Résumés 2025-LG&TBQ2

I will discuss and motivate a notion of reducible geometrically finite subgroups of the mapping class group, in analogy with Kleinian groups. This definition generalizes that of parabolic geometrical finiteness due to Dowdall-Durham-Leininger-Sisto. I will present sufficient criteria for certain RAAGs to be realized as reducible geometrically finite subgroups. This is based on joint with with Aougab, Dowdall, Hoganson, Maloni, and Whitfield.

A short introduction to spacetimes–Lorentzian manifolds–including the all-important notion of causality. Then a look at some of the forays into positing boundary constructions on spacetimes, leading up to current work on boundaries on generic spacetimes.

A link is alternating if it admits a diagram in which strands alternate between being « over » or « under » at each crossing. In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. In particular, a family of generalized alternating links which each correspond to a Euclidean or hyperbolic tiling and have a right-angled complete hyperbolic structure on their complement. We will describe a complete characterization of which of these links are arithmetic, and which are pairwise commensurable. This is joint work with David Futer.

I will discuss recent work with C. Leininger in which we produce purely pseudo-Anosov surface subgroups of mapping class groups, obtaining the first compact atoroidal surface bundles over surfaces. We do this by constructing a type-preserving representation of the figure eight knot group into the mapping class group of the thrice-punctured sphere.

Even though symplectic topology is defined in terms of smooth objects, many rigidity phenomena appear to subsist when one takes a limit of these smooth objects that converges to something merely continuous; understanding which phenomena do is precisely the study of C^0 symplectic topology. In this talk, I will present some new developments in that field regarding (limits of) Lagrangian submanifolds and how that relates to some notion of local exactness.

One perspective from which we can approach the intriguing question, « What does a random surface look like? » is by considering typical properties of translation surfaces. In 2021, Masur—Rafi—Randecker found asymptotic bounds for the covering radius – that is, the largest radius of an immersed disk – of translation surfaces for any stratum. With Anthony Sanchez and Sunrose Shrestha, we explicitly computed the covering radius for the family of doubled slit tori using methods that generalize to other slit-defined families of surfaces of any genus. In this talk, I will explain our motivations, methods, and future goals.

The 1-skeleton of an ~A_2 Bruhat-Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from ”triangle presentations.” This abstract group either embeds into PGL(3, Fq((x))) or PGL(3, Qq), or else is exotic. Currently, the complete list of triangle presentations is only known for projective planes of orders q=2 or 3. However, one abstract group that embeds into PGL(3,Fq((x))) for any prime power q is known via the trace function corresponding to the finite field of order q^3. I have found a different method to derive this group as well as others via perfect difference sets. This new method demonstrates a previously unknown connection between difference sets and ~A_2 buildings. Moreover, this new method makes the final computation of triangle presentations easier, which is computationally valuable for large q.

Given an equivariant map from one dynamical system to another, it is natural to ask for a characterization of the invariant subsystems on which the map restricts to a conjugacy. I will explain how this kind of problem shows up in the a priori quite different field of information theory, in which one wants to find the maximum rate at which information can be transmitted with acceptably low error across a given noisy channel.

My goal is for audience members to leave with a feeling for how to think about topological and dynamical objects in information-theoretic terms. The talk should be accessible to graduate students who have some knowledge of metric spaces and either Markov chains or ordinary differential equations.

Anosov flows are a fundamental example of chaotic dynamical systems. In dimension 3, they exhibit deep interactions between dynamics and the topology of the underlying manifold. For certain classes of 3-manifolds, topology almost entirely determines the dynamics of the Anosov flows they can support, whereas some 3-manifolds admit multiple non equivalent Anosov flows. In this talk, I will present this intriguing family of dynamical systems and discuss topological and dynamical construction methods (such as surgery and gluing) that illustrate both the flexibility of these flows and their strong ties to 3-manifold topology.

Fix a closed hyperbolic orientable surface X and consider a uniformly random degree-N covering space X_N. We prove that the smallest new eigenvalue of the Laplacian on X_N is at least 1/4-o(1) with high probability, which is basically best possible. The proof involves a variety of tools, and I will try to give a flavor of the ones related to hyperbolic geometry.
This is joint work with Michael Magee and Ramon Van-Handel.

Weak containment is an intensively studied tool from ergodic theory. Recently, several authors have developed an analogous theory of weak containment for topological actions. I will survey some of these developments.

Deligne-Mostow lattices are distinguished by their geometric and topological properties, most notably the rigidity of their complex hyperbolic structure. However, representations of a particular class of Deligne-Mostow lattices into PGL⁡(3,C) have been found to be outside the scope of a complex hyperbolic structure. This presentation will address the rigidity of these representations and the intriguing dynamical properties of the discovered groups.

The Marsden-Meyer-Weinstein reduction theorem for contact manifolds
has brought significant attention of many researchers. The first generalisation of the Marsden-Meyer-Weinstein reduction theorem to contact manifolds was proven by C. Albert in 1989. However, this result applies only to coorientable contact manifolds and depends on the choice of the contact form within its conformal class. In my talk, I will review various approaches to the contact reduction theorem and focus on the technical condition introduced by C. Willett, demonstrating its necessity. I will then present a new more general version of the reduction theorem, applying when Willett’s condition is not satisfied.