EHU, October 29th, 2026, 12:00 - 13:00
Ruting Wang - Northwest University
The Talbot effect was first observed in optics and has since been studied in various dispersive equations. At rational times, dispersive revival is closely related to finite exponential sums and their number-theoretic properties.
In this talk, we mainly study points of constancy for the solution $u$ of the periodic linear Schrödinger equation with step-function initial data and its squared modulus $|u|^2$. Using quadratic Gauss sums, generalized half Gauss sums, and other number-theoretic tools, we establish criteria for points of constancy and obtain classification results under suitable conditions. Our results show that $|u|^2$ may have additional points of constancy that do not occur for $u$, revealing differences in their local revival structures.
We also briefly discuss jumps, logarithmic cusps, and nonlinear effects for the Benjamin–Ono equation, and mention a possible direction for future work related to the Talbot effect in the vortex filament equation.
BCAM, October 22nd, 2026, 17:00 - 18:00
A nonlinear and probabilistic approach to the periodic Carleson convergence problem
Daniel Eceizabarrena - BCAM
Given the solution $u = u(x,t)$ to the Schrodinger equation with initial data $f$, when can we say that $\lim_{t \to 0} u(x,t) = f(x)$ pointwise in $x$? More precisely, what Sobolev regularity do we require for $f$ for this property to hold? This is known as Carleson's convergence problem.
I will discuss recent results in the periodic setting, both for the linear and non-linear Schrödinger equations, and how a probabilistic approach can dramatically decrease the Sobolev regularity required for the initial data.
This talk is in the context of the recent work https://arxiv.org/abs/2608.17100 in collaboration with Pablo Merino (BCAM).
EHU, October 15th, 2026, 12:00 - 13:00
Controllability of hyperbolic system
Hoai-Minh Nguyen - Sorbonne Université
In this talk, I will review results on the null-controllability, exact controllability of hyperbolic systems in one dimensional space for the optimal time. Both time-independent and time-dependent coefficients are discussed. Interesting and surprising phenomena occur. There is a finite gap for the optimal time in these cases even the time is involved only in the zero-order coefficients and even these coefficients are infinitely smooth. When the analyticity in time of the coefficients enters, the gap disappears. This is based on joint work with Jean-Michel Coron.
BCAM, October 8th, 2026, 17:00 - 18:00
From Diffuse Layers to Moving Interfaces: Quantitative Limits in Fluid-Phase Systems
Hanifah Mumtaz - University of Regensburg
A vinaigrette gives a simple picture of the problem: oil and vinegar form two phases separated by a thin mixing layer. In diffuse-interface models, this layer has a small but nonzero thickness. We ask what happens when this thickness tends to zero: does the diffuse layer converge to a moving sharp interface, and can we measure the error?
This talk studies this question for Allen–Cahn/Navier–Stokes systems. In the mass-conserving model, the total amount of each phase is fixed. This adds a nonlocal term to the phase equation and leads, in the limit, to a fluid-driven interface motion with volume-preserving curvature flow. I will explain how inner and outer expansions describe the layer and the bulk, how the interface law arises from the profile equation, and how these expansions are combined into an approximate solution.
The talk also compares two recent frameworks. Our work of Abels–M. (AM) proves quantitative convergence for the mass-conserving problem in two dimensions, where the main issue is the nonlocal mass constraint. The work of Abels–Fei–Liu–Moser (AFLM) treats the local, non-conserved problem in two and three dimensions and develops weighted estimates that capture the small length scale near the interface. Together, these results show how geometry, conservation laws, and interface-scale estimates enter rigorous sharp-interface limits using the techniques of matched asymptotics.
Sources: AM, Sharp Interface Limit for a Mass-Conserving NS/AC System, arXiv:2604.11597; AFLM, Higher Order Convergence for the Sharp Interface, arXiv:2606.30532.
EHU, October 1st, 2026, 12:00 - 13:00
Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations.
Claudia Peña - BCAM
The purpose of this talk is to present the article [1] in which we establish two results:
On the one hand, we study the long-wave linear (in)stability of the two-dimensional viscous, resistive Magnetohydrodynamic (MHD) equations on the periodic domain T_\alpha × T = R/(2π\alpha Z) × R/(2πZ) in the regime α ≪ 1, around a periodic shear flow (U(y), 0) coupled with a constant background magnetic field b = (b1, b2). It is a non-trivial extension of the recent paper [2] for the Navier-Stokes equations to the MHD setting.
On the other hand, as a dynamical consequence of the spectral analysis carried out along the proof, we obtain a splitting of the phase space into unstable and stable subspaces, on which solutions grow or decay exponentially in Sobolev norm.
[1] Feola, R., Franzoi, L., Montalto, R., Peña, C. Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations. arXiv:2607.25836.
[2] Colombo, M., Dolce, M., Montalto, R. and Ventura, P.: Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations. arXiv:2509.18070.
BCAM, September 17th, 2026, 17:00 - 18:00
The Hausdorff dimension of sets containing circles or line segments in many directions.
Antonio Córdoba - Universidad Autónoma de Madrid
We establish a lower bound for the Hausdorff dimension of a set K ⊂ ℝⁿ that contains a translated copy of every meridian, that is, of every (d − 2)-dimensional sphere passing through the poles of 𝕊ᵈ⁻¹, a (d − 1)-dimensional sphere, with 3 ≤ d ≤ n, and 𝕊ᵈ⁻¹ ⊂ ℝⁿ.
Under this assumption, we have
dimₕ(K) ≥ d − 1.
This result is closely related to, and reminiscent of, the classical Kakeya set problem. We build upon this connection, employing similar techniques to show that if K ⊂ ℝᵈ contains a unit straight line segment in every direction corresponding to a smooth curve on 𝕊ᵈ⁻¹, then its Hausdorff dimension is ≥ 2.