In this talk, H. C. Wang’s quantitative study of Zassenhaus neighborhoods will be extended to the exceptional Lie groups. The first application is an improved upper bound for the sectional curvature of a semi-simple Lie group. The second application is a uniform lower volume bound for orbifold quotients of symmetric spaces of non-compact type. This is joint work with M. Wang and G. Wei.

In this talk, we will relate a McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. In order to do so we introduce `orientable' fully irreducible outer automorphisms, motivated by pseudo-Anosov homeomorphisms that admit transversely oriented stable foliation. We also relate the homological stretch factor of such automorphisms to their geometric stretch factor. This is joint work with Dowdall and Taylor.

This is joint work with Hee Oh. We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in \(\mathrm{SO}(d,1)\) acting on the space \(\Gamma\backslash\mathrm{SO}(d,1)\), assuming that the associated hyperbolic manifold \(M=\Gamma\backslash\mathbb{H}^d\) is a convex cocompact manifold with Fuchsian ends. For \(d = 3\), this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any \(k\geq 1\),

- the closure of any \(k\)-horosphere in \(M\) is a properly immersed submanifold;
- the closure of any geodesic \((k+1)\)-plane in \(M\) is a properly immersed submanifold;
- an infinite sequence of maximal properly immersed geodesic \((k+1)\)-planes intersecting \(\mathrm{core} M\) becomes dense in \(M\).

The Sullivan dictionary provides a conceptual framework for understanding the connections between dynamics of rational maps and Kleinian groups. In this talk, I will explain how an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps can be established. As an application, I will show how this dynamical correspondence allows us to solve mating problems. I will further establish a connection on the parameter/deformation space and proves an analogue of Thurston’s compactness theorem for acylindrical hyperbolic \(3\)-manifolds in the rational map setting. If time permits, I will also talk about some open questions and further studies.

The geodesic length spectrum and eigenvalue spectrum of a locally symmetric space are basic geometric invariants of the space. In this talk, I will discuss some recent work on when these invariants determine the manifold. This is joint work with Justin Katz.

The Lorentzian Lichnerowicz Conjecture is a Lorentzian analogue of the Ferrand-Obata Theorem on conformal transformation groups of Riemannian manifolds. I will discuss my verification of the conjecture in dimension three, for real-analytic metrics, in recent joint work with C. Frances.

An old, fundamental problem is classifying closed \(n\)-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature \(-1\), that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will describe how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics inspired by groundbreaking work of Margulis on superrigidity to prove our characterization. I will also mention some open problems and report on recent progress on effective results by my PhD students Khanh Le and Rebekah Palmer and by Elon Lindenstrauss and Amir Mohammadi.

Translation surfaces are geometric objects that can be defined as a collection of polygons with sides identified in parallel opposite pairs by translation. If we generalize slightly and allow for polygons with sides identified by both translation and dilation, we get a new family of objects called dilation surfaces. While translation surfaces are well-studied, much less is known about dynamics on dilation surfaces and their moduli spaces. In this talk, we will survey recent progress in understanding the topology of moduli spaces of dilation surfaces as well as dynamics on dilation surfaces themselves, including recent work with undergraduate students.

We investigate the structure of the tri-pants graph, a simplicial graph introduced by Maloni and Palesi, whose vertices correspond to particular collections of homotopy classes of simple closed curves of the twice-punctured torus, called tri-pants, and whose edges connect two vertices whenever the corresponding pants differ by suitable elementary moves. In particular, by examining the relationship between the tri-pants graph and the dual of the Farey complex, we prove that the tri-pants graph is connected and it has infinite diameter.

Topological complexity is an invariant of spaces which measures the difficulty in continuously assigning paths to pairs of endpoints. We investigate generalizations of the combinatorial complexity introduced by Tanaka, and the simplicial complexity introduced by Gonzales to the equivariant setting. Time permitting, we discuss the computational tractability of these generalizations. This is joint work with Rebecca Bell, Allison Eckert, and Avery Schweitzer.