##### Department of Mathematics,

University of California San Diego

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### Department Colloquium

## Wenyu Pan

#### U Chicago

## Exponential mixing of flows for geometrically finite hyperbolic manifolds with cusps

##### Abstract:

Let ${\mathbb{H}^n}$ be the hyperbolic *𝑛*-space and Γ be a geometrically finite discrete subgroup in Isom$_+$(${\mathbb{H}^n}$) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T$^1$ (Γ \ ${\mathbb{H}^n}$) (resp. the frame bundle F(Γ \ ${\mathbb{H}^n}$)) mixes exponentially. This result has many applications, including spectral theory, prime geodesic theorems, orbit counting, equidistribution, etc.

I will start with a survey of the past results, methods, and related problems on this topic. Along the way, I will present the joint work with Jialun Li, Pratyush Sarkar.

Host: Amir Mohammadi

### January 20, 2022

### 4:00 PM

Zoom ID: **964 0147 5112**

Password: **Colloquium**

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