# Dr. Santiago Saglietti

## Overview

Santiago obtained his PhD in Probability from the University of Buenos Aires in 2014, under the supervisition of advisors Pablo Groisman and Pablo Ferrari. Afterwards, he continued his research as a postdoc in Universidad de Buenos Aires (2015, Argentina) and Pontificia Universidad Católica (2016, Chile). He currently holds a postdoc fellowship at Technion University, under supervision of Professor Dmitry Ioffe.

Santiago has also worked as an Assistant Professor in Universidad Torcuato Di Tella (2015, Buenos Aires) and as a Teaching Instructor in Universidad de Buenos Aires (2008-2014, Buenos Aires).

Santiago and his co-workers focus on four separate lines of research. A first line of research is centered on statistical mechanics and, more precisely, on perfect simulation algorithms for extremal Gibbs measures of interacting particle systems. Santiago's work in this topic consisted in realizing these extremal Gibbs measures as invariant distributions of some explicit Markovian dynamics in finding perfect simulation algorithms to sample these invariant distributions exactly. As a consequence of this algorithms, one obtains a powerful simulation tool which can also be exploited from the theoretical point of view as a means of studying Gibbs measures via methods from the theory of stochastic processes. Recently, Santiago has also started doing in research on Random Walks in Random Environments, which is also a popular model with ties to statistical mechanics.

A second line of research is the study of metastability phenomena in stochastic partial differential equations. Namely, if one starts with the solution of a (deterministic) PDE which converges to some equilibrium state as time tends to infinity, it is a relevant problem to understand how this convergence is modified upon the addition of a small random noise. Under some particular assumptions on the unperturbed system, it can be shown that the perturbed system behaves in a metastable way: it spends a long time near this equilibrium and then makes an abrupt transition towards another a different state. Santiago focused on studying metastability in this context for the particular case of PDE's with blow-up, in which many technical problems arise and the standard theory to address this questions cannot be used directly.

A third line of research is centered on the study of branching Markovian dynamics. More precisely, Santiago has worked on establishing laws of large numbers for the empirical measures of supercritical branching processes in which particles not only branch (at a supercritical rate), but also evolve according to some prescribed Markovian motion.

The fourth and final line of research is the study of random (homogeneous) fractal measures. Santiago has focused on understanding the macroscopic geometric properties of these random measures, like dimension and/or regularity, since they are helpful to understand better the structure of non-homogeneous (but deterministic) fractal measures, which are central objects in geometric measure theory.

## Selected Publications

Groisman, Pablo; Saglietti, Santiago. Small random perturbations of a dynamical system with blow-up. J. Math. Anal. Appl. 385 (2012), no. 1, 150-166.

Galicer, Daniel; Saglietti, Santiago; Shmerkin, Pablo; Yavícoli, Alexia. $\dpi{300}\inline L^q$dimensions and projections of random measures. Nonlinearity 29 (2016) 2609-2640.

Fernández, Roberto; Groisman, Pablo; Saglietti, Santiago. Stability of gas measures under perturbations and discretizations. Rev. Math. Phys.  28 (2016), no. 10, 1650022, 46 pp.

Groisman, Pablo; Saglietti, Santiago; Saintier, Nicolas. Metastability for small random perturbations of a PDE with blow-up. To appear in Stochastic Processes and their Applications.

## Research

Metastability, Probabilistic methods in Statistical Mechanics, Branching Processes and Ergodic Geometric Measure Theory.

## Contact Info

Room 317 Cooper Building
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