Seminars from until

We consider determinantal point processes $\Lambda_\phi$ arising from generalized Fock spaces $\mathcal F_\phi$, defined by a doubling subharmonic weight $\phi$. We provide a complete characterization of when these processes are almost surely separated with respect to the euclidean metric on the plane. As a consequence, we also obtain a characterization of when these processes are almost surely interpolating for the classical Fock spaces. Furthermore, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $\Lambda_\phi$ with the Poisson process with the same first intensity. Additionally, we demonstrated that, with probability one, the point process $\Lambda_\phi$ fails to be separated with respect to the distance induced by the reproducing kernel $K_\phi$.

This is a joint work with X. Massaneda.

In this seminar, we will explore a fractional version of a semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980s. The problem arises when considering steady states of the Keller–Segel model with nonlocal diffusion of chemical concentration. In the fractional setting, there are different possible notions of 'nonlocal Neumann conditions'. We will study the system with both spectral conditions, following Stinga and Volzone, and integral conditions (introduced by Dipierro, Ros-Oton, and Valdinoci), as investigated by Cinti and Colasuonno. While an extension theorem analogous to that of Caffarelli and Silvestre exists for the spectral boundary condition, no such theorem is available for integral boundary conditions. This makes it more challenging to obtain Liouville-type theorems and to prove the nonexistence of non-trivial positive solutions when the concentration parameter is sufficiently large.

Based on joint projects with Eleonora Cinti (UNIBO), Marco Ghimenti (UNIPI), Juncheng Wei (CUHK), and Tobias Weth (GUF).

We construct Einstein metrics with positive scalar curvature on two infinite families of 4-manifolds. Via Wick rotation, we obtain static axisymmetric Lorentzian spacetimes with positive cosmological constant and arbitrarily many horizons of equal surface gravity. As the topology increases, the metrics collapse to an explicit 2-disk while bubbling off periodic gravitational instantons with Kasner asymptotics, one of which is new.

What really happens when we shuffle a deck of cards — and what mathematics lies behind this seemingly simple act?

Persi Diaconis explores the mathematics of three ways that real people shuffle real cards.

Remarkably, these three methods call for three quite different kinds of mathematics.

Riffle shuffling leads to elementary combinatorics; overhand shuffling involves sophisticated coupling methods; and smooshing leads all the way to stochastic partial differential equations.

Behind these familiar ways of mixing cards lie challenging questions in analysis and probability.

Yet the problems themselves are easy to understand, making them an engaging gateway to surprisingly deep mathematics.

Along the way, the course will reveal connections to card tricks, gambling, probability, and other areas of mathematics.

In this talk, we consider partial differential equations of parabolic $p$-Laplace type in noncylindrical domains $U = \bigcup_{t \in [0,T]} \{t\} \times \Omega_t$, where the underlying spatial set $\Omega_t \subset \mathbb{R}^n$ may change in time. Our goal is to characterize time-dependent boundary values that yield the existence of a solution, focusing on domains $U$ that satisfy a Lipschitz condition with respect to a parabolic metric. In particular, we discuss the extension to $U$ of certain functions that are only defined on the parabolic boundary of $U$, and conversely the trace space for functions defined on $U$ and contained in a suitable parabolic function space. The talk is based on joint work with Daniel Campbell and Sebastian Schwarzacher.

The goal of this talk is to present the construction of new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the ordinary double point singularity in dimension 3. These metrics have tangent cone $\mathbb{C} ×(\mathbb{C}^2/\mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a singular Calabi-Yau metric with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point, thereby providing a new metric realisation of the Atiyah flop. As we will explain, this construction is meant to provide a local adiabatic model for certain degenerations of compact Calabi-Yau manifolds.

Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in $\ell^1$ if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in $\ell^1$. We prove an analog of Baxter's theorem for the SU(2)-valued nonlinear Fourier transform. The proof of Baxter's theorem is qualitative in nature. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates. In particular, we construct a counterexample for certain such estimates