PDE Afternoon

The PDE Afternoon is a seminar for scientists and guests of TU Wien and the University of Vienna. It takes place on Wednesday afternoon.

Summer 2026

Location at TU Wien: Seminarraum 107/1, Wiedner Hauptstraße 7.
Location at Uni Wien: HS02 (ground floor), Oskar-Morgenstern-Platz 1.

24 June 2026, 13:15h, Uni Wien

Student's seminar

24 June 2026, 15:00h, TU Wien

No Country for Old Proofs: The rise of AI4Math

Frieder Simon, University of Oxford, UK

AI4Math is emerging as a new field at the intersection of mathematics, machine learning, and software engineering, aimed at 1) equipping mathematicians with new AI-driven tools, and 2) autonomously generating new mathematical results. This landscape moves mathematics from a predominantly cerebral discipline toward a domain of engineering. This talk showcases the machine-learning foundations of AI4Math (training and benchmarking of LLMs for mathematics), and surveys recent developments in AI4Math, including contributions by the author. Perspectives are presented on the changing role of mathematicians, and the new background knowledge required. The talk concludes with open problems in the domain of AI4Math.

10 June 2026, 14:00h, TU Wien

Generalizations of the Forelli-Alexandrov theorem via microlocal analysis

Rami Ayoush, Warsaw University of Technology, Poland

This talk surveys the results concerning various generalizations of the Forelli-Alexandrov theorem about regularity of pluriharmonic measures. Those theorems are established by using tools from microlocal analysis, suitably modified for dealing with singularities understood in the sense of measure theory. This technique is inspired by the method of microlocalizing spectral properties of measures due to R. G. M. Brummelhuis and allows one to transfer Fourier analytic results of this type to various manifold settings. In this way we establish Forelli-Alexandrov type theorems for quaternionic spheres, tori, the Cayley plane, Sasakian manifolds, etc. This is joint work with Michał Wojciechowski.

06 May 2026, 14:00h, TU Wien

Efficient time-domain scattering synthesis via frequency-domain singularity subtraction

Oscar Bruno, Dvision of Engineering and Applied Science, Caltech University, USA 

Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier transform. However, in the case of scattering by trapping obstacles, the Helmholtz solution exhibits nearly-real complex resonances--which significantly slows the convergence of numerical inverse transform. To address this difficulty this paper introduces a frequency-domain singularity subtraction technique that regularizes the integrand of the inverse transform and efficiently computes the singularity contribution via a combination of a straightforward and inexpensive numerical technique together with a large-time asymptotic expansion. Crucially, all relevant complex resonances and their residues are determined via rational approximation of integral equation solutions at real frequencies. An adaptive algorithm is employed to ensure that all relevant complex resonances are properly identified.

15 April 2026, 14:00h, TU Wien

Shape Holomorphy of Boundary Integral Operators with Applications to Uncertainty Quantification

Fernando Henriquez Henriquez Barraza, Institute of Analysis and Scientific Computing, TU Wien

We consider a family of boundary integral operators (BIOs) arising from the boundary reduction of well-posed problems—either Laplace or Helmholtz—defined on a collection of parametrically described domains. Our goal is to establish the holomorphic (analytic) dependence of these BIOs, as well as of the solutions to associated boundary integral equations, on perturbations of the domain shape, a property commonly referred to as shape holomorphy.- To date, various results have addressed shape holomorphy under differing assumptions, particularly concerning the physical dimension of the problem and, most notably, the smoothness of the domain deformations. In this talk, we present recent results on the shape holomorphy of BIOs under Lipschitz-regular deformations. - We also discuss the implications of these findings for the development of computational methods in forward and inverse uncertainty quantification, as well as model order reduction.

15 April 2026, 15:00h, TU Wien

Boundary layers in stochastic homogenization - treatment of fluctuations via large-scale regularity

Claudia Raithel, Institute of Analysis and Scientific Computing, TU Wien

In the quantitative homogenization of linear elliptic PDEs the goal is to approximate the heterogenous coefficient operator $-\nabla\cdot a\nabla$, $a: \mathbb{R}^d \to\mathbb{R}^{d\timesd}$, with a homogeneous coefficient operator $-\nabla \cdot a_{hom} \nabla$ at large enough scales - "large enough" being quantified through, e.g., homogenization rates. On a domain with boundary conditions, due to the enforcement of the boundary data, there is a boundary layer in which the homogenization process is qualitatively different than in the interior. In order to obtain higher-order homogenization rates it is necessary to quantify this layer. In the setting of (stationary and ergodic) random coefficients, a main step of this is estimating the fluctuations of certain random variables which depend on $a$ both nonlocally and nonlinearly. We will discuss these fluctuation estimates in the case of homogeneous Dirichlet boundary data -in particular, the role played by a large-scale regularity theory for the operator $-\nabla \cdot a \nabla $ on domains. This talk is based on joint work with P. Bella, J. Fischer, and M. Josien.

18 March 2026, 14:00h, TU Wien

Asymptotic problems for the Keller-Segel system with density cut-off

Mingyue Zhang, Institute of Analysis and Scientific Computing, TU Wien

In this talk, I will discuss the incompressible limit and other distinct limits of the Keller-Segel system with porous medium-type nonlinear diffusion and logistic sensitivity. The Keller-Segel system is a fundamental model for describing chemotaxis, the migration of cells in response to chemical gradients. We prove that as the diffusion exponent tends to infinity, for large cell density, the limiting system becomes a Hele-Shaw type free boundary problem. For small cell density, we justify that the stiff pressure effect vanishes, resulting in the limiting system being a hyperbolic Keller-Segel system. The mathematical methods include the energy formulation, the entropy equality, the kinetic formulation, and \(L^1\) compactness.

18 March 2026, 15:00h, TU Wien

Rational Extended Thermodynamics and its Development for Polyatomic Gases

Takashi Arima, Kanagawa University, Department of Mechanical Engineering, Yokohama, Japan

Rational Extended Thermodynamics (RET) provides a systematic framework for modeling non‑equilibrium flows by enlarging the set of independent fields beyond the classical hydrodynamic variables to include dissipative fluxes such as viscous stress, dynamic pressure, and heat flux. The resulting balance‑law models can be formulated as symmetric‑hyperbolic systems and, in suitable relaxation limits, recover the Navier–Stokes–Fourier theory. Local constitutive relations are constrained by universal principles, in particular the entropy principle together with Galilean invariance and stability requirements. RET also admits a kinetic‑theory support: for rarefied gases, the balance laws and closures can be made consistent with moment equations connected to the Boltzmann equation via maximum‑entropy closures. In the talk I will review the two complementary viewpoints—phenomenological RET and molecular RET—and highlight recent developments for polyatomic gases. I will conclude with a brief outlook on multicomponent extensions, and with a few open questions aimed at stimulating discussion.

11 March 2026, 13:15h, Uni Wien

Brief organizational meeting with the students from Uni Wien