Building Research Interconnections in Dynamics and Geometry

BRIDGe 2026  

July 13-17 

Tentative Schedule (abstracts below)

Monday:  

• AM (9-12): Dick Canary

• PM (2-5): Nathaniel Sagman

Tuesday: 

• AM (9-12): Sebastian Hurtado-Salazar

• PM (2-5): Emily Dumas

Wednesday:

• AM (9-12): Simion Filip

Thursday: 

• AM (9-12): Richard Wentworth

• PM (2-5):  Carolyn Abbott

Friday: 

• AM (9-12): James Farre

• PM (1:30-4:30): Chris Leininger

Titles and Abstracts:

• Speaker: Carolyn Abbot

Title:  Hierarchically hyperbolic groups

Abstract: Hierarchically hyperbolic groups (HHGs) are a generalization of hyperbolic groups that include mapping class groups, most CAT(0) cubical groups (including right-angled Artin and Coxeter groups), many 3-manifold groups, and various combinations of such groups.  HHGs are defined axiomatically, and the axioms are meant to capture the Masur-Minsky hierarchy machinery of mapping class groups.  In this way, one can think of HHGs as groups that have a similar coarse geometry to mapping class groups.  The first half of this talk will focus on introducing the class of HHGs through examples and exploring their basic properties. In the second half, I will describe some more advanced properties of HHGs, including a notion of geometrically finite subgroups, and recent results on HHGs, including a classification of their Morse elements.

• Speaker: Dick Canary

Title:  An introduction to Anosov representations

Abstract: We will begin by surveying properties of convex cocompact representations into SO(n,1) as a way of motivating the definition of Anosov representations. We will then attempt to demystify the dynamical definition of Anosov representations and use it to derive basic properties of Anosov representations. We will define Sambarino flows associated to Anosov representations and survey their applications. We will finish by discussing some of our favorite open questions.

• Speaker: Emily Dumas

Title: Geometric theory of Anosov subgroups acting on flag varieties

Abstract: We will survey results on the construction and properties of quotient manifolds associated to Anosov subgroups of a Lie group acting on the flag variety G/P.  We will give a brief introduction to a theory developed by Guichard, Kapovich, Leeb, Porti, and Wienhard which describes explicit domains of discontinuity in the flag variety so that that the assocated G/P-manifold is compact. An interesting feature that emerges in higher rank is that a single Anosov subgroup typically has many different cocompact domains of discontinuity in the flag variety, and we will discuss the combinatorial objects that describe and classify them.

Finally we will survey some results on the topology and geometry of these flag variety manifolds associated to Anosov subgroups, including work of Alessandrini, Maloni, Tholozan, and Wienhard which shows in particular that Anosov subgroups isomorphic to a surface group pi_1(S) give rise to quotient manifolds that are smooth fiber bundles over S. If time allows we will discuss some joint results with Sanders which describe the homology of a flag variety quotient manifold and properties of its complex structure in cases where G is a complex Lie group.

• Speaker: James Farre

Title: Convex projective structures in dimension 3

Abstract: Many familiar three-dimensional geometries have projective models. Hyperbolic space, anti-de Sitter space, and Minkowski space can all be realized inside projective geometry, with their isometry groups appearing as subgroups of SL(4,R). Classical quasi-Fuchsian surface groups in these geometries preserve convex sets in projective space on which they act properly and cocompactly. This property is stable under deformation in SL(4,R) leading to a rich class of surface subgroups of SL(4,R) preserving convex domains in RP^3.

A central theme will be the relationship between convex cocompactness in projective geometry and the Anosov property. In dimension 3, strongly convex cocompact subgroups of SL(4,R) are a source of projective-Anosov representations, while still retaining enough geometric structure to be studied using tools from Kleinian groups, Teichmüller theory, and dynamics.

I will present examples, current results, and questions regarding convex cocompact projective structures and phenomena that appear near the boundary of this deformation space.  This is based on joint work with Marit Bobb. 

• Speaker: Simion Filip

Title: Linear representations of surface groups and algebraic geometry

Abstract: I will discuss two related but distinct directions where representations of fundamental groups of surfaces relate to algebraic geometry. In the first part, we will look at the case when the target is a noncompact Lie group, and in the second part, the target will be a compact Lie group.

Part 1: Anosov representations and Hodge theory 

Abstract 1: I will discuss one source of representations of fundamental groups of Riemann surfaces (or more generally algebraic manifolds) to Lie groups, namely those coming from the monodromy of families of algebraic manifolds. They provide a class of representations that can be studied by both topological techniques, and also by analytic techniques such as Higgs bundles (in their simplest forms, variations of Hodge structure). I will discuss some examples when these questions connect to Anosov representations, and what one might hope for in general.

Part 2: Algebro-geometric subgroups of the mapping class group and their action on character varieties of compact groups

Abstract 2: Let us call a subgroup \Gamma of the genus g mapping class group "algebro-geometric" if it arises as the monodromy of an algebraic family of Riemann surfaces. Fix a compact Lie group G and consider the action of \Gamma on the character variety X of representations of the fundamental group of a surface into G. In joint work with Brown, Eskin, and Rodriguez-Hertz, we show that the action of \Gamma on X exhibits various strong rigidity properties: all orbit closures are (immersed) orbifolds, with special properties coming from algebraic geometry.

I'll provide the necessary background and discuss some further directions that are suggested by the points of view taken in the talks.

• Speaker: Sebastian Hurtado-Salazar

Title: Infinite vs Finite Covolume in Higher Rank

Abstract: First part: Our main motivation is to find effective criteria to know when a subgroup of a Lie group (for example, given by a set of generators, or satisfying some assumptions) is a lattice. We will discuss some old problems, known criteria, wishes, and some of the available tools.

Second part: We will talk about work in progress with Subhadip Dey and Mikolaj Fraczyk.

We construct examples of infinite covolume discrete subgroups in G = SO(n,2)  (n \geq 3) which act ergodically on the Furstenberg boundary of G, giving a counterexample to a conjecture of Margulis for G = SO(n,2), n>2. The examples arise from lattices in SO(n,1) and their deformations inside G. More important than the actual examples, we describe a new way of studying these deformations of lattices, which relate them to deformations of the smooth (left multiplication) action of SO(n-1,1) in the finite volume quotient SO(n,1)/\Gamma. This correspondence allows us to use powerful tools from smooth dynamics. Some potential applications of the correspondence are understanding local rigidity of these groups inside G, and lower bounds on critical exponents of hyperbolic manifolds with totally geodesic boundary.

• Speaker: Chris Leininger

Title:  Coarse geometry of surface bundles.

Abstract: I will begin by briefly recalling the basic theory of surface bundles and their classification via the monodromy homomorphism.  Motivated by Thurston's Hyperbolization Theorem for surface bundles over a circle, I will then introduce Farb and Mosher's definition of convex cocompactness for subgroups of the mapping class group and its relation to Gromov hyperbolicity of general surface bundles via the action of the monodromy on Teichmüller space.  I'll end the first half with several open questions related to these concepts, in particular a question independently posed by Mosher and Hamenstädt about geometric finiteness in the mapping class group.  In the second half, I will describe recent progress in developing a notion of geometric finiteness in the mapping class group, and how these notions relate to the coarse geometry of a surface bundle via the monodromy.  In addition to the work of Farb-Mosher, Hamenstädt, and others, I will discuss my own joint work with Dowdall, Durham, Kent, Sisto, and Russell.

• Speaker: Nathaniel Sagman

Title: Harmonic maps in non-positive curvature

Abstract: One way to understand a homotopy class of maps is to produce and study a representative that solves a PDE or an extremal problem. When the target is a Riemannian manifold of non-positive curvature, harmonic maps are a natural choice: they often exist, and they encode topological and geometric information. This perspective has been especially useful in Teichmüller theory--in the classical setting, through harmonic maps between surfaces; and in higher Teichmüller theory, through equivariant harmonic maps to symmetric spaces. 

In this talk, I’ll give a broad introduction to harmonic maps in non-positive curvature. The first part of the talk will cover the basic definitions and some applications, old and new. In the second part, I’ll discuss equivariant harmonic maps and minimal immersions (i.e., conformal harmonic maps) to symmetric spaces of non-compact type. In particular, I’ll present results on the existence, uniqueness, and geometry of such maps, along the way explaining the connection with Higgs bundles, and I’ll share some questions and conjectures.

• Speaker: Richard Wentworth

Title: Higgs bundles, isomonodromy and minimal surfaces

Abstract: In a joint work with Brian Collier and Jeremy Toulisse, we construct a joint moduli space of Higgs bundles, where the Riemann surface structure is allowed to vary in the Teichmüller space of the underlying smooth surface. The nonabelian Hodge correspondence defines a foliation on this joint moduli space whose holonomy recovers the natural mapping class group action on the character variety. This formalism allows one to address issues of special surface group representations that have an algebraic origin. It also has some strong implications for spaces of equivariant minimal surfaces.

Participants

• Carolyn Abbott

• Francisco Arana-Herrera 

• Harry Bray

• Corey Bregman

• Aaron Calderon

• Dick Canary

• Brian Collier (organizer)

• Michelle Chu

• Jeff Danciger (organizer)

• Emily Dumas

•  James Farre

• Simion Filip

• Sebastian Hurtado-Salazar

• Chris Leininger

• Beibei Liu

• Sara Maloni (organizer)

• Giuseppe Martone

• Filippo Mazzoli

• Yair Minsky

• Alex Nolte

• Kasra Rafi

• Charles Ried

• Nathaniel Sagman

• Peter Smillie

• Wouter Van Limbeek

• Teddy Weisman

• Richard Wentworth

• Brandis Whitfield

• Mike Wolf