Program and abstracts:
Tuesday, August 5th
9:30 Luen-Chau Li (Penn State University)
Title: On Infinite Periodic Band Matrices and the Periodic Toda Flow
Abstract: In this talk, we will consider an isospectral deformation of infinite periodic band matrices with period n, with lower bandwidth equal to k, and upper bandwidth equal to k’, and subject to the conditions that 1 ≤ k, k’ ≤ n − 1, k + k’< n. We will show that this flow, which was introduced by van Moerbeke and Mumford in the late 1970s, translates into a corresponding flow on matrix loops in gl(n, C), which turns out to be a natural extension of the periodic Kostant-Toda flow. We will then show that this flow is Liouville integrable on generic coadjoint orbits of a finite-dimensional Lie group. Finally, we will discuss the correspondence between an open collection of loops corresponding to the infinite periodic band matrices, and the collection of extended algebro-geometric data. The additional variables in the extended algebro-geometric data allow us to set up a bijection between the two sets of objects, thus extending the dictionary in the work of van Moerbeke and Mumford.
11:00 André Nachbin (Worcester Polytechnic Institute)
Title: Water waves on graphs
Abstract: We present a weakly nonlinear, weakly dispersive Boussinesq system for water waves propagating on a 1D branching channel, namely for studying reflection-transmission on a metric graph. Our graph model uses a new nonlinear compatibility condition at the vertex which improves reflection transmission properties, and therefore generalizes the well-knownNeumann-Kirchhoff condition. The model includes forking-angles in a systematic fashion. Our vertex condition is formulated by looking also at solutions of the 2D (parent) fattened graph model, namely a graph-like domain with a small lateral width. We present numerical simulations comparing solitary-wave-propagation on the 1D (reduced) graph model with the respective results of the 2D model, where a compatibility condition is not needed at the forked region. We will comment on ongoing studies, in particular regarding the importance of including angle-information, a feature not present in most waves-on-graph models.
13:30 Charles Epstein (Flatiron Institute)
Title: Fredholm, Sommerfeld, Carlos and I
Abstract: As a graduate student, Carlos spent a great deal of time studying Fredholm integral equations of second kind. I remember when he finally found a proof that his equation had a trivial null-space, which allowed him to proceed with his work on inverse scattering for the Boussinesq equation. As a student I was totally ignorant of this subject, but ironically, have spent the past decade working almost exclusively with such equations, and my principal concern has been establishing uniqueness.
Today I’ll speak about the Sommerfeld radiation conditions for the Helmholtz and Maxwell equations, and how they are generalized to give uniqueness results for scattering problems involving open wave-guides. These are beyond the “elementary” classical theory because the potentials involved do not have compact support. These generalizations were first worked out by Isozaki in the setting of N-body Schrodinger equations. I’ll describe how subsequent developments due to Melrose and Vasy lead to uniqueness results for open wave-guides. If time permits, I’ll explain a recent method I introduced that uses Fredholm integral equations of second kind for solving these types of scattering problems, and the role played by the PDE uniqueness theorems in the proof that these Fredholm equations have trivial null-space.
14:30 Kenneth McLaughlin (Tulane University)
Title: Asymptotic analysis of multisoliton solutions of integrable partial differential equations and the kinetic theory of soliton gasses
Abstract: The kinetic theory of solitons and soliton gases originates from the discovery of the complete integrability of the Korteweg–de Vries (KdV) equation in the 1960s, leading to the identification of solitons as fundamental nonlinear phenomena. The concept of a soliton gas was introduced by Zakharov in 1971 and further developed by El in 2003, modeling solitons as interacting particle-like structures. In recent years, rigorous analytical results have been established that provide confirmation of the qualitative theory. In this talk, I will describe some of these advances, including (1) a rigorous derivation of kinetic equations governing soliton gases in KdV-type systems without randomness, as well as (2) the analysis of random collections of solitons, in which both mean behavior and fluctuation results are established. This is joint work with several teams, including Manuela Girotti, Aikaterini Gkogkou, Tamara Grava, Robert Jenkins, Guido Mazzuca, Oleksandr Minakov, and Maxim Yattselev.
16:00: Percy Deift (New York University)
Title: Applications of a Commutation Formula II
Abstract: In 1978 I described a variety of applications of the commutation formula
λ(λ + AB)^{−1} + A(λ + BA)^{-1} B = 1
for operators A, B in a Banach space, to problems in mathematics and physics. These problems included
isospectrality, KdV, regularization of PDE’s, and operator estimates in statistical mechanics and quantum field theory.
Over the intervening 50 years or so many other applications of the commutation formula have been recognized. In this lecture I will describe some of these appli-
cations in various fields of mathematics and physics and in which commutation plays a key role. These applications include determinants and dimension reduction, random matrix theory and dimension
expansion, integrable operators, Poncelet’s porism, interpolation theory without complex analysis, eigenvalue computation, and KPZ.
This is joint work with Fritz Gesztesy.
Wednesday, August 6th
9:30 Roberto Imbuzeiro (IMPA)
Title: Barycenters in metric spaces
Abstract: If X is a random variable taking values in a metric space, a Fréchet mean of X is any (deterministic) point m that is a minimizer of the expectation of squared distance to X. This talk considers the problem of estimating Fréchet means from an i.i.d. sample from that is possibly contaminated by adversarial noise. The goal is to obtain optimal or-nearly optimal high probability bounds under weak moment assumptions. We show that, for a broad class of Alexandrov spaces of positive or nonnegative curvature, we have strong rates improving upon the literature. In particular, these results apply to Fréchet means in Wasserstein space. Extensions for other classes of spaces, and relations to classical topics in Robust Statistics will also be discussed.
11:00 Boyan Sirakov (PUC-Rio)
Title: Uniform (or not) a priori estimates for the Lane-Emden system in the plane
Abstract: The Lane-Emden equation has been used as a basic example of a semilinear elliptic PDE for more than 50 years. Yet various open problems persist. In this talk we review a recent result on uniform boundedness of positive solutions and its surprising consequences. Further, we show that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. On the other hand, the boundedness may fail if the exponents are not comparable, a surprising incidence of a situation in which the sub-critical Lane-Emden system behaves differently from the scalar equation.We highlight various open problems too.
13:30 Alessandro Alla (Sapienza, Università di Roma)
Titulo: Data-Driven Methods for PDE-Based Dynamical Systems and Control
Abstract: The talk focuses on data-driven methods for modeling and control of dynamical systems governed by PDEs. In the first part, we present an efficient, non-intrusive regression approach for constructing reduced-order models of reaction–diffusion systems that exhibit pattern formation. Inspired by Dynamic Mode Decomposition with control (DMDc), our method augments surrogate models with polynomial terms to accurately capture strongly nonlinear dynamics without intrusive access to the governing PDEs. The learning problem is formulated as a regularized least-squares regression in a reduced-order subspace identified via Proper Orthogonal Decomposition (POD). Numerical experiments on classical models, including Schnakenberg and Mimura-Tsujikawa, show that higher-order surrogates significantly improve prediction accuracy while maintaining low computational cost. In the second part, we propose online algorithms to identify and control PDEs with unknown parameters, assuming the system is observable given a control input and initial condition. Starting from an initial parameter guess, the control is computed via the State-Dependent Riccati Equation (SDRE), while the observed trajectory is used to iteratively update parameter estimates through Bayesian Linear Regression. Numerical results highlight the effectiveness of these methods even in challenging scenarios with parameter uncertainty.
14:30 Nicolau Saldanha (PUC-Rio)
Title: Domino tilings in dimension 3
Abstract: A domino is the union of two adjacent unit squares (or cubes). Domino tilings in dimension 2 have been extensively studied and there are several deep and remarkable theorems. Almost without exception, similar problems in dimension 3 or higher are much harder. In this talk we consider the simplest local move among domino tilings of a given compact region: a flip consists of removing two dominoes and placing them back in a different position. In dimension 2, Thurston proved that any two tilings of a simply connected region can be joined by a finite sequence of flips. In higher dimensions, the corresponding question is far subtler.
There exists an invariant under flips known as the twist. The twist can be defined in terms of the relative helicity of a divergence-free vector field. It can also be considered a variant of the Hopf invariant. For domino tilings in dimension 3, the twist assumes integer values. For domino tilings in dimension at least 4, the twist assumes values in Z/(2). In dimension 3, a slab is a box of dimensions 2x2x1; for slab tilings in dimension 3, there exists a version of the twist assuming values in Z^3. For many regions, there are explicit examples of tilings which admit no flip, and these give us examples of pairs of tilings with the same twist but in different connected components under flips.
A cylinder of dimension n is the cartesian product of a contractible region of dimension n-1 (the base) and an interval. Given a base D, we construct a CW complex, the domino complex, and study its fundamental group, the domino group. The base D is called regular if the domino group has a certain structure. We prove that many bases are regular. For instance, tileable rectangles of sides at least 3 are regular. If D is regular it follows that it is almost always (but not always) true that, if two tilings have the same twist then they are in the same connected component.
For dimension 3, if D is regular then the sizes of flip connected components follow a normal distribution. For each value of the twist, there is a giant component.
For higher dimensions, the numbers of tilings with each value of the twist are almost equal. Thus, there are two twin giant components of almost equal size.
This includes joint work with Carlos Tomei, B. Khesin, C. Klivans, J. Freire, R. Marreiros, P. Milet, A. Vieira and many others.
16:00-17:30 Discussion on Undergraduate Mathematics Teaching
Débora Mondaini (PUC-Rio), Humberto Bortolossi (UFF), João Paixão (UFRJ), Renata Rosa (PUC-Rio), Ricardo Martins (Unicamp), Yuriko Yamamoto Baldin (UFSCAR)
Thursday, August 7th
9:30 Marcus Sarkis (Worcester Polytechnic Institute)
Title: Discretization reduction for elliptic problems with rough coefficients
Abstract: We consider multiscale finite element methods to approximate the solution of an elliptic partial differential equation with rough coefficients. The methods follow the Variational Multiscale and the Localized Orthogonal Decomposition–LOD methods and the adaptive domain decomposition method denoted by BDDC-Balancing Domain Decomposition with Constraints in order to select local modes based on localized generalized eigenvalue problems. On the first stage of the proposed method, the degrees of freedom of the multiscale basis functions are based on the corners of a coarse triangulation, and on the second stage modes on the edges of coarse triangulation are chosen based on these local generalized eigenvalue problems. As a result, optimal error energy estimate is achieved which is mesh and coefficient independent and with localized multiscale functions, and without assuming any regularity of the solution beyond the H1 norm. Numerical experiments are provided. This is a joint work with Alexandre Madureira, LNCC, Brazil.
11:00 Peter Gibson (York University)
Title: Solution of the scalar Riccati equation
Abstract: The scalar Riccati equation, which was first formulated three centuries ago, relates to a diversity of models in mathematical physics. Despite its long history, there is no known method to solve the Riccati equation explicitly in its most general form. In this talk, we present a new integral operator, called the bivariate exponential, that generalizes in a very natural way the standard exponential primitive operator, i.e., the operator that is inverse to logarithmic differentiation. The bivariate exponential operator enables solution of the Riccati equation with arbitrary coefficients. This in turn leads to a number of applications, including explicit solution of the Schrödinger equation, inversion of the Miura transform, and a new formula for Airy functions.
13:30 Antônio Leitão (UFSC)
Title: A piecewise constant levelset approach for semi-blind deconvolution: Application to barcode decoding
Abstract: We consider the semi-blind deconvolution problem modelling the decoding of linear barcodes.The Piecewise Constant Level Set (PCLS) ansatz in [2013, De Cezaro et al., Inv. Probl.29015003] is used as starting point to propose and analyze an iterative method for solving the nderlying inverse problem.
14:30 Helena Lopes (UFRJ)
Title: Beyond the resolution of the Onsager Conjecture
Abstract: In a seminal 1949 paper Lars Onsager conjectured, using dimensional analysis, that it might be possible for some incompressible inviscid flows not to conserve energy, as long as they were sufficiently rough. This is in line with the Kolmogorov 1941 theory of turbulence. More precisely, Onsager conjectured that, if a solution of the Euler equations were more regular than Holder continuous with exponent 1/3 then energy would be conserved; otherwise energy might not be balanced. Research on the Onsager Conjecture developed intensely in the early 2000s, following DeLellis and Sezekelyhidi's introduction of the use of convex integration to fluid dynamics. The Conjecture was fully resolved by Isett in 2018 but there are still many issues to be understood. In this talk I will give a brief account of the resolution of the Onsager Conjecture and then concentrate on ongoing subsequent research, particularly regarding two dimensional flows. I aim to discuss recent results for vanishing viscosity solutions in the supercritical case.
16:00 Edgar Pimentel (Universidade de Coimbra)
Title: Numerical methods for fully nonlinear free boundary problems
Abstract: We propose finite difference numerical methods for fully nonlinear free boundary problems of transmission type. Our methods address some genuine difficulties in the numerical treatment of this class of models. In the degenerate setting, our approach ensures monotonicity and derives stability from a comparison-like argument. In the uniformly elliptic case, the main novelty in our work is a fixed-point argument leading to the convergence of double-layer regularisation. We illustrate our findings with numerical examples and conclude with a few open problems in the field.
Friday, August 8th
11:00 David Torres (Universidad Politécnica de Madrid)
Title: Integrable systems and Poisson geometry of compact type
Abstract: Under some topological conditions a (complete) integrable system with no singularities is a Lagrangian fibration with torus fibers. Their global geometry was described by Nekrasov and Duistermaat, and their analogs in the non-commutative setting -- proper isotropic fibrations -- were described by Dazord and Delzant. In this talk we shall discuss non-commutative integrable systems from a Poisson geometric viewpoint. More precisely, we shall discuss our attempt to describe what Poisson geometry of compact type suggests as the non-commutative integrable systems with the most elementary singularities. This is joint work (in progress) with M. Crainic and R. Fernandes.
12:00 Michael Shapiro (Michigan State University)Michael Shapiro
Title: Noncommutative integrability
Abstract: We will discuss integrability of a discrete dynamical system on a noncommutative space. The main example is provided by the Grassmann pentagram map on $Gr(k,3k)$. We discuss the generalization of Poisson brackets with values in an associative algebra and the generalization of Liouville integrability. The talk is based on collaborative projects with N. Ovenhouse, S. Arthamonov, L. Chekhov, and I. Bobrova.
12:00 TBA