Abstracts Spring Semester 2026

Title: The Caffarelli-Silvestre extension phenomenon for complete Bernstein functions of the Laplacian 

Abstract: In their renowed paper, Caffarelli and Silvestre showed that, for 0<s<1, the fractional Laplacian (-\Delta)^s is the Dirichlet-to-Neumann operator for the differential operator div(t^{1-2s}\nabla u(x,t)) on the upper half-space.

In this talk we present a generalization of the extension techniques due to M. Kwaśnicki and J. Mucha, to the case of operators that are complete Bernstein functions of the Laplacian.

Title: Diffeomorphism-based convex integration schemes for incompressible fluid flows 

Abstract: We will review how one can design convex integration schemes using diffeomorphisms to address problems that are typically not amenable to more traditional schemes. To illustrate this philosophy, we will consider the construction of rough steady Euler flows with certain prescribed topological properties and of Hölder continuous dissipative solutions to ideal MHD. The talk is based on joint work with Daniel Peralta-Salas and Javier Peñafiel-Tomás. 

Title: C^{1+alpha} regularity for fractional p-harmonic functions

Abstract: The fractional p-Laplacian is the natural nonlocal counterpart to the classical p-Laplacian, arising as the first variation of the fractional Gagliardo seminorm. In this talk, we present a recent result, obtained in collaboration with David Jesus (KAUST) and Luis Silvestre (University of Chicago), establishing that solutions to the homogeneous equation enjoy Hölder differentiability, extending the well-known regularity theory from the local to the nonlocal setting.

Title: A gravitational collapse singularity theorem without global hyperbolicity assumptions 

Abstract: We present a gravitational collapse singularity theorem which, contrary to Penrose's, does not assume global hyperbolicity and works even in presence of closed timelike curves. The theorem reconciles the phenomenon of black hole evaporation, which is incompatible with global hyperbolicity, with the prediction of singularities. 

Title: Green kernels and a new proof of the stable Bernstein theorem in R^4

Abstract: In this talk, we will give a new proof of the stable Bernstein theorem in R^4 , i.e. of the fact that complete, 2-sided stable minimal hypersurfaces M^3 → R^4  are hyperplanes. By some groundbreaking works in the past 5 years, the stable Bernstein theorem is now proved for M^n → R^(n+1) up to dimension n ≤ 5, while it fails for dimension n ≥ 7. Its validity for n = 6, in full generality, is a challenging open problem. Our proof uses very few tools, and might provide some insight to address the case n = 6. The argument hinges on integral formulas and a sharp pointwise estimate for the Green kernel of the Laplacian. Part of the arguments are based on the fact that stable minimal hypersurfaces in Euclidean space are examples of manifold with a spectral Ricci lower bound. Time permitting, I will describe further results obtainable via such interplay, including a spectral splitting theorem and a characterization of the 3D-catenoid in R^4.

This is joint work with X. Cabré, G. Catino, P. Mastrolia and A. Roncoroni.

Title: A new notion of weak solution to a sharp interface model for a two-phase flow of incompressible viscous fluids with different densities and viscosities

Abstract: In this talk, I would like to introduce a model for the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier–Stokes and Mullins–Sekerka type parts, and can be obtained from the asymptotic limit of the diffuse interface model introduced by Abels, Garcke, and Grün in 2012. I will present a new notion of weak solutions and show its global in time existence, together with a consistency result. This new notion of solution allows to include different densities of the two fluids, a sharp energy dissipation principle à la De Giorgi, as well as a weak formulation of a constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by Abels and Röger in 2009. This is a joint work with Helmut Abels. 

Title: About cyclicity in weighted Dirichlet spaces 

Abstract: Given a bounded operator T acting on a Hilbert space H, a vector x ∈ H is called cyclic if {p(T)x : ppolynomial} is dense in H. This talk is concerned with the case when T is the forward shoift on a class of Dirichlet-type spaces on the unit disc and pertains also to the case of several complex variabes, where the definition of cyclicity is modified accordingly. It is quite obvious that in such Hilbert spaces of analytic functions, cyclic elements cannot have zeros in the domain in question, but sufficient conditions for cyclicity involves more subtle ideas. For example, one could consider the appropriate capacity of the set of zeros of the radial limits of the function in question a path which leads to the very natural, but still unsolved Brown-Shields conjecture.

Alternatively, the lack of ”boundary zeros” of the function f is also implied if 1 f , or log f belongs to the space and it turns out that these conditions imply cyclicity for such Dirichlet-type spaces. They can also be considered for Dirichlet type spaces on the unit ball. The aim of the talk is to give an account about some recent results in this direction.

The material is based on joint work with S. Richter, and also with K.M. Perfekt, S. Richter, C. Sundberg, and J. Sunkes.

Title: Asymptotically sharp inequalities in convex sets: Improved estimates for the ratio of buckling and Dirichlet eigenvalues

Abstract: Many spectral and geometrical inequalities within the class of convex sets do not admit an optimizer, yet are known to be asymptotically sharp on sequences of thinning sets. Motivated by the growing interest in the study of quantitative versions of such inequalities, we investigate and refine a spectral inequality, due to Payne, involving the ratio of the buckling and Dirichlet eigenvalues, which is thought to attain its maximum on the infinite slab.

Title: Mathematical analysis of a variational model for ferronematics in two dimensions

Abstract: Ferronematics are composite materials in which the magnetic susceptibility of a liquid crystal is enhanced through the inclusion of magnetic nanoparticles. From a mathematical perspective, the system is described by two coupled order parameters: a tensor-valued field Q, representing the liquid crystal configuration, and a vector field M, accounting for the average magnetization induced by the nanoparticles.

The total free energy consists of Ginzburg–Landau-type contributions for both Q and M, together with a coupling term that energetically favors their alignment.

In this talk, we investigate the asymptotic behavior of minimizers and critical points of the energy functional as a small parameter $\varepsilon to 0$. The analysis focuses on the emergence of singular structures, including defects of codimension one and two, across different regimes of the model.

The talk is based on joint papers with G.Canevari, F.Dipasquale, A.Majumdar and Y.Wang.

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