Geometry & Topology seminars at Unimib is a new cycle of postdoc seminars planned and organised by Gianluca Faraco and Andrea Galasso. With the present initiative we aim to spread the knowledge of topics arising in low-dimensional topology, CR-geometry and geometric quantization in our department by hosting speakers from other departments around the world.

The first cycle took place from March 1st to May 17th, 2023.

Organisers: Gianluca Faraco & Andrea Galasso

Past seminars

•  01.03.23  --  Yohann Bouilly, (University of Strasbourg) 

Title: Torelli subgroup action on character varieties

Abstract: The character varieties of a surface encode geometric structures as Teichmüller space or holomorphic vector bundles. The mapping class group of the surface acts on the character varieties and preserves a measure. This action is known to be properly discontinuous on the Teichmüller space but its behavor is different for character varieties in a compact group (the action is known to be ergodic) and some ergodicity conjecture and results exist for exotic components. In this talk we will explain why the Torelli subgroup, which is the kernel of the action of the mapping class group on the first homology space, acts ergodically on character varieties in a compact Lie group and what happen for exotic components for SL_2-character varieties.

• 08.03.23  --  Lorenzo Ruffoni, (TUFTS)

Title: Manifolds without real projective structures

Abstract: Every surface admits a metric of constant curvature. Similarly, every 3-manifold can be decomposed into pieces that admit a geometric structure locally modeled on a homogeneous space. In both cases, the most common model is the real hyperbolic space, and each model geometry can be realized inside the real projective space. On the other hand, in dimension at least 4 there are smooth manifolds that cannot be modeled on the real projective space. Previously known examples of such “non-geometric” manifolds are small, in the sense that their fundamental groups are finite or virtually cyclic. In this talk I will show how to construct examples in dimension at least 5 whose fundamental groups are non-elementary Gromov hyperbolic groups.

• 21.03.23  --  Yongquan Zhang, (Stony Brook)

Title: Geodesic planes in quasifuchsian manifolds and geodesic rays for measured laminations

Abstract: A geodesic plane in a hyperbolic 3-manifold is a totally geodesic isometric immersion from the hyperbolic plane. In a complete hyperbolic 3-manifold of finite volume, any geodesic plane is closed or dense, independently due to Ratner and Shah. Recently, there has been some exciting progress on generalizing this result to the infinite volume setting. In this talk, I will restrict to the case of a geodesic plane contained in an end of a quasifuchsian manifold. In this case, the topological behavior of the plane depends on the bending lamination. In particular, I will discuss the existence of exotic roofs, which are geodesic planes that accumulate on the boundary of the end, but cannot be separated from it by a support plane. A necessary condition is the existence of exotic rays for the bending lamination, which are geodesic rays that intersect the lamination recurrently, but are not asymptotic to any leaf of the lamination.

• 04.04.23  --  Alessio Savini (University of Geneva)

Title: Cohomology of semisimple Lie groups

Abstract: There are several ways to define a cohomology theory for a semisimple Lie group G. For instance one can consider either the usual De Rham cohomology, the continuous cohomology or the continuous bounded cohomology. In this seminar we will discuss those cohomology theories, focusing our attention on the comparison conjecture by Monod. If time allows, we will talk about some recent results about Monod's conjecture by myself and Michelle Bucher.  

• 18.04.23  --  Hendrik Herrmann (Bergische University of Wuppertal)

Title: On Some Embedding Results for CR Manifolds

Abstract: I will give a brief exposition of some fundamental results on the embeddability of strongly pseudoconvex  CR manifolds due to Boutet de Monvel, Lempert and others. In this context I will talk about a refinement concerning the Reeb dynamics of the underlying manifold and of the spheres in the complex Euclidean space. If  time permits, I will say a few words about the proof. This is a joint work with Chin-Yu Hsiao, George Marinescu and Wei-Chuan Shen.

• 03.05.23  --  Wei-Chuan Shen (University of Cologne)

Title: Semiclassical spectral asymptotics of Toeplitz operators on CR manifolds

Abstract: This talk deals with semiclassical spectral asymptotics of Toeplitz operators on CR manifolds. First, we recall the notions of compact strictly pseudoconvex embeddable CR manifolds and Szeg\H{o} kernel expansion. We then review some classical and recent developments for Toeplitz operators and their functional calculus. Finally, we study the spectral operator $\chi_k(T_P)$ constructed by the functional calculus of the first-order Toeplitz operator, which considers a set of eigenvalues in $k\supp\chi$ and weights them according to the values of a good cutoff function $\chi_k$ at each  eigenvalue in the set. We will show that the kernel of the spectral operator $\chi_k(T_P)$ is a semi-classical Fourier integral modulo a $k$ - negligible smooth kernel. Time permitting, I will give some applications of such asymptotic expansion, which provides CR analogues (without group action assumption on our CR manifolds) of high power line bundle results in complex geometry. 

• 17.05.23  --  Weixia Zhu (University of Vienna)

Title: Heat kernel asymptotics for Kohn Laplacians on CR manifolds

Abstract: In complex geometry, Demailly and Bismut used the heat kernel asymptotics for Kodaira Laplacians to establish Morse inequalities on complex manifolds. It is natural to ask whether one can establish analogies to Morse inequalities on CR manifolds by studying the asymptotic behaviour of the heat kernel. In this talk, I will show how to establish the heat kernel asymptotics for Kohn Laplacian (particularly in the Heisenberg case) with values in high tensor power of line bundle. As an application, it provides a heat kernel proof of Morse inequalities on compact CR manifolds. This is a joint work with Chin-Yu Hsiao.