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.  

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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.

This session is aimed at first-year undergraduates preparing for Prelims exams. A panel of lecturers and current students will share key advice on exam technique and revision strategies, offering practical tips from their own experience.

One of the problems posed by Kadison in 1967 asks whether every separably acting von Neumann algebra is generated by a single element. The problem remains open in its full generality but significant progress has been made since. One can of course ask the same question in the C*-algebraic setting where, however, counterexamples are abundant even among commutative C*-algebras. I will give an overview of the history of the problem and then discuss some recent results on single generation of C*-algebras associated to graphs and C*-algebras with Cartan subalgebras.

The stable Andrews-Curtis conjecture remains one of the most notorious unsolved problems in group theory. It proposes that every balanced presentation of the trivial group can reduced to the standard presentation (with one generator and one relation) using a sequence of simple moves. In my talk, I will focus on group presentations that are ‘thickenable’, which means that their associated 2-complex embeds in a 3-manifold. For such presentations, the stable Andrews-Curtis conjecture is known to hold. In my talk, I will explain how one can also get an explicit exponential-type upper bound on the number of stable Andrews-Curtis moves that are required. This is in sharp contrast to what is known about non-thickenable presentations.

Theta operators are weight-shifting differential operators on  automorphic forms. They play an important role in studying congruences between Hecke eigenforms and their p-adic variation. For instance, the classical theta operator, which acts on q-expansions of modular forms as q·(d/dq), is used crucially in Edixhoven’s proof of the weight part of Serre’s conjecture, Katz’s construction of p-adic L-functions over CM fields, and Coleman’s classicality theorem.

Recent years have witnessed extensive works on understanding theta operators over general Shimura varieties, from both geometric and representation-theoretic perspectives. In this talk, I will hint at some aspects of this fascinating area of research. If time permits, I will discuss my ongoing work on overconvergent theta operators over Siegel Shimura varieties.

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We introduce a new spanning tree model which we call Random Spanning Trees in Random Environment (RSTRE), which was introduced independently by A. Kúsz. As the inverse temperature beta varies in the underlying Gibbs measure, it interpolates between the uniform spanning tree and the minimum spanning tree. On the complete graph with n vertices, we show that with high probability, the diameter of the random spanning tree is of order n^1/2 when β=o(n/log n), and is of order n^1/3 when β > n^4/3 log n. We conjecture that the diameter exponent linearly interpolates between these two regimes as the power exponent of beta varies. Based on joint work with L. Makowiec and M. Salvi.

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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.

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For more than a hundred years, scientists have carefully analysed the apparently random fluctuations in Brownian trajectories to learn about soft systems. In a more general sense, however, the information hidden within experimental fluctuations is typically underexploited, due to challenges in unambiguously linking fluctuation signatures to underlying physical mechanisms. In this talk, I will discuss our recent work developing new approaches to interpreting fluctuations in experimental data from a variety of soft systems, and thereby turn ‘noise’ into signal. In particular, I will share some recent results taking a fresh look at fluctuations in equilibrium colloidal monolayers. Here, we have combined experiment, simulation and theory to explore how simply counting colloids can reveal details of self and collective dynamics in interacting systems [1,2,3]. I will then discuss ongoing work to extend this understanding to confined driven systems [4], with the long-term goal of elucidating characteristic fluctuations in our synthetic nanopore experiments [5].

[1] E. K. R. Mackay, B. Sprinkle, S. Marbach, A. L. Thorneywork, Phys. Rev X. (2024)

[2] A. Carter, ALT et al., Soft Matter, 21, 3991, (2025)

[3] E. K. R. Mackay, ALT et al., arXiv:2512.17476, (2025)

[4] S. F. Knowles, E. K. R. Mackay, A. L. Thorneywork, J. Chem. Phys., (2024)

[5] S. F. Knowles, A. L. Thorneywork et al., Phys. Rev. Lett, 127, 137801, (2021)

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.

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

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The dynamic nature of first order methods can be interpreted by means of continuous time models. In this survey talk, we explain how physical concepts like acceleration, inertia or momentum have been used to improve the performance of convex optimization algorithms. 

We give special attention to the historical evolution of complexity results, especially in the form of convergence rates, under the light of this connection. We also discuss different ways in which acceleration schemes can be applied when the smoothness or strong convexity parameters are unknown, and how these ideas extend to saddle point and constrained problems. 

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Core-annular flows are often proposed to reduce frictional losses in industrial pipeline transport processes. Traditionally, a low-viscosity lubricating film is placed around a more viscous core to reduce the drag on the core. However, maintaining stable pipelining, where the core and the lubricant remain separated has proved challenging.
In this talk we present an alternative approach using three-layer, horizontal core-annular pipe flow, in which two fluids are separated by a deformable elastic solid. In the experiments, an elastic solid created by an in-situ chemical reaction maintains the separation of the core and annular fluids. Corrugations of the elastic interface are observed and stable pipelining, where the elastic shell created separating the two fluids remains intact, is successfully demonstrated even when the core fluid is buoyant. We also develop a theoretical model combining lubrication theory for the fluids with standard shell theory for the elastic solid, to predict the buckling states resulting from radial compression of the shell.
The self-sculpting of the shell by buckling cannot by itself generate hydrodynamic lift owing to symmetry in the direction of flow. Instead, we demonstrate that hydrodynamic lift can be achieved by other elastohydrodynamic effects, when that symmetry becomes broken during the bending of the shell.

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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