Título: Quantitative results in Poincaré recurrence for mixing processes

Palestrante: Alexsandro Gallo (UFSCar)
Data: 16/01/2023
Horário: 15:00h
Local: Transmissão Online

Confira AQUI o link para a transmissão.

Resumo: A very classical question in stochastic processes is to quantify the time it takes to first enter a given set, or the number of visits to this set in a fixed time window. The aim of the talk is to present some recent quantitative results concerning first hitting/return times and number of visits to rare sets (small/vanishing measure) in mixing processes. We will focus on asymptotic results (essentially convergence in distribution and large deviations). Most of the presentation will be about the classical case where the sets of interest are strings of symbols of increasing size. At the end, if time allows, we will give some glimpses of the case in which we target general sets having vanishing measure.

All the talks are held in English.

More complete information about the seminars will be available at HERE.

Título: KPZ fluctuations of the Wick ordered planar stochastic heat equation

Palestrante: Alejandro Ramírez (PUC-Chile)
Data: 09/12/2022
Horário: 15:00h
Local: Transmissão online

Confira AQUI o link para a transmissão.

Resumo: We give a rigorous formulation of the Wick ordered stochastic heat equation, with planar white noise, whose solution is the free energy of an undirected random polymer, using a version of the Skorokhod integral. The solution is represented as an L^1 limit of a martingale given by the Feynman-Kac formula. We also show that the fundamental solution far from the center has fluctuations given by the 1+1 KPZ equation. This is a joint work with Jeremy Quastel and Balint Virag.

The talk will be held in English.

More complete information about the seminars will be available HERE.

Título: Cliques quase-maximais em grafos aleatórios
Data: 09/01/2023, às 15:30h
Local: CT - Instituto de Matemática/UFRJ, Bloco B - sala B106-A
Palestrante: Simon Griffiths (PUC/RJ)

Resumo: Consideramos a questão de quantos cliques quase-maximais disjuntos podem ser encontrados num grafo aleatório G(n,p). Em 1992, Alon e Spencer conjecturaram que existe uma família de ordem n^2/(log n)^2 tais cliques disjuntos. Recentemente Acan e Kahn mostraram que a conjectura é falsa, provando uma cota superior da forma O(n^2/(log n)^3). Nós fornecemos uma cota inferior da mesma ordem.

Informações sobre todo o programa dos seminários de probabilidade clique AQUI 

Title: Chasing patterns: diffusion-driven instabilities in networked and high-order systems

Speaker: Riccardo Muolo (Université de Namur)
October 10 from 3:30 p.m. to 4:30 p.m.
Local: Sala B106-a – Bloco B – CT – Instituto de Matemática – UFRJ

Abstract: Many natural and artificial systems exhibit collective behaviors, which show in the form of spatio-temporal patterns. This has triggered the interests of scholars, who have proposed several theories to account for such diversity. One of the most popular mechanisms of pattern formation is due to Alan Turing, who showed that diffusion can disrupt a homogeneous stable state, triggering an instability [1]. The original theory has been conceived in the framework of reaction-diffusion PDEs, but it has been recently extended on networked systems [2]. Moreover, it has been shown that a Turing-like mechanism occurs in the framework of synchronized coupled oscillators, as diffusion can lead to a loss of synchronization [3]. 
In this seminar I will present an overview of Turing theory in networked systems, showing that the Turing framework is indeed not too far from that of coupled chaotic oscillators [4]. I will then focus the attention on two results we have recently obtained. The first one, about the study of reaction-diffusion systems on top of non-normal networks, i.e., networks whose adjacency matrix is non-normal [5]. Such topology makes the system more sensible to perturbation, leading to a loss of stability even when a linear stability analysis predicts otherwise [6, 7]. Finally, I will briefly introduce high-order structures, i.e., hypergraphs and simplicial complexes, and show a recent extension of Turing theory on such topologies [8].

References
[1] A M Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B, 237:37, 1952.
[2] Hiroya Nakao and Alexander S Mikhailov. Turing patterns in network-organized activator-inhibitor systems. Nature Physics, 6:544, 2010.
[3] J Challenger, D Fanelli, and R Burioni. Turing-like instabilities from a limit cycle. Phys. Rev. E, 92:022818, 2015.
[4] Louis M Pecora and Thomas L Carroll. Master stability functions for synchronized coupled systems. Physical Review Letters, 80(10):2109, 1998.
[5] Malbor Asllani, Renaud Lambiotte, and Timoteo Carletti. Structure and dynamics of non-normal networks. Sci. Adv., 4:Eaau9403, 2018.
[6] Riccardo Muolo, Malbor Asllani, Duccio Fanelli, Ph K Maini, and Timoteo Carletti. Patterns of non-normality in networked systems. Journal of Theoretical Biology, 480:81, 2019.
[7] Riccardo Muolo, Timoteo Carletti, James P Gleeson, and Malbor Asllani. Synchronization dynamics in non-normal networks: the trade-off for optimality. Entropy, 23:36, 2021.
[8] Riccardo Muolo, Luca Gallo, Vito Latora, Mattia Frasca, and Timoteo Carletti. Turing patterns in systems with high-order interaction. arXiv preprint arXiv:2207.03985, 2022.

All the talks are held in English.

More complete information about the seminars will be available at Here.

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