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Speaker Prof Dr Maria Lukacova will talk about 'Numerical analysis of oscillatory solutions of compressible flows'

Oscillatory solutions of compressible flows arise in many practical situations.  An iconic example is the Kelvin-Helmholtz problem, where standard numerical methods yield oscillatory solutions. In such a situation,  standard tools of numerical analysis for partial differential equations are not applicable. 

We will show that structure-preserving numerical methods converge in general to generalised solutions, the so-called dissipative solutions. 
The latter describes the limits of oscillatory sequences. We will concentrate on the inviscid flows, the Euler equations of gas dynamics, and mention also the relevant results obtained for the viscous compressible flows, governed by the Navier-Stokes equations.

We discuss a concept of K-convergence that turns a weak convergence of numerical solutions into the strong convergence of
their empirical means to a dissipative solution. The latter satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress.  We will also discuss suitable selection criteria to recover well-posedness of the Euler equations of gas dynamics. Theoretical results will be illustrated by a series of numerical simulations.  

Cells make decisions across developmental biology, immunology, and synthetic biology. These processes are typically described using systems of ordinary differential equations, where mathematical analysis focuses on steady-state solutions. However, understanding how the timing of cell decisions is controlled requires moving beyond this paradigm. In this talk, I will discuss two complementary molecular mechanisms for controlling dynamical speed. First, I will show how timing can be regulated through critical slowing down, and how combining different bifurcations can generate emergent temporal behaviours even in small gene regulatory networks. Secondly, I will address developmental tempo, where embryos from different species execute remarkably similar genetic programmes at different speeds. I will present a mathematical framework based on orbit invariance that allows us to explore potential molecular mechanisms underlying species-specific differences in developmental timing.

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It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.

Professor Colbrook is going to talk about: 'A Computational Framework for Infinite-Dimensional Nonlinear Spectral Problems' 

Nonlinear spectral problems -- where the spectral parameter enters operator families nonlinearly -- arise in many areas of analysis and applications, yet a systematic computational theory in infinite dimensions remains incomplete. In this talk, I present a unified framework based on a solve-then-discretise philosophy (familiar, for example, from Chebfun!), ensuring that truncation preserves convergence. The setting accommodates unbounded operators, including differential operators with spectral-parameter-dependent boundary conditions. 
In the first part, I introduce a provably convergent method for computing spectra and pseudospectra under the minimal assumption of gap-metric continuity of operator graphs -- the weakest natural setting in which the resolvent norm remains continuous. 
In the second part, I develop a contour-based framework for discrete spectra of holomorphic operator families, with a complete analysis of stability, convergence, and randomised sketching based on Gaussian probes. This perspective unifies and extends many existing contour integral methods. Examples throughout highlight practical effectiveness and subtle phenomena unique to infinite dimensions, including the perhaps unexpected sensitivity to probe selection when seeking to avoid spectral pollution.

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Heterogeneity is a fundamental feature of biological systems. Oncology is one of the fields in which this feature is most evident, as its key players are characterised by mutability, plasticity, and often “uncontrolled” dynamics. Whether heterogeneity arises from spatial structure, environmental variability, or cellular traits, effective therapeutic strategies must explicitly account for it in order to eradicate or control tumours.

From a modern perspective, this requires balancing the hit-hard / keep-it-sensitive trade-off, while also considering not only medical but also broader patient-related side effects of treatments. Contemporary medicine is increasingly exploring ways to exploit the very characteristics that have historically made cancer so dangerous, turning them into potential advantages for therapy.

The multiscale nature of tumour systems, together with the need to predict the combined effects of multiple, non-parallelisable processes, makes the development of optimised mathematical tools particularly compelling. Such tools can address questions that are both scientifically challenging and highly relevant from a clinical and humanitarian perspective.

In this seminar, we will analyse tumour masses from a structured population perspective, focusing on the role of heterogeneity in shaping therapeutic strategies. We will first discuss how heterogeneity in phenotypic composition and nutrient distribution influences the eco-evolutionary dynamics of tumour growth. We will then consider more specifically its impact on radiotherapy.

In particular, we will highlight the advantages of mathematically rigorous modelling in bridging theory and biology. We will also adopt a more exploratory perspective, using these models to illustrate how mathematics can serve as a potential decision-support tool for the selection and optimisation of treatment protocols, within an image- and model-driven framework.

The final part of the seminar will focus on potential future developments, with the aim of fostering an open and collaborative discussion on novel perspectives to improve understanding, prediction, and therapeutic optimisation.

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Professor Luis Nunes Vicente will talk about 'Reducing Sample Complexity in Stochastic Derivative-Free Optimization via Tail Bounds and Hypothesis Testing';

We introduce and analyze new probabilistic strategies for enforcing sufficient decrease conditions in stochastic derivative-free optimization, with the goal of reducing sample complexity and simplifying convergence analysis. First, we develop a new tail bound condition imposed on the estimated reduction in function value, which permits flexible selection of the power used in the sufficient decrease test, q in (1,2]. This approach allows us to reduce the number of samples per iteration from the standard O(delta^{−4}) to O(delta^{-2q}), assuming that the noise moment of order q/(q-1) is bounded. Second, we formulate the sufficient decrease condition as a sequential hypothesis testing problem, in which the algorithm adaptively collects samples until the evidence suffices to accept or reject a candidate step. This test provides statistical guarantees on decision errors and can further reduce the required sample size, particularly in the Gaussian noise setting, where it can approach O(delta^{−2-r}) when the decrease is of the order of delta^r. We incorporate both techniques into stochastic direct-search and trust-region methods for potentially non-smooth, noisy objective functions, and establish their global convergence rates and properties. 

This is joint work with Anjie Ding, Francesco Rinaldi, and Damiano Zeffiro.

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Many cells exhibit exponential growth not only at the population level but also at the single-cell level. However, single-cell growth rates fluctuate over time. We distinguish between two conceptually distinct sources of growth rate fluctuations: intrinsic continuous fluctuations resulting from intracellular processes, and fluctuations that originate at division events, which we refer to as kicks. We use a simple model to describe single-cell growth and identify the signatures of continuous noise and division kicks. To infer the true biological behavior reliably from experiments, it is crucial to account for measurement noise. We derive analytical expressions for the statistics of meaningful observables, accounting for continuous fluctuations, division kicks, and measurement noise. Importantly, we find that ignoring measurement noise can lead to incorrect biological conclusions. Our results provide insights into how different sources of growth rate variability and measurement errors influence observed cell size dynamics, offering an interpretable framework for analyzing experimental data in cellular biology. 

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In the context of Fourier analysis on the real line, a \textit{one-sided problem} involves deducing properties of a function $f$ from some information about the restriction of its Fourier transform $\widehat{f}$ to a half-line, for instance to $\mathbb{R}_- := (-\infty, 0)$. A prototypical result, which is foundational to the theory of Hardy spaces on $\mathbb{R}$, asserts that if $f \in L^2(\mathbb{R})$ is non-zero and $\widehat{f}$ vanishes on a half-line, then $f$ satisfies the \textit{Szeg\H{o} condition} $\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty$. 

Various problems in operator theory involve the study of functions $f$ satisfying a weaker condition of decay of $\widehat{f}$ on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local Szeg\H{o}-type conditions hold: if the decay of $\widehat{f}$ is strong enough on a half-line, then the mass of the function $f \in L^2(\mathbb{R})$ must concentrate enough for the integral $\int_E \log |f(x)| dx$ to converge on a "massive" set $E$. 

In his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.

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