IME-USP

# Eventos

### Webinar on Evolution Equations and Dynamical Systems:

Daniel Marroquin (UFRJ)

Stochastic two-scale Young measures and homogenization of stochastic conservation laws

Abstract: We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Our homogenization method is based on the notion of stochastic two-scale Young measures, whose existence we establish. This is a joint work with Hermano Frid and Kenneth Karlsen.

Dia 15.06 (quarta-feira)

Às 14 horas

### Webinar on Evolution Equations and Dynamical Systems

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalr ordinary differential equations

Abstract: The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor. This is a join paper with Juan Campos and Carmen Nuñez.

Dia 1º.06.2022 (quarta-feira)
Às 14 horas

ID da reunião: 879 0486 3474
Senha de acesso: 728120

### NeuroMat webinars 2022

pathways to the 2023 IHP thematic program
Random processes in the Brain

Single-neuron model in cortical context
Markus Diesmann, Jülich Research Centre

Abstract: In the preparation of the 2023 IHP thematic program “Random Processes in the Brain” the question came up how relevant the single-neuron model is for cortical dynamics and function. Given the plethora of single-neuron models available, insight into their differential effects on the network level would give theoreticians guidance on which model to choose for which research question. The purpose of this talk is to outline a small project approaching this question which could be carried out in the framework of the thematic program in a collaboration of several labs. The talk first presents a well-studied full-density network model of the cortical microcircuit as a suitable reference network. The proposal is to replace the original single-neuron model by alternative common single-neuron models and to quantify the impact on the network level. For this purpose the presentation reviews a range of common single-neuron models as candidates and a set of measures like firing rate, irregularity, and the power spectrum. It seems achievable that all relevant neuron models can be formulated in the domain-specific language NESTML and data analysis be carried out in the Elephant framework such that a reproducible digital workflow for the project can be constructed. A minimal scope of the investigation covers a static network in a stationary state. However, there are indications in the literature that the conventional constraints on network activity are weak. Furthermore, hypotheses on the function of the cortical microcircuit depend on the transient interaction between cortical layers, synaptic plasticity, and a separation of dendritic and somatic compartments. Therefore, we need to carefully debate how the scope of an initial exploration can usefully be restricted.

Tuesday, May 31, 2022, at 10 am (S.Paulo local time) / 3pm (Paris local time)

### Colóquio do IME

Edson de Faria (IME-USP)
Os passeios de Sullivan pelos bosques da Matemática

Resumo: Nesta palestra apresentaremos um panorama das principais contribuições matemáticas de Dennis Sullivan, ganhador do Prêmio Abel deste ano.

No Auditório Antonio Gilioli, 2o andar do Bloco A do IME-USP, seguido de um café de confraternização.

Transmissão ao vivo no link https://youtu.be/clmSCRmX00M.

Dia 20.05.2022 (sexta-feira)
Às 14 horas

### Webinar on Evolution Equations and Dynamical Systems

Xiaoying Han (Auburn University)
The lottery competition model in random environments

Abstract: We will introduce a mathematical model that describes the competition for resources among different ecological species in random environments. Starting from a discrete time model over observation periods, continuous models are developed using diffusion approximation. The resulting system is a system of nonlinear stochastic differential equations, with complex nonlinear structures. Dynamics of the SDE system will be presented to show how environmental fluctuations affect coexistence among species. In particular, we will discuss a two species system under non-stationary environments, and an N species system (N > 2) under stationary environments.

Dia 20.05.2022 (sexta-feira)
Às 14 horas

ID da reunião: 879 0486 3474
Senha de acesso: 728120

### NeuroMat webinars 2022

pathways to the 2023 IHP thematic program
Random processes in the Brain

Disorganization of mental activity in psychosis
By Peter F Liddle, Institute of Mental Health, University of Nottingham Many patients with psychotic illnesses including schizophrenia, suffer persisting disability despite treatment of delusions and hallucinations with
antipsychotic medication. There is substantial evidence that disorganization of mental activity makes a major contribution to persisting disability, by disrupting thought, emotion and behaviour. Evidence suggests that this disorganization involves impaired recruitment of the relevant brain systems required to make sense of sensory input and achieve our goals. There is diminished engagement of relevant brain circuits, together with failure to suppress task-irrelevant brain activity.
We propose that disorganization of mental activity reflects imprecision of the predictive coding that shapes perception and action. The brain generates internal models of the world that are successively updated in light of sensory information. What we perceive is determined by adjusting predictions to minimise discrepancy between prediction and sensory input. Motor actions are controlled by a forward model of the state of brain and body as intended action is executed. Action is continuously adjusted to minimize discrepancy between prediction and sensory input. Disorganization is associated with both imprecise timing and imprecise content of predictions. We need models that incorporate the interactions between excitatory and inhibitory neurons in local circuits with
parameters representing long range communication between brain regions to help us understand the pathophysiological mechanism responsible for imprecise predictive coding in psychotic illness.

Tuesday, April 26, 2022, at 10 am (S.Paulo local time)

### Webinar on Evolution Equations and Dynamical Systems

Tiago Pereira (ICMC-USP)
Coherence resonance in networks

Abstract: Complex networks are abundant in nature and many share an important structural property: they contain a few nodes that are abnormally
highly connected (hubs). Some of these hubs are called influencers because they couple strongly to the network and play fundamental dynamical and structural roles. Strikingly, despite the abundance of networks with influencers, little is known about their response to stochastic forcing. Here, for oscillatory dynamics on influencer networks, we show that subjecting influencers to an optimal intensity of noise can result in enhanced network synchronization. This new network dynamical effect, which we call coherence resonance in influencer networks, emerges from a synergy between network structure and stochasticity and is highly nonlinear, vanishing when the noise is too weak or too strong. Our results reveal that the influencer backbone can sharply increase the dynamical response in complex systems of coupled oscillators.

Dia 20.04.2022 (quarta-feira)
Às 14 horas

ID da reunião: 879 0486 3474
Senha de acesso: 728120

### Webinar on Evolution Equations and Dynamical Systems

Antoine Laurain (IME-USP)
Optimal control of volume-preserving mean curvature flow

Resumo: We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate tracking-type cost functional. The gradient of the cost functional is obtained via a formal sensitivity analysis of the space-time tube generated by the mean curvature flow. We show that the perturbation of the tube may be described by a transverse field satisfying a parabolic equation on the tube. We propose a numerical algorithm to approximate the optimal control and show several results in two and three dimensions demonstrating the efficiency of the approach.

Dia 06.04.2022 (quarta-feira) Às 14 horas
A partir deste semestre, as reuniões serão feitas pelo Zoom.

ID da reunião: 879 0486 3474
Senha de acesso: 728120

### Webinar on Evolution Equations and Dynamical Systems

Björn Sandstede (Brown University)
Homoclinic and heteroclinic bifurcations: from theory to applications

Abstract: I will provide an overview of homoclinic and heteroclinic bifurcations in differential equations, geometric and analytic approaches to study them, and their applications to traveling waves in spatially extended systems. I will start with bifurcations of codimension one, which lead to periodic orbits or to chaotic dynamics, and then discuss bifurcations of codimension two, which often serve as organizing centers for more complex dynamics in such systems. Amongst the approaches I will discuss are homoclinic center manifolds and homoclinic Lyapunov-Schmidt reduction. The applications will focus on the FitzHugh-Nagumo system and provide an overview of homoclinic orbits that arise in this model.

Dia 24.11.2021 (quarta-feira) Às 11 horas

### Webinar on Evolution Equations and Dynamical Systems

Piotr Kalita (Jagiellonian University)
On attractors of non-autonomous subquintic wave equation

Abstract: We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. We show the existence, smoothness, upper-semicontinuous dependence on the perturbation of the nonlinearity, and relations between pullback, uniform, and cocycle attractors for the class of Shatah–Struwe weak solutions, which, as we prove, coincides with the class of the solutions obtained by the Galerkin method. Talk will be based on the joint article with J. Banaśkiewicz (https://link.springer.com/article/10.1007/s00245-021-09790-8).

Dia 10.11 .2021 (quarta-feira)
Às 11 horas

### Webinar on Evolution Equations and Dynamical Systems

Danielle Hillhorst (Université Paris-Sud)
Singular limit of an Allen-Cahn equation with nonlinear diffusion and a mild noise

Abstract: The objective of this talk is to understand the generation and the propagation of the interface of an Allen-Cahn equation with nonlinear diffusion perturbed by a mild noise. The Allen-Cahn equation was introduced to understand the motion of the interface separating two different phases of a metal alloy. Instead of the usual linear diffusivity of the classical Allen-Cahn equation, we consider a density dependent diffusion term and perturb the reaction term by a mild noise. In particular, we describe how the nonlinear diffusion term affects the motion of the interface. This is joint work with Perla El Kettani, Yong Jung Kim and Hyunjoon Park.

Dia 27.10 (quarta-feira)
Às 11 horas

### Colóquio do IME

Palestrante: Maria Luisa Sandoval Schmidt, IP-USP
Data: sexta, 15/10/2021, 14h
Local: sala do Google Meet a ser divulgada na semana do evento
Mais informações sobre o Colóquio do IME em https://www.ime.usp.br/coloquio/.

### Webinar on Evolution Equations and Dynamical Systems:

Eiji Yanagida (Tokyo Institute of Technology)
Solvability of  the heat equation with a dynamic Hardy-type potential

Abstract:  Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time.
In the subcritical case, when the motion of the singularity is not so quick, it is shown that there exist two types of positive solutions.  On the other hand, when the singularity moves like a fractional Brownian motion, there exists a positive solution in a wider range of parameters.

Dia 13.10.21 (quarta-feira)
Às 09 horas

### Webinar on Evolution Equations and Dynamical Systems:

Sergey Zelik (University of Surrey)
Smooth extensions for inertial manifolds of semilinear parabolic equations

Abstract:  We discuss the problem of smoothness  of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\epsilon}$-regularity for such manifolds (for some positive, but small $\epsilon$). Nevertheless, under some  natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\Bbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the $C^{1,\epsilon}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.

Dia 29.09.21 (quarta-feira)
Às 11 horas

### Webinar on Evolution Equations and Dynamical Systems

Pedro T. P. Lopes (IME-USP)
On the Cahn-Hilliard equation on manifolds with conical singularities

Abstract: The Cahn-Hilliard equation has been widely studied. Results on global solutions, existence of attractors and convergence to the equilibrium can be found in classical papers and monographs. Those results have been generalized in many directions, for instance, for different potentials and boundary conditions. However, a topic that has not yet been fully explored is how the regularity of a surface or a domain affects the dynamical properties of the solutions. In this presentation, we restrict our study to the most classical version of the Cahn-Hilliard equation and to one of the simplest types of singularities on a surface or a domain: the conical singularity. We show how new results on complex interpolation and Lp-regularity, obtained using pseudodifferential operators, can be combined with classical approaches in order to understand the dynamics of the solutions in this context. Joint work with Nikolaos Roidos (Patras University).

Dia 1º.09 (quarta-feira)
Às 11 horas

### Seminário de Sistemas Dinâmicos

Jernej Činč (University of Vienna and IT4Innovations Ostrava)
Genericity of the pseudo-arc in a measure-theoretic setting

Abstract: The pseudo-arc is besides the arc the only planar continuum (i.e. compact connected metric space) so that every of its proper subcontinua is homeomorphic to itself (Hoehn and Oversteegen, 2020). Its first description appeared in the literature about hundred years ago and due to many of its remarkable properties it is an object of much interest in several branches of mathematics. There are results indicating that the pseudo-arc appears as a generic continuum in very general settings. For instance, Bing has proven that in any manifold M of dimension at least 2, the set of subcontinua homeomorphic to the pseudo-arc is a dense residual subset of the set of all subcontinua of M (equipped with the Vietoris topology). In this talk I will present a result which reveals that the pseudo-arc is a generic object also in a certain measure-theoretic setting; namely, I will show that the inverse limit of the generic Lebesgue measure preserving interval map is the pseudo-arc. Building on this result I will construct a family of planar homeomorphisms with attractors being the pseudo-arc with several interesting topological and measure-theoretical properties.

Dia 20.07 (terça-feira)
Às 14 horas

### Webinar on Evolution Equations and Dynamical Systems

Artur Alho (Instituto Superior Técnico de Lisboa)
Dynamical systems in scalar field cosmology

Abstract: In this talk I will give an overview of dynamical systems in cosmology with scalar fields. This includes models of early inflation, quintessence and quintessential inflation.

Dia 07.07 (quarta-feira)
Às 16 horas

### Seminário A5

Introdução aos grafos quânticos
Nataliia Goloshchapova (IME- USP)

Resumo:
Um grafo quântico é uma rede composta de arestas e vértices, nos quais as funções são definidas e um operador diferencial linear atua. Os grafos fornecem modelos simplificados em matemática, física, química e engenharia, quando se considera a propagação de ondas de vários tipos através de um sistema quasi-dimensional que se parece com uma vizinhança fina de um grafo.
Primeiramente, apresentaremos uma breve descrição dos objetos e dos conceitos básicos da Teoria dos Grafos. Em seguida, discutiremos o problema de Cauchy, estabilidade das ondas solitárias das equações não lineares de Schrodinger, Klein-Gordon, e Korteweg-de Vries na reta e no grafo métrico.

Dia 01.07 (quinta-feira)
Às 16 horas

Site: https://www.ime.usp.br/~acinco/

### Seminário de Modelagem Matemática

Larissa Sartori (IME-USP)
Time-scale analysis and parameter fitting for vector-borne diseases
Abstract: Vector-borne diseases are progressively spreading in a growing num-ber of countries, and it has the potential to invade new areas and habitats.From  the  dynamical  perspective,  the  spatial-temporal  interaction  of  models that try to adjust to such events are rich and challenging. The first challenge is to address the dynamics of the vectors (very fast and local) and the dynam-ics of humans (very heterogeneous and non-local). The objective of the present paper is to use the well-known Ross-Macdonald models, identifying different time scales, incorporating spatial movements and estimate in a suitable way the parameters. We will concentrate on a practical example, a simplified space model, and apply it to dengue spread in the state of Rio de Janeiro, Brazil.

Dia 29/06 (terça-feira), às 18h

### Webinar on Evolution Equations and Dynamical Systems

Approximating solutions of the 3D Navier-Stokes equations on R^3 using large periodic domains
James Robinson (University of Warwick)

Dia 23.06 (quarta-feira)
Às 16 horas

### Webinar on Evolution Equations and Dynamical Systems

On the connections of parabolic equations with discontinuous nonlinearity
José Valero (Universidad Miguel Hernández de Elche)

Abstract: We consider a parabolic equation of reaction-diffusion type with a discontinuous nonlinearity, which can be expressed by means of a Heaviside functions as a differential inclusion. We show first that under mild assumptions this equation generates a multivalued semiflow that possesses a global attractor and study its structure, which is an interesting and challenging problem. It is noticeable that our problem is the limit of a sequence of Chafee-Infante problems that undergo an infinite sequence of bifurcations, so it is reasonable to expect that it inherits the structure of the attractor of the Chafee-Infante equation. In fact, we prove that there is an infinite (but countable) number of equilibria and that the sequence of equilibria of the approximative problems converges to the equilibria of the limit problem. Since a Lyapunov function exists, the attractor is characterized by the fixed points and their heteroclinic connections, so a full description of the dynamics is got if we determine which connections exist. We give a partial answer to this question. Finally, if we restrict the semiflow to the positive cone, then nice regularity properties of solutions are obtained. In particular, the structure of the global attractor, in this case, is fully understood.

Dia 09.06 (quarta-feira)
Às 14 horas

### Webinar on Evolution Equations and Dynamical Systems:

Multiple radial solutions for some Neumann problems
Francesca Colasuonno (Università di Bologna)

Abstract:  In this seminar, I will talk about some semilinear or quasilinear problems under Neumann boundary conditions. In the first part of the seminar, I will focus on supercritical p-Laplacian problems, set in a ball of R^N, and describe some existence results obtained via variational methods. In the second part, I will present some multiplicity results obtained via the shooting method for ODEs. In particular, I will describe a recent result on a one-dimensional mean-curvature problem, for which we find several positive oscillating BV-solutions. These results are contained in some joint papers with Alberto Boscaggin, Colette De Coster, and Benedetta Noris.

Dia 26.05 (quarta-feira)
Às 16 horas

### Webinar on Evolution Equations and Dynamical Systems

Stability for wave equation with localized damping revisited

Abstract: In this talk we revisit some classical problems involving the wave equation to show another way to prove the stability of the problems. We start considering the n-dimensional linear wave equation in a bounded domain subject to a locally distributed linear damping term. In this case, we proved, via semigroup results (Gearhart Theorem), that the energy decays exponentially. For our surprise, this result has never been proved by this methodology so far. After this, using the linear case, we proved the stability to the wave equation now subject to a locally distributed nonlinear damping term. The second case considered is when the domain is whole space $\mathbb{R}^N$. In this situation, we proved two results. First, using semigroups results with full damping out of a compact set. The second case (when the domain is whole space) the goal here is that we removed damping. Precisely, given a positive real number $M>0$, we show that for any compact set $K$ of $\mathbb{R}^N$, it is possible to build a region $\Xi$ free of damping, with $meas(\Xi)=M$ (measure of $\Xi$), such that $\Xi$ is globally distributed. Finally, we also considered a case with an unbounded domain with finite measure. In both cases where the undamped region is unbounded it possesses a finite measure we considered microlocal analysis tools combined with Egorov’s theorem.

### Webinar on Evolution Equations and Dynamical Systems

Spontaneous periodic orbits in the Navier-Stokes flow
Prof. Jean-Philippe Lessard  (McGill University)

Abstract: In this talk, we introduce a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier-Stokes equations on the three-torus. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton-Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier-Stokes equations with Taylor-Green forcing.