BCAM, Thursday, Feb 12th, 2026, 17:00 - 18:00
Title: Fractional dispersion properties of solutions to the Helmholtz equation with periodic scattering data
Javier Canto - EHU
In this talk, we explore some properties of solutions to the Helmholtz equation. More precisely, we show that under mild regularity conditions, an asymptotic fractional uncertainty principle holds. We apply this result to a particular solution with periodic scattering data, and we obtain a very intriguing object with fluctuating behaviour that has been previously observed in solutions to the Schrödinger equation. We claim that this is tied to the Talbot effect.
This talk is based on joint work with N. M. Schiavone (UPM) and L. Vega (EHU & BCAM)
UPV/EHU, Thursday, January 29th, 2026, 12:00-13:00
Title: Coherent energy cascades in random Hamiltonian systems.
Anxo Biasi - Instituto Galego de Física de Altas Enerxías, Universidade de Santiago de Compostela.
The problem of Sobolev norm growth—namely, the transfer of energy from low to arbitrarily high modes—has been extensively studied in Hamiltonian systems with a deterministic structure, such as the cubic nonlinear Schrödinger equation. In this talk, I will present an extension of this problem to Hamiltonian systems dominated by random nonlinear interactions. I will first introduce analytic solutions describing three types of energy cascades that lead either to unbounded growth or to finite-time blow-up of Sobolev norms. After, I will present numerical simulations demonstrating the rapid emergence of these dynamics from incoherent initial conditions. Taken together, these results demonstrate coherent energy cascades as robust mechanisms of energy transfer in systems dominated by random structures.
UPV/EHU, Thursday, January 22nd, 2026, 12:00 - 13:00
Title: On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R3
Claudia Peña Vázquez - BCAM
The purpose of this talk is to present the article [1], in which we establish two results: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in R3 with initial data (u_0, b_0) of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference u_0 − b_0. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field b under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.
References
[1] Lucà, R. and Peña, C., On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R3. Preprint: arXiv:2505.09340.
BCAM, Thursday, January 15th, 2026, 17:00--18:00
Title: Finite-time singularity via pendula for the forced 2D Boussinesq equation
André Laín Sanclemente - ICMAT
The blow-up takes place in a well-posedness regime, i.e., the force is uniformly bounded in time in some space where there is local existence of solutions. The singularity follows from a "vorticity layer cascade" mechanism based on an accumulated hysteresis effect on the amplitude of the vorticity layers caused by the deformation of the density layers. The displacement and deformation of those layers is governed by an infinite number of copies of the ODE that describes the movement of a pendulum. Furthermore, the density vanishes uniformly in space at a sequence of times that accumulate at the blow-up point, giving the singularity a flickering nature. Besides, the vorticity, the density, the velocity, the force of the vorticity equation and the force of the density equation of our solution will be compactly supported in space and their support will be uniform in time. In this talk, we will strive to describe the construction in detail and provide some ideas of the proof. Moreover, if time allows it, we will also comment how this construction may be used to prove a blow-up for the 2D non-homogeneous incompressible Euler equations.
BCAM, Tuesday, January 13th, 2026, 17:00--18:00
Title: Estimates for oscillatory integrals with damping factors
Sanghyuk Lee - Seoul National University
Let $\sigma$ be the surface measure on a smooth hypersurface $\mathcal H \subset \mathbb{R}^{d+1}$. A fundamental subject in harmonic analysis is to determine the decay of $\widehat{\sigma}$. For nondegenerate $\mathcal H$, the stationary phase method yields the optimal decay, while sharp bounds in the degenerate case are known only in limited situations. In this talk, we are concerned with the oscillatory estimate
$$
|(\kappa^{1/2}\sigma)^\wedge(\xi)| \le C|\xi|^{-d/2},
$$
for convex analytic surfaces $\mathcal H$, where $\kappa$ is the Gaussian curvature. The damping factor $\kappa^{1/2}$ is expected to recover the optimal decay, as suggested by the stationary phase expansion, but the work of Cowling–Disney–Mauceri–Müller shows that such bounds fail in general for $d \ge 5$ even when the surface is convex and analytic. However, it has remained open whether the estimate holds in lower dimensions $2\le d\le 4$. We establish it for $d=2,3$, and with a logarithmic loss for $d=4$. Our approach is inspired by the stationary set method of Basu–Guo–Zhang–Zorin-Kranich. We also discuss applications to convolution, maximal, and adjoint restriction operators. This talk is based on joint work with Sewook Oh.
BCAM, Thursday, January 8th, 2026, 17:00--18:00
Title: Maximal averages over hypersurfaces
Sewook Oh - KIAS, South Korea
Stein's spherical maximal function can be generalized to maximal averages over general hypersurfaces. Although this topic has been extensively studied, many problems still remain open.
In this talk, I will present Lp estimates for maximal averages associated to hypersurfaces, with particular emphasis on their relationship to the decay rate of the Fourier transform of measures supported on the hypersurfaces.
BCAM, Thursday, December 18th, 2025, 17:00--18:00
Title: The non-homogeneous Euler equations below the Lipschitz threshold
Francesco Fanelli - BCAM
UPV/EHU, Thursday, December 11th, 2025, 12:00 - 13:00
Title: Low regularity well-posedness of nonlocal dispersive perturbations of Burger's equation
Didier Pilot - University of Bergen
We show that the Cauchy problem associated with a class of dispersive perturbations of Burgers’ equations is locally well-posed. This class includes the low-dispersion Benjamin–Ono equation, also known as the low-dispersion fractional KdV equation:
∂ₜu − Dₓ^α ∂ₓu = ∂ₓ²(u).
We prove local well-posedness in the Sobolev space H^s(K), where K = ℝ or 𝕋, for s > s_α = 1 − (3α)/4, when 2/3 ≤ α ≤ 1. Moreover, we obtain a priori estimates for solutions at a lower regularity threshold, namely s > s̃_α > 1/2 − α/4. The result also extends to other values of s_α when 0 < α < 2/3. As a consequence of these results, and using the Hamiltonian structure of the equation, we obtain global well-posedness in H^s(K) for s > s_α when α > 2/3, and in the energy space H^{α/2}(K) when α > 4/5.
In the first part of the talk, I introduce the equations, explain their connection to fluid mechanics, and review several existing mathematical results as well as open problems.
In the second part, I give an overview of the proof. The argument combines: an energy method for strongly non-resonant dispersive equations, introduced by Molinet and Vento, refined Strichartz estimates, and modified energy methods.
In addition, we use a full symmetrization of the modified energy both for the a priori estimates and for estimating the difference between two solutions. This symmetrization yields crucial cancellations in the associated symbols, which are essential to close the estimates.
BCAM, Thursday, December 4th, 2025, 17:00 - 18:00
Title: The initial-to-final-state inverse problem with Strichartz potentials
Alberto Ruiz - UAM
UPV/EHU, Thursday, November 27th, 2025, 12:00 - 13:00
Title: Bilinear cone multiplier in two dimensions
Linfei Zheng - BCAM
The bilinear cone multiplier is a natural extension of the cone
multiplier proposed by Stein and the bilinear ball multiplier
introduced by Grafakos and Li. In this talk, we will discuss some
recent progress on the boundedness of the bilinear cone multiplier in
two dimensions.
This is based on a joint work with Luz Roncal, Saurabh Shrivastava and Kalachand Shuin.
BCAM, Thursday, November 20th, 2025, 17:00 - 18:00
Title: Endpoint estimates for the fractal circular maximal function and related local smoothing
Luz Roncal - BCAM
Abstract: The spherical maximal function is a relevant object in harmonic analysis, connected to the solutions of the wave equation and related smoothing properties, and variants of it have been widely studied in the literature. In recent times, there has been an increasing interest in understanding sharp forms of L^p-L^q estimates for the spherical maximal function when the supremum is taken over dilation sets of fractal dimensions of different nature. In this talk we will prove missing endpoint estimates for the fractal spherical maximal function which were open when d=2, and study closely related L^p-L^q local smoothing estimates for the wave operator over fractal dilation sets. Our approach relies on bilinear restriction estimates for the cone due to T. Wolff and T. Tao.
This is joint work with Sanghyuk Lee, Feng Zhang, and Shuijiang Zhao.
UPV/EHU, Thursday, November 13th, 2025, 12:00 - 13:00
Title: Fourier analytic methods in the spectral theory of Schrödinger operators
Dr. Konstantin Merz - Institute for Theoretical Physics, Department of Physics, ETH Zürich.
Estimating the location and accumulation of eigenvalues for Schrödinger operators is a classical problem at the intersection of spectral theory and mathematical physics. In this talk, we illustrate how Fourier analysis, in particular Fourier restriction, yields sharp eigenvalue estimates for Schrödinger-type operators with short-range potentials. We highlight the limitations of such methods for long-range potentials and show how they can be overcome by introducing randomness. Our results shed new light on the critical temperature for superconductivity in BCS theory and on the energy and lifetime of resonances in nuclear physics.
BCAM, Tuesday, November 11th, 2025, 17:00 - 18:00
Title: A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform
Rahul Sethi - Georgia Tech
BCAM, Thursday, November 6th, 2025, 17:00 - 18:00
Title: Boundary value problems for 2-D Dirac operator on corner domains
Fabio Pizzichillo - Universidad Politécnica de Madrid
We present recent results on self-adjoint extensions of Dirac operators on planar domains with corners under infinite-mass boundary conditions. Corners are shown to hinder the elliptic regularity valid for smooth boundaries.
We then extend the analysis to unbounded domains with infinitely many corners: the operator is self-adjoint when no concave corners are present, while in the concave case self-adjoint extensions arise. Among these, we single out a distinguished extension whose domain lies in a Sobolev space H^s, with s ≥ 1/2 depending on the corner opening.
Finally, we briefly present another related model involving delta-shell interactions and mention a forthcoming result in this direction.
The results presented come from different works in collaboration with Hanne Van Den Bosch and Miguel Camarasa.
BCAM, Thursday, October 30th, 2025, 17:00 - 18:00
Title: Volume Doubling in graphs with nonnegative curvature
Emmanuel Russ - Aix-Marseille Université
Let G be a discrete infinite graph with bounded geometry, endowed
with a standard Laplacian. Assuming that the curvature of G (in the
sense of Bakry–Émery) is nonnegative, we show that the volume of balls
satisfies the doubling property, as in the case of Riemannian manifolds.
This is a joint work with Hervé Pajot.
UPV/EHU, Thursday, October 23rd, 2025, 12:00 - 13:00
Title: An optimal fractional Hardy inequality on the discrete half-line
Rubén de la Fuente - BCAM
Hardy type inequalities have been a widely studied field since the first proofs in the early 20th century. The interest in these inequalities comes from their many applications in functional analysis, PDEs, spectral theory, or probability.
In this context, we prove a discrete fractional Hardy’s inequality. In particular, a Hardy inequality concerning the fractional Laplacian in the discrete half-line with a weight that is optimal, in the sense that it cannot be substituted by any pointwise larger weight and its decay is as slow as possible. The strategy of the proof relies mainly on spectral properties of the aforementioned operator and criticality theory for graph Laplacians. As an immediate application, we derive some unique continuation results for positive Schrödinger operators involving the fractional Laplacian on the discrete half-line. Joint work with Ujjal Das.
BCAM, Thursday, October 16th, 2025, 17:00 - 18:00
Title: Convex integration for the Monge-Ampere system.
Marta Lewicka - University of Pittsburgh
The Monge-Ampere system (MA) is the multi-dimensional version of the Monge-Ampere equation, arising from the prescribed curvature problem and closely related to the problems of isometric immersions and the minimization of elastic energies of thin shells.
In case of dimension d=2 and codimension k=1, (MA) reduces to the classical Monge-Ampere equation as the prescription of the Gaussian curvature of a shallow surface in R^3, whereas (MA) prescribes the full Riemann curvature of a shallow d-dimensional manifold in R^{d+k}.
(MA) takes its weak formulation, called the Von Karman system (VK). When d= 2, k= 1, (VK) arises in the theory of elasticity as the the Von Karman stretching content a thin film.
Closely related to (MA) and (VK) is the system (II) for an isometric immersion of the given d-dimensional Riemannian metric into R^{d+k}. (II) yields (VK) when equating the leading order terms along a perturbation of the Euclidean metric.
This lecture will concern the ongoing study of existence, regularity, and multiplicity of solutions to systems (MA), (VK), (II) through the method of convex integration, building on the prior fundamental results due to Nash, Kuiper, Kallen, Borisov, the more recent approach due to Conti, Delellis and Szekelyhidi, and the parallel analysis of Cao, Hirsch and Inauen.
We will also explore relation to the scaling of the non-Euclidean energies of elastic deformations and the quantitative isometric immersion problem.
UPV/EHU, Thursday, October 9th, 2025, 12:00 - 13:00
Title: Continuous density variations in the ocean
Théo Fradin - University of Bordeaux
The aim of this talk is to study the well-posedness of the incompressible Euler equations in an oceanic setting, including continuous density variations. Because of specific features of our setting (presence of a free-surface, small scale ratios, i.e. the shallow water parameter), a first approach will consist in studying the case of small density variations, which will lead to the justification of the well-known non-linear shallow water equations. We then move to a case where large density variations can be studied, which is when the density is strictly decreasing with height. This setting of so-called stable stratification is widely used in the study of geophysical flows, and is one of the main ingredients that contribute to the global oceanic circulation, regulating the Earth' climate.
UPV/EHU, Thursday, October 2nd, 2025, 12:00 - 13:00
Title: Integrability of projective equations and applications.
Dmitry Sinelshchikov - Biofisika Institute
This talk consists of two parts. In the first part we consider integrability of a family of second-order differential equations that is a projection of a geodesic flow of a two-dimensional (pseudo) Riemannian manifold. We develop two approaches for establishing integrability of the projective equations. The first one is based on finding solution of equivalence problems for the family of projective equations and its integrable subcases. The second one is connected with the classification of quasi-polynomial invariants for the projective equations.
In the second part of the talk we discuss a problem of mathematical modelling of wound healing in Drosophila embryos. We propose a mathematical model of this process, which is a system of nonlinear PDEs, demonstrate its correspondence to experimental data and discuss some properties of its solutions.
UPV/EHU, Tuesday, September 30th, 2025, 12:00 -13:00
Title: Water wave radiation by a submerged disc
Juliana Sartori Ziebell - Federal University of Rio Grande
The interaction between water waves and objects is of great importance for the study of structures used in offshore and coastal engineering. In this talk, a thin plate is submerged below the free surface of deep water. The problem is reduced to a hypersingular integral equation of the second kind over the surface of a unit disc. This problem will be analysed for particular cases. Numerical results will be presented for the heave added mass and damping coefficients.
BCAM, Thursday, September 25th, 2025, 17:00 - 18:00
Title: Self-organisation of magnetic nanoparticles in a viscous fluid via dipolar interactions
Ludovic Godard-Cadillac - Institut Polytechnique de Bordeaux
The purpose of this talk is to present recent investigations for the dynamics of an ensemble of monodomain magnetic nanoparticles suspended in a viscous fluid. The particles interact via dipolar magnetic forces, with their magnetic moments constrained along their individual easy axes of magnetization. The physical model incorporates particle inertia, viscous drag exerted by the fluid, and a repulsive interaction to account for collisions and prevent overlap. This talk presents both numerical investigations on the dynamics of the particles and some theoretical results on the steady states.
BCAM, Thursday, September 18th, 2025, 17:00 - 18:00
Title: Traveling wave behavior for KPP equation on the hyperbolic space
Irene Gonzálvez - BCAM
In this talk we present the Cauchy problem on hyperbolic space for the heat equation with a KPP type forcing term. Our goal is to understand how hyperbolic geometry affects the dynamics of solutions. We address the question of propagation versus extinction, including the critical case. In the case of propagation, we show that if the initial datum has a certain symmetry, the solution converges asymptotically to a traveling wave of minimal speed in a moving frame. The choice of this moving frame depends on the symmetries of the initial datum, which, in turn, is closely related to the three types of isometries in hyperbolic space: elliptic, hyperbolic, and parabolic.
UPV/EHU, Tuesday, September 16th, 2025, 11:30 - 13:00
Title: Unique Continuation Principles of Nonlinear Dispersive Equations
Gustavo Ponce - University of California - Santa Barbara
This talk is concerned with unique continuation properties (UCP) of some non-linear dispersive equations. We study both the local UCP and the asymptotic at infinity UCP. The dispersive model to be considered include the Korteweg-de Vries eq., the NLS nonlinear Schrödinger eq., the Camassa-Holm eq. and Benjamin-Ono eq (and related models). We will see that some of the known results have been shown to be sharp and others are only known under special restrictions.
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Title: Unique continuation for nonlocal dispersive equations
Carlos Kenig - University of Chicago
We will discuss recent works on unique continuation for nonlocal nonlinear dispersive equations, including the Benjamin-Ono and water wave equations as well as variable coefficient versions of them. This describes joint works with Pilod, Ponce and Vega.
UPV/EHU, Thursday, September 11th, 2025, 12:00 - 13:00
Title: Large deviations in the semi-classical limit of quantum spin systems
Christiaan J.F. van de Ven - Friedrich-Alexander University Erlangen-Nürnberg
The continuous $C^*$-bundle generated by Berezin-Toeplitz quantization on a symplectic manifold provides a rigorous framework for describing the semiclassical limit of quantum systems. The rate of convergence in this limit, often quantified by a rate function or entropy, is naturally formalized through the theory of large deviations. In this work, I investigate the specific case of the complex projective line, which serves as the phase space for a single quantum spin system. The spin, which characterizes the system's quantum nature, defines the semiclassical parameter and corresponds to the dimension of the associated Hilbert space. More precisely, we establish a full large deviation principle for the local Gibbs state and explicitly characterize the corresponding rate function. This is done by using a sophisticated argument based on the Feynman-Kac integral.