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# The Joy of Geometry

# The Joy of Geometry, a pre-Christmas workshop in Geometry

# EPFL, December 5-6, 2013

Organisers Marc Troyanov and Peter Buser

**Location : Thursday in the new building of the Centre Bernoulli **

** Friday in the MA-building, room MAA110**

**Thursday December 5 (****Centre Bernoulli****)**

**10:00-10.10 Opening**

**10:15-11.10 Bruno Colbois, Neuchâtel**

*Upper bounds for the spectrum of the Laplacian: a metric approach, *

I will first make a general presentation of the geometric spectral theory (relations between the spectrum of the Laplacian on a domain or on a manifold, and the geometry of this domain or of this manifold). In the second part of the talk, I will give as example to type of metric upper bounds for the spectrum of submanifolds of the Euclidean space: one related to the isoperimetric ratio (join work with El Soufi and Girouard), and the other related to a notion of indices (join work with Dryden and El Soufi). I will also make a comment on the geometry of submanifolds with large eigenvalues (join work with Savo).

**11:30 - 12:20 Daniele Bartolucci, University of Rome **

*On the Best Pinching Constant of Conformal Metrics on S ^{2} with One or Two Conical Singularities*

We answer a long-standing open question asked by Thurston concerning the best pinching constant for Riemannian metrics on the 2-sphere with one or two conical singularities.

**14:15 - 15:05 Daniele Valtorta, EPFL**

*Sharp estimates for the first eigenvalue of the p-Laplacian under Ricci and diameter conditions*

In this talk we will discuss sharp estimates for the first eigenvalue of the p-Laplace operator on compact Riemannian manifolds (for any real p>1). Following the gradient comparison technique introduced by Kroger, we obtain the sharp estimates assuming a lower bound on the Ricci curvature and an upper bound on the diameter using a generalized Bochner formula for the p-Laplacian.

**15:30 - 16:20 Viktor Schroeder, Zürich**

Möbius structure on the boundary of complex hyperbolic space.

The boundary at infinity of the complex hyperbolic space carries a natural Moebius structure. We characterize this Moebius structure purely in terms of metric Moebius geometry. This is joined work with S. Buyalo.

**17:15 - 18:15 EPFL Colloquium (at **** auditorium CM-4****) :** **Emmanuel Kowalski, ETHZ **

*Écarts entre nombres premiers, d'après Y. Zhang et J. Maynard*

L'exposé rappellera certaines des questions classiques concernant les nombres premiers et présentera les approches de théorie analytique des nombres qui ont été découvertes pour étudier ces questions. On parlera ensuite tout spécialement des extraordinaires résultats récents qui permettent de montrer l'existence d'une infinité de paires de nombres premiers à distance bornée l'un de l'autre. [link ->]

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**Friday December 6 (****MA-building, room MAA110****)**

**9:15 - 10:05 François Fillastre, Université de Cergy-Pontoise**

*Future convex sets in Minkowski space*

We look at convex sets in **R**^{d+1} (endowed with its standard Lorentzian metric) which are such that their Gauss map is a surjection onto the hyperbolic space. Analogously to the convex bodies case, we define area measures for those sets, and study the Christoffel problem (prescription of the area measure of order one) and the Minkowski problem (prescription of the area measure of order d).A class of example of such convex sets is given by flat Lorentzian manifolds coming from General Relativity.In a simple case, we get convex sets invariant under a group of linear isometries. If the group is fixed, one can develop a theory analog to the mixed-volumes for convex bodies. Work partially joint with Francesco Bonsante and Giona Veronelli.

**10:30 - 11:20 Vincent Emery, EPFL**

*Hyperbolic manifolds of small volume*

I will discuss a conjecture stating that except in dimension 3, the complete hyperbolic n-manifold of the smallest volume is noncompact.

**11:30 - 12:20 Craig J. Sutton, Dartmouth College**

*Can you hear the length spectrum of a manifold?*

Inverse spectral geometry is the study of the relationship between the geometry of a Riemannian manifold and the spectrum of its associated Laplace operator. Motivated in part by considerations from quantum mechanics, it is a long-standing folk-conjecture that the spectrum of a manifold determines its length spectrum (i.e., the set consisting of the lengths of the closed geodesics). Using the trace formula of Duistermaat and Guillemin one can see that this conjecture is true for sufficiently ``bumpy'' Riemannian manifolds. However, our understanding of the conjecture in the homogeneous setting---where closed geodesics occur in large families---is rather incomplete. In this talk, we will demonstrate that the conjecture is true for compact simple Lie groups equipped with a bi-invariant metric and, more generally, compact irreducible symmetric spaces of splitting rank by showing that the so-called Poisson relation is an equality in this setting.