Upcoming talks:

A geometric transition is a continuous path of geometries/geometric structures which changes in isomorphism type, generalizing the transition from hyperbolic to Euclidean geometry as curvature goes to zero. In this talk I will give an overview of a new formalism for discussing geometric transitions, and use this to construct examples of transitions relevant to geometric topology. In particular, we will see a transition between hyperbolic and de Sitter geometry which does not occur within real projective geometry, and a transition of complex hyperbolic geometry which does not occur within complex projective space.

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, in particular regarding the kind of representation varieties that stem out of this approach.

I will explain how the concept of Ruelle resonances for Anosov flows can be generalized to higher rank Anosov action such as Weyl chamber flows on higher rank locally symmetric spaces. In a first part of the talk I will introduce the notion of Ruelle resonances for Anosov flows as well as some motivation to study them. I will then explain how the concept can be extended to higher rank actions using the theory of Taylor spectra.

Past talks:

Convex projective geometry may be understood as a generalization of hyperbolic geometry, but one key difference is that some convex projective domains admit flat subspaces. Benoist proved strong properties about the behavior of these flat subspaces in dimension 3, and we will discuss a generalization of this result to all dimensions. Namely, codimension-1 flats in a convex divisible domain (in dimension 3 or greater) form a discrete set which projects to a finite collection of virtual tori. The complement of these tori is a union of cusped convex projective manifolds (in the sense of Cooper-Long-Tillman).

Recently, work of Long and Thistlethwaite, Weir, and Alessandrini-Lee-Schaffhauser generalized some of the theory of higher Teichmüller spaces to the setting of orbifold surfaces. In particular, they compute the dimension of Hitchin components for triangle groups, and find that this dimension is 0 only for a finite number of low-dimensional examples. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with the diagonal embedding in PGL(2n,R) or PSp(2n,R) is always locally rigid.

In this talk, we will describe some fundamental domains for subgroups of PU(2,1) in the complex hyperbolic plane and in its boundary at infinity, and we will focus on the examples constructed by Parker-Will and Deraux-Falbel. We will recall some geometric properties of the complex hyperbolic plane, as well as the hypersurfaces called bisectors. The fundamental domains that we consider can be used for proving the discreteness of some groups, and provide geometric structures on manifolds.

We give effective estimates for the number of saddle connections on a translation surface that have length less or equal than L and are in a prescribed homology class modulo q. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum. This is joint work with Michael Magee, and builds on recent advances by Magee on Selberg's eigenvalue conjecture for moduli spaces of abelian differentials, and Rodolfo Gutierrez-Romo's work on Zorich's conjecture on Zariski-density of Rauzy-Veech groups.

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