Abstracts Fall  Semester 2025

Title: Quantitative stability for the critical p-Laplace equation and beyond

Abstract: The critical p-Laplace equation arises in the context of determining the optimal constant in the Sobolev inequality and is also related to the Yamabe problem when p=2.

Since the seminal work of Gidas, Ni, and Nirenberg [Math. Anal. Appl. Part A, 1981], considerable effort has been devoted to the classification of positive solutions to this equation and, more broadly, to the study of symmetry and monotonicity properties of positive solutions to equations modeled on the critical one.

Following Struwe’s qualitative stability result [Math. Z., 1984], a number of works have addressed quantitative aspects of stability of the critical equation for p=2 and functions in the energy space D^{1,2}(R^n).

In this seminar, we present recent stability results for the general case 1<p<n, as well as quasi-symmetry and monotonicity properties of both energy and non-energy solutions for a broader class of semilinear equations.

Title: Recent progress on the structure of metric currents

Abstract: The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for metric 1-currents as superposition of 1-rectifiable sets in complete and separable metric spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents and a more refined analysis in the Banach space setting. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).

Title: Relativistic and Discrete Eigenvalue Problems

Abstract: In this seminar, I will present several distinct eigenvalue problems and explore their interconnections. The first problem concerns the existence of ground states for the nonlinear Dirac equation with power-type nonlinearities. I will demonstrate that, in the nonrelativistic limit, these Dirac ground states converge to the nonlinear Schrödinger ground state, thereby providing a rigorous foundation for certain physical principles. The second problem involves the eigenvalue problem of the discrete Laplacian on a newly introduced random graph model, which features a higher number of connected graphs. I will examine bounds for the first eigenvalue of the discrete Laplacian and discuss an application within hyperbolic geometry.

Title: Curvature-Dimension for Autonomous Lagrangians 

Abstract: The talk will introduce a new curvature-dimension condition for autonomous Lagrangians on weighted manifolds. We will see that, like its Riemannian counterpart, this condition is equivalent to displacement convexity of entropy along cost-minimizing interpolations in an L^1 sense, and that it implies various familiar consequences of lower Ricci curvature bounds. As examples, we will consider classical (mechanical and electromagnetic) Lagrangians on Riemannian manifolds. In particular, we will state a generalization of the horocyclic Brunn-Minkowski inequality to complex hyperbolic space of arbitrary dimension, and a new Brunn-Minkowski inequality for contact magnetic geodesics on odd-dimensional spheres. 

Title: p-Wasserstein barycenter: a multimarginal optimal transport problem 

Abstract: The talk will be about the barycenters of N probability measures with respect to the p-Wasserstein metric (p>1), which generalizes the notion of Wasserstein barycenters for p=2, introduced by Agueh and Carlier. In particular it will be showed that

- p-Wasserstein barycenters of absolutely continuous functions are unique and again absolutely continuous;

- p-Wasserstein barycenters admit a multi-marginal OT formulation;

- the optimal multi-marginal plan is unique and of Monge form if the marginals are absolutely continuous and it has an explicit  parametrization as a graph over any marginal space.

This last result can be inserted in the more general, and still not fully understood, framework of Monge problems for multi marginal OT. In this case, a key ingredient is a quantitative injectivity estimate for the (highly non-injective) p- barycenter map on the support of an optimal multi-marginal plan.

Finally, we will discuss the asymptotic behavior as p goes to 1 or infinity and the regularity of the density of the barycenter.

This is a joint work with G. Friesecke and T. Ried. 

Title: The operator curl acting on a closed 3-manifold: spectral analysis 

Abstract: We study the spectrum of the operator curl acting on a connected oriented closed Riemannian 3-manifold. The spectrum is asymmetric about zero and this spectral asymmetry is the focus of our analysis.

Spectral asymmetry is a major subject in pure mathematics and theoretical physics. The traditional measure of spectral asymmetry is the so-called eta invariant, a real number. The classical definition of the eta invariant is by means of analytic continuation of the eta function, an analogue of the Riemann zeta function for non-semi-bounded operators. We prove that the eta invariant for the operator curl can equivalently be obtained as the trace of the difference of positive and negative spectral projections, appropriately regularised. Our construction is direct, in the sense that it does not involve analytic continuation, and is based on the use of pseudodifferential techniques.

This is joint work with Giovanni Bracchi and Matteo Capoferri.

Title: Harmonic unit vector fields on 3-manifolds 

Abstract: In this talk, we will discuss harmonic unit vector fields on compact 3-manifolds whose integral curves are totally geodesic. Under mild curvature assumptions, we will see how both the vector fields and the underlying manifolds can be classified. Our approach is inspired by Carriere’s classification of Riemannian flows on compact 3-manifolds and generalizes several previous results. 

Title: Four-dimensional gradient shrinking Ricci solitons 

Abstract: In this talk, we will discuss 4-dimensional complete (not necessarily compact) gradient shrinking Ricci solitons. We will show that a 4-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual (or anti-self-dual) part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton R^4, or S^3×R, or S^2×R^2. Moreover, we will show that if the quotient of norm of the self-dual part of the Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half- conformally flat or locally Kähler. This talk is based on joint works with Xiaodong Cao, Hung Tran, Huai-Dong Cao and Detang Zhou. 

Title: Phase-field tumour growth models: analysis, control and inverse reconstruction 

Abstract: Mathematical models for tumour growth are becoming increasingly common in the recent scientific literature, due to a newfound interest in mathematical applications to biological phenomena. Most importantly, one of the main contributions of Mathematics in this area is the development of patient-specific tumour growth models that can help clinicians’ decisions through personalised tumour forecasts. 

Here, we mostly consider tumour growth models of phase-field type, mainly concerning young avascular tumours. Thus, we describe a tumour through a phase variable, representing the difference in volume fractions between cancerous cells and healthy cells in a given tissue. More specifically, our models are systems of partial differential equations of Cahn-Hilliard type, coupled to additional reaction-diffusion equations for other key quantities, such as nutrients used by the tumour cells to proliferate.

During the course of the talk, we will present some recent results and challenges in the mathematical analysis of such models. In particular, we will focus on the well-posedness of the PDE systems and the regularity of their solutions, as well as problems related to optimal control of therapies and inverse reconstruction of earlier states.