Silvia Sabatini: Topological properties of (tall) monotone complexity one spaces

In symplectic geometry it is often the case that compact symplectic manifolds with large group symmetries admit indeed a Kähler structure. For instance, if the manifold is of dimension 2n and it is acted on effectively by a compact torus of dimension n in a Hamiltonian way (namely, there exists a moment map which describes the action), then it is well-known that there exists an invariant Kähler structure. These spaces are called symplectic toric manifolds or also complexity-zero spaces, where the complexity is given by n minus the dimension of the torus.

In this talk I will explain how there is some evidence that a similar statement holds true when the complexity is one and the manifold is monotone (the latter being the symplectic analog of the Fano condition in algebraic geometry), namely, that every monotone complexity-one space is simply connected and has Todd genus one, properties which are also enjoyed by Fano varieties. These results are largely inspired by the Fine-Panov conjecture and are in collaboration with Daniele Sepe [2].

Moreover, with Isabelle Charton and Daniele Sepe [1], we completely classify monotone complexity one space that are "tall" (no reduced space is a point), and prove that the torus action extends to a full toric action, that each of these spaces admits a Kähler structure and that there are finitely many such spaces, up to a notion of equivalence that will be introduced in the talk.

References:
[1] I.Charton, S.Sabatini, D.Sepe, "Compact monotone tall complexity one T-spaces", arXiv:2307.04198 [math.SG].
[2] S.Sabatini, D.Sepe, "On topological properties of positive complexity one spaces", Transformation Groups 9  (2020).

Sheila Sandon: Non-squeezing in symplectic and contact topology

The symplectic non-squeezing theorem, discovered by Gromov in 1985, has been the first result showing a fundamental difference between symplectic transformations and volume preserving ones. A similar but more subtle phenomenon in contact topology, the odd dimensional cousin of symplectic topology, has been found by Eliashberg, Kim and Polterovich in 2006, and refined by Fraser in 2016 and Chiu in 2017: in this case non-squeezing depends on the size of the domains, and only appears above a certain quantum scale. In my talk I will describe these fundamental results of symplectic and contact topology and, if time permits, outline the main geometric ideas behind a proof of the contact non-squeezing theorem using generating functions, a classical method just based on finite dimensional Morse theory (joint work with Maia Fraser and Bingyu Zhang).

Neza Zager Korenjak: Constructing proper affine actions

Inspired by infinitesimal strip deformations of Danciger-Gueritaud-Kassel, we construct higher strip deformations, allowing us to find explicit proper actions of free groups on 4n+3-dimensional space.