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The hot spots conjecture asserts that as time goes to infinity, the hottest and coldest points in an insulated domain will migrate towards the boundary of the domain. In this talk, I will describe joint work with Jaume de Dios Pont and Alex Hsu where we find the exact failure of the hot spots conjecture in every dimension.

A volume conjecture relates a certain asymptotical growth of a given quantum topological invariant of a hyperbolic 3-manifold to the hyperbolic volume of this manifold.

In this talk I will mention several of these volume conjectures, their common points and differences, notably those associated to the Baseilhac-Benedetti invariants and to the Andersen-Kashaev TQFT.

A general strategy to prove a volume conjecture is to use the combinatorial properties of a given triangulation of the manifold to simplify the expression of the quantum invariant, and hopefully to successfully apply the saddle point method in the desired asymptotics.

I will use the figure-eight knot complement as a recurring example, as it is the simplest member of two infinite families, the hyperbolic twist knots and the once-punctured torus bundles over the circle.

No prerequisite in quantum topology or hyperbolic geometry will be needed.

(This talk will cover joint works with François Guéritaud, Stéphane Baseilhac and Ka Ho Wong)