# Research

## Supervision

Current:

**Alejandro Gárriz Molina**. Postdoctoral “Juan de la Cierva” researcher (since January 2024).**Inmaculada Benítez**. PhD student since January 2023. Co-supervised with María José Cáceres.**Prasanta Kumar Barik**. IMAG Postdoctoral researcher since 2023.**Niccolò Tassi**. PhD student since October 2022.**Alejandro Ramos Lora**. PhD student since 2020. Co-supervised with María José Cáceres.

Past:

**Antonio Camunga Tchikoko**. PhD student, co-supervised with Jesús Montejo.**Alejandro Gárriz Molina**. Postdoctoral researcher (March-July 2021).**Sebastian Throm**. Postdoctoral researcher funded by the German DFG. (2018-2020).**Havva Yoldaş**. PhD student at Universidad de Granada / BCAM, finished in October 2019. Thesis avaiable at Teseo online repository. Now assistant professor at TU Delft.**Daniel Balagué**. PhD student at Universitat Autònoma de Barcelona, finished in June 2013. Co-supervised with José A. Carrillo. The thesis is available at the Teseo online repository. Daniel is now a lecturer at TU Delft.

## Recent publications

Below you can find some recent papers. A full list can be found here.

### Preprints

**On the asymptotic behavior of the NNLIF neuron model for general connectivity strength**. 2024.We prove new results on the asymptotic behavior of the nonlinear integrate-and-fire neuron model. Among them, we give a criterion for the linearized stability or instability of equilibria, without restriction on the connectivity parameter, which provides a proof of stability or instability in some cases. In all cases, this criterion can be checked numerically, allowing us to give a full picture of the stable and unstable equilibria depending on the connectivity parameter and transmission delay. We also give further spectral results on the associated linear equation, and use them to give improved results on the nonlinear stability of equilibria for weak connectivity, and on the link between linearized and nonlinear stability.

**The sequence of pseudo-equilibria describes the long-time behaviour of the NNLIF model with large delay**. 2024.There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate and fire (NNLIF) models that describe neuronal activity at the level of the membrane potential of neurons. We introduce a set of novel states, which we call “pseudo-equilibria”, and give evidence of their defining role in the behaviour of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system’s parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a new strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudo-equilibria. Unlike with the direct entropy dissipation method, this technique allows us to see how a large delay favours convergence. We also present a detailed numerical study to support our results. This study explores the overall behaviour of the NNLIF system and helps us understand, among other phenomena, periodic solutions in strongly inhibitory networks.

**Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review**. 2024.We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number $N$. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-$N$ limit for locally integrable (that is, less singular) potentials.

**One-dimensional inelastic Boltzmann equation: Regularity & uniqueness of self-similar profiles for moderately hard potentials**. 2023.We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form $| \cdot |^\gamma$ for $\gamma > 0$ small enough (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $\gamma = 0$). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated linearised operator. In particular, a part of the paper is devoted to the trend to equilibrium for the Maxwell model in suitable weighted Sobolev spaces, an extension of results which are known to hold in weaker topologies. Our results can be seen as a first step towards a full proof, in the one-dimensional setting, of a conjecture in Ernst & Brito (2002) regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation.

### Some recent papers

**Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups**.*Journal of Functional Analysis*284(7):109830, 2023.We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.

**The scaling hypothesis for Smoluchowski’s coagulation equation with bounded perturbations of the constant kernel**.*Journal of Differential Equations*270:285–342, 2021.We consider Smoluchowski’s coagulation equation with a kernel of the form $K=2+\epsilon W$, where $W$ is a bounded kernel of homogeneity zero. For small $\epsilon$, we prove that solutions approach a universal, unique self-similar profile for large times, at almost the same speed as the constant kernel case (the speed is exponential when self-similar variables are considered). All the constants we use can be explicitly estimated. Our method is a constructive perturbation analysis of the equation, based on spectral results on the linearisation of the constant kernel case. To our knowledge, this is the first time the scaling hypothesis can be fully proved for a family of kernels which are not explicitly solvable.

## Some informal mathematical notes

## Funding

I have been co-PI of several research projects funded by the Spanish Ministerio de Economía y Competitividad (MTM2014-52056-P from 2015 to 2017; MTM2017-85067-P from 2018 to 2021; and PID2020-117846GB-I00 from 2021 to 2024). I am currently on the research board of the María de Maeztu funding held by the Institute of Mathematics of the University of Granada. Until 2015 I was the principal researcher of a Marie-Curie CIG project based at the University of Birmingham. Here is a short page with a summary of the aims and results of this project.

## Editorial work

I am currently an editor of Communications in Pure and Applied Analysis and Journal of Statistical Physics.