Marcos Alexandrino (USP)
Title: Singular Riemannian foliations, variational problems and principles of symmetric criticalities
Absrtact: A singular foliation on a complete Riemannian manifold M is called a Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of M into orbits of an isometric action. In this talk, we investigate problems of Calculus of Variations in compact Riemannian manifolds equipped with SRFs with special properties. Examples of such SRF being considered include isoparametric foliations, SRF on Euclidean fiber bundles, and the partition of M into the orbits of a Lie group acting by isometries. This lecture is based on collaborative work with Leonardo F. Cavanaghi, Diego Corro, and Marcelo K. Inagaki. Its target audience is the general public interested in Differential Geometry and Analysis.
Emília Alves (UFF)
Title: Intersections of Real Bruhat Cells: Homotopy Type and New Developments
Absrtact: Bruhat cells provide a fundamental decomposition in Lie theory. While individual cells are topologically simple, their intersections can exhibit surprisingly rich behavior. In this talk, we will discuss recent developments in the study of intersections of real Bruhat cells, with particular emphasis on the homotopy type of the intersection of two cells.
A useful description of these intersections is given by explicit submanifolds of the group of real unipotent lower triangular matrices, which have the same homotopy type. This perspective allows us to combine geometric, topological, and combinatorial techniques to investigate these spaces.
For matrices of size (n+1)×(n+1) with n ≥ 4, these submanifolds decompose as a disjoint union of contractible connected components. In higher dimensions, however, new phenomena emerge. We will describe recent methods developed to study these cases, present new results, and discuss several open problems and conjectures.
This is joint work with J. Lambert (UFF), G. Leal (PUC-Rio), N. Saldanha (PUC-Rio), B. Shapiro (Stockholm University), and M. Shapiro (MSU).
Américo Cunha (LNCC)
Title: The Generalized Gaussian Iterated Map: Dynamics, Bifurcations, and Statistical Properties
Abstract: Nonlinear feedback mechanisms are ubiquitous in mathematical models of physical, biological, and engineered systems. Gaussian nonlinearities, in particular, appear in contexts ranging from signal processing and machine learning to neural and population dynamics. Motivated by these applications, this seminar presents a mathematical study of the generalized Gaussian iterated map, a smooth discrete-time dynamical system generated by repeated Gaussian transformations. We discuss the map's fundamental dynamical properties, including invariant intervals, fixed points, stability, and bifurcation structure, as well as aspects of its long-term statistical behavior. The analysis reveals how a simple Gaussian response function can produce a rich spectrum of nonlinear phenomena, providing insight into the interplay between local nonlinear mechanisms and global dynamical organization.
Pedro Duarte (Universidade de Lisboa)
Title: Statistical Properties and Lyapunov Exponents in Dynamical Systems
Abstract: The law of large numbers, the central limit theorem, and the large deviations principle are classical results from probability theory, originally proved for independent and identically distributed (i.i.d.) random processes. In this talk, I will first survey the origins of these statistical properties and discuss how they extend to more general stationary random processes and to ergodic, measure-preserving dynamical systems. In the second part, I will focus on large deviation estimates and central limit theorems for a specific class of dynamical systems known as linear cocycles, and explore their connection with the regularity of the cocycle's Lyapunov exponents.
Roberto Imbuzeiro (IMPA)
Title: Recovering structure from binary graphical model dynamics
Abstract: We report on work in progress with Guilherme Ost (UFRJ) and Guilherme Reis (UFF) on a class of toy models for neuronal dynamics. These models consist of sets of units taking 0/1 values, which interact pairwise via a matrix of weights. The main question we address is how much of the weight structure can be recovered from the dynamics. We will focus on "mean field" cases where individual pairwise interactions may be quite weak. Interestingly, this property turns out to be helpful.
Liliane Maia (UnB)
Title: A crucial role of the spectrum of the Schrödinger operator in Nonlinear Elliptic Problems
Abstract: We will present some recent results concerning the existence of weak solutions to nonlinear Schrödinger equations -Δu + V(x)u = f(u), u ∈ H¹(R^N), under very mild assumptions on the nonlinear term, which satisfy only conditions around zero and at infinity. Our approach highlights the crucial role of the spectrum of the operator associated with the potential V, thereby generalizing and unifying several previous results. Using variational methods, specifically a modified linking structure combined with the monotonicity trick for linking. This is a work in collaboration with Romildo Lima (UFCG, Brazil) and Mayra Soares (UnB, Brazil).
Letícia Mattos (Universität Heidelberg)
Title: On almost Gallai colourings in complete graphs
Abstract: We introduce a natural extension of Gallai edge-colourings of the complete graph Kn, in which rainbow triangles, and more generally rainbow t-cliques, are allowed if they are edge-disjoint. In this setting, we are interested in maximising their number. Our main motivation comes from a lemma of Berkowitz concerning bounds on the modulus of the characteristic function of clique counts in random graphs.
In the case of triangles (that is, when t = 3), our problem can be viewed as a rainbow analogue of the celebrated Ruzsa-Szemerédi problem, which asks for the maximum number of blue triangles in a red-blue edge-colouring of Kn, under the condition that all blue triangles are edge-disjoint. Our proof combines various applications of the probabilistic method and a generalisation of the edge-isoperimetric inequality for the hypercube.
Based on a joint work with A. Grebennikov and T. Szabó.
David Martínez Torres (Universidad Politécnica de Madrid)
Title: Topology of tridiagonal isospectral manifolds and the weak Bruhat order
Abstract: A tridiagonal isospectral manifold is a conjugacy class of symmetric matrices with simple spectrum. In 1984, Tomei showed how properties of the permutation group and the group of sign matrices (both Coxeter groups) entered in the study of the topology of the tridiagonal isospectral manifold. We shall recall these results and discuss the case of arbitrary spectrum. This is joint work with C. Tomei.
João Marcos do Ó (UFPB)
Title: Functional Inequalities with Boundary Terms
Abstract: Weighted Hardy-Friedrichs and Hardy-Sobolev type inequalities with boundary terms in unbounded domains are studied. Based on these inequalities, Liouville-type theorems are established, and applications to quasilinear elliptic problems are discussed.
Paulo Orenstein (IMPA)
Title: Robust and Efficient Mean Estimation via Soft Shrinkage
Abstract: Abstract: How should we weight observations when estimating a mean? The sample mean is efficient and minimax but fragile under heavy tails; robust estimators resist heavy tails but typically sacrifice efficiency. We show a soft-shrinkage calibration improves on both fronts: observations are smoothly downweighted by their distance from a pilot estimate, with the shrinkage scale calibrated so that the total rejected weight equals a prescribed budget. Any reasonable pilot is upgraded to sub-Gaussian concentration with the minimax constant sqrt(2) under finite variance alone, and the budget doubles as an exact breakdown point. The mechanism trades variance for bias, and the bias can be quantified exactly. This yields a second-order theory establishing an MSE expansion that identifies when shrinkage beats the sample mean, Berry–Esseen and moderate deviation approximations, and asymptotically exact confidence intervals. Beyond the theory, the results give concrete guidance for designing mean estimators and bridge modern sub-Gaussian estimation with classical robust statistics. This is joint work with Antônio Catão.
Sebastián Pérez (PUC Valparaíso)
Title: Robust heterodimensional cycles of co-index two
Abstract: In dimension four, we study co-index two heterodimensional cycles, i.e., cycles associated with two saddles whose unstable indices differ by 2. In a partially hyperbolic setting, we introduce the notion of non-escaping for such cycles and prove that they can be C¹ approximated by diffeomorphisms exhibiting robust heterodimensional cycles of co-index one and two. This is joint work with P. Barrientos, L. Díaz, Y. Ki and C. Lizana.
Yulia Petrova (USP)
Title: Traveling waves in mathematical biology and fluid dynamics
Abstract: Many natural phenomena can be described as propagation fronts: from the displacement of oil by water in petroleum reservoirs to the spread of species in ecosystems. From a mathematical point of view, these fronts can be approximated by special solutions called traveling waves.
In this talk, I will present the connection between the search for traveling waves and the study of heteroclinic orbits in dynamical systems. I will discuss how these ideas arise in models of species invasion in mathematical biology and in models of fluid motion in porous media. In the case of fluids under the action of gravity, when a heavier fluid lies above a lighter one, interesting instabilities arise and lead to the formation of propagation patterns. The talk is based on work in progress and on a joint paper with S. Tikhomirov (IME-USP) and Ya. Efendiev (Texas A&M), published in SIMA in 2025.
Makson Santos (Universidade de Lisboa)
Title: Regularity phenomena in nonlocal equations
Abstract: In this talk, we investigate the regularity properties of solutions to a class of fully nonlinear nonlocal equations, in both the elliptic and degenerate cases. For the elliptic scenario, we establish optimal regularity results for equations with a right-hand side in L^p, where the regularity space depends on the value of p. In the degenerate case, we prove the existence of at least one viscosity solution belonging to the class C_{loc}^{1, \alpha} for some constant \alpha \in (0,1). Additionally, when the order of the operator is sufficiently close to 2, we derive regularity estimates in Hölder spaces for the gradient of any viscosity solution.