## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Wednesday, June 29, 2016 — 1:30 PM EDT

**Speaker:** Luca Di Cerbo, International Centre for Theoretical Physics (ICTP) (Mathematics Section), Trieste, Italy

**Lecture 1:** Wednesday June 29, 1:30 p.m. - 3:00 p.m., MC 5501

The study of volumes in hyperbolic geometry, real & complex, is a classical topic in differential geometry which is beautifully interconnected with many other parts of mathematics such as geometric topology, algebraic geometry, group theory and even computational mathematics. In this first lecture, I will introduce this fascinating subject through the study of volumes of hyperbolic Riemann surfaces via the classical Gauss-Bonnet theorem. Then, I will give an overview of some modern works on hyperbolic 3-manifolds due to Thurston and his school. Finally, I will recall the necessary background for the study of volumes in higher dimensional complex hyperbolic geometry such as the Chern-Gauss-Bonnet theorem. The last part of this lecture is preparatory for the second lecture, which will be exclusively on complex hyperbolic geometry.**Lecture 2:** Thursday June 30, 1:30 p.m. - 3:00 p.m., MC 5501

In this lecture, I will focus on the study of complex hyperbolic surfaces and their volume spectrum. Recall that a complex hyperbolic surface is a complete finite volume complex surface (a manifold of real dimension four) which admits a metric of negative constant holomorphic sectional curvature. In the compact case, a problem which has attracted considerable attention is the classification of all complex hyperbolic surfaces with minimal volume. This intriguing problem, originally proposed by Mumford in the 70's, has a long and rich history which I will briefly recall. The analogous problem in the non-compact finite volume case has an interesting history as well. In fact, Hirzebruch in the 80's constructed the first example of a non-compact complex hyperbolic surface with minimal volume. After reviewing Hirzebruch's example, I will construct a new example, recently discovered in collaboration with M. Stover, and address the classification problem of all such surfaces. Finally, I will explicitly construct a tower of examples which saturates the whole admissible volume spectrum of complex hyperbolic surfaces. This is again joint with M. Stover.

**Registration for this event is required. **

For more information contact, Benoit Charbonneau.

To help us advertise, please download a copy of the event poster.

Location

MC - Mathematics & Computer Building

5501

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.