**Irving Calderón**

Department of Mathematical Sciences

Mathematical Sciences & Computer Science Building

Durham University

Upper Mountjoy Campus

Stockton Road

DH1 3LE

Email: irving.d.calderon-camacho(AT)durham.ac.uk

Office: 2044

Since October 2021, I'm a postdoc at Durham University in the team of Michael Magee. Before that I did my PhD in Orsay under the supervision of Yves Benoist. Here is mi CV

My research interests lie broadly in the interactions between geometry, number theory and dynamics. An example of an object that informs these three domains is the Laplace-Beltrami operator \(\Delta_S\) of a finite-area hyperbolic surface \(S = \Gamma \backslash \mathbb{H}^2\). The \(L^2\)-eigenvalues of \(\Delta_S\) encode important geometric and dynamical information about the surface, such as:

- its Cheeger constant \(h(S)\), which measures the connectedness of the surface;
- asymptotics of the number \(\pi_S(R)\) of closed geodesics in \(S\) of length \(\leq R\);
- mixing rates for the geodesic and horocyclic flows on \(S\).

*homogeneous dynamical system*: a Lie group \(H\) acting on a homogeneous space \(X\)—usually of finite volume—of another Lie group. Many arithmetic problems can be reformulated as understanding features of \(H \curvearrowright X\), such as the classification of invariant measures or the distribution of closed orbits. Thanks to its rich structure, homogeneous dynamical systems can be studied with a wide variety of tools from ergodic theory, algebraic groups, unitary representations of Lie groups, automorphic representations...

## Publications and preprints

**Explicit spectral gap for Schottky subgroups of \(\mathrm{SL}(2,\mathbb{Z})\),**with M. Magee (*Submitted*). arXiv version**\(S\)-integral quadratic forms and homogeneous dynamics**. To appear in*Mémoires de la SMF*. arXiv version**Rigidity of the second symmetric product of the pseudo-arc,**with R. Hernández and A. Illanes. Topology Appl. 221(2017), 440-448. Published version**Being semi-Kelley does not imply semi-smoothness,**with E. Castañeda, C. Islas, D. Maya and F. Ruiz. Questions Answers Gen. Topology 32 (2014), no. 1, 73-77. Published version

## Dissertations

In my PhD thesis I revisit two classical topics on integral quadratic forms: the problem of \(\mathbb{Z}\)-equivalence—which asks to find a practical method to decide if two given integral quadratic forms in \(d\) variables coincide up to a base change of \(\mathbb{Z}^d\)—and the finite generation of integral orthogonal groups. My work is inspired by this recent article of H. Li and G. Margulis. Let \(S_f\) be a finite set of prime numbers, \(S = S_f \cup \{\infty\}\) and \(\mathbb{Z}_S\) the ring of \(S\)-integers. My contribution is an effective criterion of \(\mathbb{Z}_S\)-equivalence of integral quadratic forms, and a result that gives an explicit finite genereting set of any \(S\)-integral orthogonal group (by means of simple inequalities in terms of the real and \(p\)-adic sizes of the coefficients of the quadratic form). These generalize results of Li and Margulis for \(S = \{\infty\}\).

**Here**is my thesis, which I defended in June 2021.

Here are my master and bachelor dissertations:

- Dynamics on the space of 2-lattices of \(\mathbb{R}^3\)
through the study of random walks. My M2
*mémoire*supervised also by Yves Benoist. - Orbitas cerradas de flujos casi-geodésicos. The
*tesina*I wrote at the end of my master at UNAM, supervised by Ana Rechtman. - Flujo de Ricci combinatorio en superficies. My bachelor thesis supervised by Pablo Suárez.